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\topmatter
\jourlogo{\eightpoint Selecta mathematica, k{\bf 2} (1996), p. }
\medskip
\medskip
\title{\bf Degree theory and BMO};\\
   {\bf Part II:  Compact Manifolds with Boundaries}
\endtitle
\author  Ha\"im Brezis${}^{(1)}$ and Louis Nirenberg${}^{(2)}$\\
\smallskip
(and an Appendix with Petru Mironescu)
\endauthor
\affil ${}^{(1)}$Universit\'e P. et M. Curie, Paris, France and Rutgers
University, New Brunswick, New Jersey\\
${}^{(2)}$New York University, 
Courant Institute, New York
\endaffil
\address 
${}^{(1)}$Analyse Num\'erique\endgraf
Universit\'e P. et M. Curie\endgraf
4, Pl. Jussieu\endgraf
75252 Paris, Cedex 05 France\endgraf
and\endgraf
Rutgers University\endgraf
Department of Mathematics\endgraf
Hill Center, Busch Campus,\endgraf
New Brunswick, New Jersey 08903\endaddress
\email  brezis\@ann.jussieu.fr; brezis\@math.rutgers.edu;
\endemail
\address
${}^{(2)}$Courant Institute\endgraf
New York University\endgraf
251 Mercer Street\endgraf
New York, New York 10012
\endaddress
\email  nirenl\@cims.nyu.edu\endemail
\rightheadtext{Part II.  Degree theory and BMO}
\leftheadtext{Haim Brezis and Louis Nirenberg}
\endtopmatter
\document
\noindent
{\bf II.0.  Introduction}
\medskip
This is a continuation of H.~Brezis and L.~Nirenberg~[1] ($ =$ [BNI]),
and we will often refer to concepts and results in that paper.  There,
we extended degree theory to VMO maps between compact $n$-dimensional
oriented manifolds without boundaries.  In this paper we consider a
class of maps $u$ from a bounded domain $\Omega\subset\Bbb R^n$ into
$\Bbb R^n$.  In classical degree theory, for $u\in C^0
(\overline\Omega, \Bbb R^n)$, the degree of $u$ at a point
$$
p\notin u(\partial\Omega) \tag 0.1
$$
is defined; it is denoted by $\deg(u,\Omega,p)$.
\medskip
The larger class of maps we consider, as in [BNI], is the class
VMO$(\Omega,\Bbb R^n)$ satisfying an appropriate variant of $(0.1)$.
To define VMO in a domain $\Omega$, we have first to define BMO.  There
are several possible definitions; they turn out, however, to be
equivalent.  Here is one:
\medskip
\definition{Definition}  A real function $f$ in $L^1_{\text{loc}}(\Omega)$
is in BMO$(\Omega)$ if
$$
\|f\|_{\text{BMO}(\Omega)} := \operatornamewithlimits{\sup}_B \Mint_B |f
- \Mint_B f| < \infty, \tag 0.2
$$
where $\sup$ is taken over all Euclidean balls with closure in
$\Omega$.
\medskip
In fact, one may use balls in any norm in $\Bbb R^n$---though this is
far from obvious---see Corollary A1.1.  Furthermore, one may consider
the $\sup$ in (0.2) over the class of balls $B$ lying ``well
inside'' $\Omega$, i.e., say $B = B_r(x)$ with $r\leq
\dfrac12$dist$(x,\partial\Omega)$.  The resulting norm is smaller than
that in (0.2), but is equivalent to it (see Theorem A1.1).
\medskip
Now VMO is the closure of $C^0(\overline\Omega)$ in the BMO norm of
(0.2).  A useful characterization of VMO$(\Omega)$ is
$$
\operatornamewithlimits{\lim}\Sb\varepsilon\rightarrow
0\\\varepsilon\leq\frac12 \text{dist}(x,\partial\Omega)\endSb
\Mint_{B_\varepsilon(x)} |f - \overline f_\varepsilon(x)| =
0\quad\text{uniformly in $x$}.
$$
Here 
$$
\overline f_\varepsilon(x) = \Mint_{B_\varepsilon(x)} f.
$$
This is the analogue of Sarason's characterization of VMO in $\Bbb
R^n$; see D.~Sarason~[1].  A surprising fact about VMO$(\Omega)$ is
that it is the closure in the BMO norm of $C_0^\infty(\Omega)$, $C^\infty$
functions with compact support in $\Omega$ (see Theorem 1; it is
proved in Appendix 1).
\medskip
The facts above about BMO and VMO in $\Omega$ are due to Peter Jones;
the proofs given here are slight modifications of his.
\bigskip
In addition to bounded domains in $\Bbb R^n$ we also consider domains
$\Omega$ in a smooth open $n$-dimensional Riemannian manifold $X_0$,
where $\overline\Omega$ is compact in $X_0$.  BMO$(\Omega)$ is defined
as in (0.2); the $\sup$ is now taken over geodesic balls
$B_\varepsilon(x)$ with $\varepsilon < r_0$, the injectivity radius of
$\overline\Omega$.  As in $\Bbb R^n$, the various possible alternate
definitions of BMO$(\Omega)$ are equivalent.  Furthermore, the space
BMO$(\Omega)$ is independent of the Riemannian metric on $X_0$ (see
Lemma 2 in \S II.1).  VMO is defined as above.  We then consider
VMO maps of $\Omega$ into an $n$-dimensional smooth open manifold $Y$
(which is smoothly embedded in some $\Bbb R^N$).  If $X_0$ and $Y$ are
oriented, and $p\in Y$ is such that (0.1) holds---in a suitable
sense---then we define
$$
\deg(u,\Omega,p);
$$
this is done again by approximation.
\medskip
In dealing with manifolds one has to consider the effect of change of
local coordinates.  A result used here, but which more
properly fits in [BNI], asserts that if the manifold $X_0$ is
compact (without boundary), and if $H$ is a smooth diffeomorphism of a
ball $B_R$ in $\Bbb R^n$ onto a subset of $X_0$, then there are positive
constants $C, \varepsilon_0$, such that
$$
|\overline{(f\circ H)_\varepsilon} (y)  - \overline f_\varepsilon
(H(y))|\leq C \|f\|_{\text{BMO}} \tag 0.3
$$
for every $f\in \text{BMO}(X_0), |y|\leq R/2$, and $\varepsilon <
\varepsilon_0$.  This is essentially Lemma A3.3.
\medskip
In II.1 BMO and VMO are introduced and their invariance under choice
of norms, as described above, is presented as well as associated
properties.
\medskip
Section II.2 takes up the definition of the degree.  The analogue we use of
condition (0.1) is that there exist a neighbourhood $U$ in
$\Omega$ of $\partial\Omega$, and a number $d_0 > 0$ such
that
$$
\Mint_{B_\varepsilon(x)} \text{dist}(u,p) \geq d_0\quad\forall
B_\varepsilon(x)\text{  in $U$ with $\varepsilon =
\frac12$dist$(x,\partial\Omega)$}. \tag 0.4
$$
\medskip
Various properties of degree are then established, including (Corollary 1), the
invariance of degree under continuous deformation in the BMO topology,
provided that, under the deformation, (0.4) holds uniformly for all maps
considered, with the same $U$ and $d_0$.  In Remark 4, an example is
given in which the stability of degree fails in case this uniformity
is dropped.
\medskip
In general, functions in VMO$(\Omega)$ do not have a well defined
trace on $\partial\Omega$. In II.3, in case $\partial\Omega$ is
smooth, we introduce a subclass of VMO$(\Omega)$ which does:  Suppose
$\varphi\in\text{VMO}(\partial\Omega)$; we may then extend $\varphi$
inside $\Omega$ to a function $\widetilde \varphi$ belonging to
VMO$(\Omega)$ with
$$
\widetilde\varphi(x) = \varphi(P(x))\qquad\text{near $\partial\Omega$}.
$$
Here $P$ is the projection to the nearest point on $\partial\Omega$.
We then say that a function\newline $f \in\text{VMO}(\Omega)$ has $\varphi$ as
trace on $\partial\Omega$, written as
$$
f\in\text{VMO}_\varphi(\Omega)
$$
provided the function
$$
g = \cases  f - \widetilde\varphi&\quad\text{in $\Omega$}\\
                 0&\quad\text{outside $\Omega$},
\endcases
$$
belongs to VMO on a neighbourhood of $\overline\Omega$. 

\medskip
Theorem 2 asserts that for $f$ in VMO$(\Omega)$,
$$
f\in\text{VMO}_\varphi(\Omega)\Longleftrightarrow
\operatornamewithlimits{\lim}\Sb x\rightarrow\partial\Omega\\
\varepsilon=\frac12\text{dist}(x,\partial\Omega)\endSb \Mint_{B_\varepsilon(x)} |f - \widetilde\varphi| =
0. \tag 0.5
$$
\enddefinition
\medskip
Various examples of VMO${}_\varphi(\Omega)$ are presented in \S II.3.
Example 2 states that $W^{1,n}(\Omega)\subset\text{VMO}_\varphi(\Omega)$. Lemma 7 asserts that for $x$ near $\partial\Omega$, if
$d(x) = \text{dist}(x,\partial\Omega)$, the function
$$
f(x) = \overline\varphi_{d(x)} (P(x))
$$
---then extended inside in the rest of $\Omega$ by smooth cutoff---belongs to
VMO${}_\varphi(\Omega)$.  Lemma 8 says that the harmonic extension of
$\varphi$ inside $\Omega$ belongs to VMO${}_\varphi(\Omega)$; this is
proved in Appendix 3.
\medskip

Recently, L.~Greco, T.~Iwaniec, C.~Sbordone and B.~Stroffolini~[1]
 introduced a notion of degree for a class of Sobolev maps which
is weaker than $W^{1,n}$ and is not contained in VMO.
\medskip

Finally, in II.3, a question of H.~Amann is answered.  In [BNI], if
$X,Y$ are compact oriented $n$-manifolds without boundaries, and
$\varphi,\psi\in\text{VMO}(X,Y)$ are connected by some homotopy $H$
which is continuous in a parameter $t$ on [0,1], with values in
VMO$(X,Y)$, then (Corollary 6 in [BNI]) $\deg\varphi = \deg \psi$.
Amann asked whether the conclusion still holds in case
$$
H\in \text{VMO}(X\times [0,1], Y).
$$
Under suitable conditions on $H$ for $t$ near $0$ and
$1$, Corollary 3 asserts that the answer is yes.
\medskip
Section II.4 extends to VMO${}_\varphi(\Omega)$ a standard result for
continuous maps $u: \overline\Omega\rightarrow\Bbb R^n$, with
$u\big\vert_{\partial\Omega} = \varphi$.  Namely, if
$\varphi\not=p\quad\forall x\in\partial\Omega$, then
$$
\deg(u,\Omega,p) = \deg(\frac{\varphi - p}{|\varphi - p|},
\partial\Omega, S^{n-1}). \tag 0.6
$$
\medskip
Appendix 1 proves a number of results of II.1.
\medskip
In Appendix 2, written with P. Mironescu, we consider Toeplitz
operators on $S^1$.  For any continuous complex-valued function
$\varphi$ on $S^1$, with $\varphi\not=0$ everywhere, there is,
classically, an associated Toeplitz operator $T_\varphi$.  It is a
Fredholm operator in $\Cal H^2$ and
$$
\text{index}(T_\varphi) = -\deg\left(\frac{\varphi}{|\varphi|}, S^1, S^1\right).
$$
In Theorem A2.1 a similar result is proved for $\varphi$ satisfying
$$
\varphi\in\text{VMO}(S^1,\Bbb C) \cap L^\infty, |\varphi|\geq a > 0\text{  on
$S^1$}.
$$
This result is essentially contained in Theorem 7.36 in R.~G.~Douglas~[1];
the proof here is different and is pretty much self contained---though
we use the fundamental $\Cal H^1$-BMO duality of C.~Fefferman~[1] (see
also C.~Fefferman and E.~Stein~[1]).

\medskip
Appendix 3 deals with properties of the harmonic extension of BMO and
VMO maps.
\medskip
The plan of the paper is:
\medskip
\noindent
II.1\qquad  BMO and VMO on domains
\smallskip
\noindent
II.2\qquad  Degree of maps on domains
\smallskip
\noindent
II.3\qquad  VMO functions having a VMO trace; VMO${}_\varphi$
\smallskip
\noindent
II.4\qquad  For $u\in \text{VMO}_\varphi$,$\deg (u,\Omega,p) = $ a
boundary degree
\smallskip
\noindent
Appendix 1  \qquad  Some properties of BMO and VMO in
domains
\smallskip
\noindent
Appendix 2 \qquad  (with P. Mironescu). Toeplitz operators and VMO
\smallskip
\noindent
Appendix 3 \qquad  The harmonic extension of VMO maps
\bigskip
\noindent
 We are especially grateful to Peter Jones and wish to express thanks
also to several colleagues for interesting conversations:  H.~Amann,
S.~Chanillo, A.~Connes, I.~Gohberg, P.~D.~Lax, F.~H.~Lin, P.~Mironescu.
\bigskip
\noindent
{\bf II.1.  BMO and VMO on domains}
\medskip
Let $\Omega$ be a bounded domain (open connected set) in $\Bbb R^n$.
Later we will consider domains in a manifold.
\medskip
There are several natural notions of BMO$(\Omega)$.
\medskip
\definition{Definition 1}  A locally integrable real function $f$ on
$\Omega$ belongs to BMO$(\Omega)$ if
$$
\|f\|_{\text{BMO}} := \sup_{B\in \Cal C} \frac{1}{|B|} \int_B |f - \bar f_B|
< \infty, \tag 1.1
$$
where $\Cal C$ is the class of all open balls $B$ whose closures lie in
$\Omega$, and
$$
\bar f_B = \underset B\to\Mint f,
$$
the average of $f$ over $B$.
\medskip
BMO$(\Omega)$ so defined forms a Banach space modulo constants.
Similarly a map $u : \Omega\rightarrow \Bbb R^N$ belongs to
BMO$(\Omega,\Bbb R^N)$ if each component of $u$ is in BMO$(\Omega)$.
Its BMO norm is also given by (1.1) where $|\;\;|$ denotes the
Euclidean norm in $\Bbb R^N$.  As in [BNI] an equivalent norm is
$$
\|u\|_\star = \sup_{B\in\Cal C}\; \Mint_B \;\Mint_B |u(y) - u(z)|
d\sigma(y) d\sigma (z); \tag 1.2
$$
in fact
$$
\|u\|_{\text{BMO}} \leq \|u\|_\star \leq 2 \|u\|_{\text{BMO}}.
\tag 1.3
$$
\enddefinition
\medskip
\definition{Definition 2}  For $0 < k < 1$ let $\Cal C_k$ denote all
balls $B_r (x)\subset \Omega$ satisfying
$$
r\leq k\text{ dist}(x,\partial\Omega).
$$
Such balls are called ``well inside'' $\Omega$.  Using $\Cal C_k$ instead
of $\Cal C$ in (1) we obtain a different smaller norm
$$
\|f\|_{\text{BMO},k}.
$$
\medskip
It is not difficult to see that for $0 < k_1, k_2 < 1$, the norms
$$
\|f\|_{\text{BMO},k_1}\quad\text{and $\|f\|_{\text{BMO},k_2}$ are
equivalent}.
$$
(see Lemma A1.1 in Appendix 1).  A more striking fact is that each of
these is equivalent to the norm (1.1), even if no regularity of
$\partial\Omega$ is required. As we show in Theorem A1.1,
this equivalence holds not just for the Euclidean norm but for any norm
on $\Bbb R^n$.  This fact is far from trivial and is due to Peter
Jones.  We present a slight modification of his proof; see Theorem A1.1.
\medskip
It is more convenient to work with Definition 2.  {\it From now on we take that as our definition of BMO, with $k$ fixed
as $1/2$, and we simply write
$$
\|f\|_{\text{BMO}, 1/2}\text{  as $\|f\|_{\text{BMO}}$ and $\Cal
C_{1/2} = \Cal C$}.
$$
We use formula (1.2) as well with balls $B$ well inside (with $k =
1/2$).
\enddefinition
\bigskip
\remark{\bf Remark 1}  In Definition 2, if we restrict the class $\Cal
C_k$ to all balls $B_r(x)$ satisfying 
$$
r \leq \min \{k\text{ dist}(x,\partial\Omega), r_0\}
$$
for some given $r_0 > 0$, we get a smaller norm which is,
however, equivalent to the original one.  This is easily seen by a
trivial covering argument.
\endremark
\remark{\bf Remark 2}  Another possible definition of BMO$(\Omega)$ is
to take as $\Cal C$ the class of all cubes with closures in $\Omega$, or
all those with edges parallel to the axes, or with cubes ``well
inside'' $\Omega$.  The corresponding norms are all equivalent to the
BMO norm above (see
the discussion after Theorem A1.1 in Appendix 1).
\endremark
\medskip
Clearly $L^\infty(\Omega) \subset \text{BMO}(\Omega)$ with continuous injection:
$$
\|f\|_{\text{BMO}} \leq 2 \|f\|_{L^\infty}.
$$
In particular $C^0(\overline\Omega) \subset \text{BMO}(\Omega)$.
\medskip

  We now define VMO$(\Omega)$.  It was first introduced by D.~Sarason [1] in
all of $\Bbb R^n$.
\medskip
\definition{Definition}  VMO$(\Omega)$ is the closure in BMO$(\Omega)$ of
$C^0(\overline\Omega)$, i.e., $f\in\text{ VMO}(\Omega)$ if there is a
sequence $(f_j)$ in $C^0(\overline\Omega)$ converging to $f$ in BMO$(\Omega)$.
\enddefinition
\medskip
In view of Lemma 1 below, if $f\in \text{VMO}(\Omega)$ then there is a
sequence $(f_j)$ in $C^0(\overline\Omega)$ converging to $f$ in BMO$(\Omega)$, in
$L^1_{\text{loc}}(\Omega)$, and a.e.
\medskip
\proclaim{Lemma 1}  Given a compact set $K$ in $\Omega$, there is a
constant $C_K$ such that
$$
\|f - \overline f_K \|_{L^1(K)} \leq C_K \|f\|_{\text{BMO}}
$$
for every $f\in \text{BMO}(\Omega)$.
\endproclaim

The proof of Lemma 1 is similar to that of Lemma A.1 in [BNI].
\medskip

To prove the assertion before the lemma, observe first that given any
$\varepsilon > 0$ and any compact set $K\subset\Omega$, there is a
$g\in C^0(\overline\Omega)$ such that
$$
\|f - g\|_{\text{BMO}} < \varepsilon,\quad \|f - g\|_{L^1(K)} <
\varepsilon.
$$
This uses Lemma 1.  The assertion then follows by choosing
$\varepsilon=\dfrac1j, j = 1,2,\ldots,$ and
$$
K = \{x\in\Omega; \text{dist}(x,\partial\Omega) \geq \dfrac{1}{j}\}.
$$.
\medskip
It is clear that if $f \in \text{VMO}(\Omega)$ then
$$
\lim_{\varepsilon\rightarrow 0} \;\Mint_{B_\varepsilon(x)}
|f-\overline f_\vare (x)| = 0\text{\qquad  ``uniformly in $x$''}.\tag 1.4
$$
where
$$
\overline f_\vare(x) = \operatornamewithlimits{\Mint}_{B_\vare(x)} f.
$$
\medskip
More precisely, (1.4) means that for every $\delta > 0$, there exists
$\varepsilon_0$ such that, for all $x\in \Omega$,
$$
\Mint_{B_\varepsilon(x)} |f - \bar f_\varepsilon(x)| < \delta
$$
for all $\varepsilon\leq \min\{\varepsilon_0, \dfrac12\text{
dist}(x,\partial\Omega)\}$.
\medskip

The converse is true; this is far from obvious.  In fact, a much
stronger result holds.  It is due to Peter Jones (private
communication):
\medskip
\proclaim{Theorem 1 (P. Jones)}  The following are all equivalent for
$f$ in BMO$(\Omega)$:
$$
f \in \text{VMO} (\Omega).
\tag 1.5
$$
$$
\operatornamewithlimits{lim}\Sb\varepsilon\rightarrow 0\\
\varepsilon\leq \frac12\text{dist}(x,\partial\Omega)\endSb\;
\Mint_{B_\varepsilon (x)} |f - \bar f_\varepsilon(x)|= 0\text{\quad
uniformly in $x$}
\tag 1.6
$$
in the sense above.
$$
\text{There exists a sequence $(f_j)$ in $C^\infty_0(\Omega)$
converging to $f$ in BMO$(\Omega) \cap L^1_{\text{loc}}(\Omega)$}.
\tag 1.7
$$
\endproclaim

The proof of Theorem 1 is in Appendix 1.
\medskip
\example{\bf Example 1}  $W^{1,n}(\Omega)\subset \text{VMO}(\Omega)$.
\endexample
To see this, observe first that $W^{1,n}(\Omega)\subset \text{BMO}
(\Omega)$, with continuous injection. This follows from Poincar\'e's
inequality in any ball $\overline B\subset\Omega$,
$$
\Mint_B | f - \overline f_B|\leq C(n) \left(\int_B (\nabla
f|^n\right)^{1/n}. \tag 1.8
$$
This implies that (1.6) holds and thus, by Theorem 1, $f$ is in
VMO$(\Omega)$.
\medskip
\remark{\bf Remark 3}  Theorem 1 asserts that $C_0^\infty(\Omega)$ is
dense in VMO$(\Omega)$.  Recall that it is {\it not} dense in
$W^{1,n}(\Omega)$.
\endremark
\medskip
More generally, we have as in [BNI]:
\medskip
\example{\bf Example 2}  $W^{s,p}(\Omega)\subset \text{VMO}(\Omega)$
in the limiting case of the Sobolev embedding:\newline  $sp = n, 0 < s < n$,
($s$ may or may not be an integer).
\endexample

In [BNI] we discussed functions involving $\log|x|$:

$(a)$\quad  $\log|x|$ is in BMO$(\Omega)$ but not in VMO$(\Omega)$ if $0
\in \Omega$,

$(b)$\quad $\log|\log|x||$ is in VMO$(\Omega)$.

$(c)$\quad $|\log|x||^\alpha, 0 < \alpha < 1$, is in VMO$(\Omega)$.
\bigskip
Consider now a domain $\Omega$, having compact closure in a smooth
manifold $X$ without boundary.  In order to define BMO$(\Omega)$ and VMO$(\Omega)$, one first puts a
smooth Riemannian metric on $X$, the notions above of BMO$(\Omega)$
and VMO$(\Omega)$ extend except that we use geodesic balls
$B_\varepsilon(x)$ and always assume that $\varepsilon < 
r_0$, the injectivity radius of $\overline\Omega$.  The
definitions are independent of the choice of metric.  In fact, there is a more general result:
\medskip
\proclaim{Lemma 2}  Let $\Omega_1, \Omega_2$ be two bounded domains in
$\Bbb R^n$ and let $H$ be a $C^1$ diffeomorphism of a neighbourhood of
$\overline\Omega_1$ onto a neighbourhood of $\overline\Omega_2$.  If
$f\in\text{BMO}(\Omega_2)$ \text{(}respectively VMO$(\Omega_2)$\text{)} then $f\circ
H$ is in BMO$(\Omega_1)$ \text{(}respectively VMO$(\Omega_1)$\text{)} and
$$
\|f \circ H\|_{\text{BMO}} \leq C\|f\|_{\text{BMO}}.
$$
\endproclaim

This is proved in Appendix 1.  Furthermore, Theorem 1 holds in this
situation, with no change.
\medskip
\example{\bf Example 3}  Let $\Omega$ be such a domain on a manifold
$X$.
\endexample
\medskip
\proclaim{Lemma 3}  The function
$$
\varphi(x) = \log(1/\text{\rm dist}(x,\partial\Omega))
$$
is in BMO$(\Omega)$.  Here {\rm dist} could be measured using any metric on
$\Omega$ which is equivalent to the Riemannian metric.
\endproclaim

\proclaim{Lemma 4}  With $\varphi$ as in Lemma 3,
$|\varphi|^\alpha\in\text{VMO}(\Omega)$ for $0 < \alpha < 1$.
\endproclaim

Lemmas 3 and 4 are proved in Appendix 1.
\bigskip
\medskip
\noindent
{\bf II.2.  Degree of maps on domains}
\medskip
Let $\Omega$ be a general bounded domain in $\Bbb R^n$, let $u\in
\text{VMO}(\Omega,\Bbb R^n)$ and let $p$ be a point in $\Bbb R^n$.
Our goal is to define $\deg(u,\Omega,p)$ and prove that it has the
standard properties of a degree.
\medskip

In the usual case, when $u\in C^0(\overline\Omega)$, one assumes that
$$
p\notin u(\partial\Omega). \tag 2.1
$$
General functions in VMO$(\Omega)$ have no trace on the boundary.
(Later we shall introduce a subclass of VMO functions with a
trace---the notion of trace is delicate and the subclass is somewhat
restricted.)  Thus the condition (2.1) has to be given a different
form.
\medskip
\noindent
{\bf Notation.}  We denote by $\Cal D$ the class of balls
$B_\varepsilon(x)$ in $\Omega$ with
$$
\varepsilon = \frac12  \text{dist}(x,\partial\Omega).
$$
\medskip
In place of (2.1) we use the condition:
$$
\cases
\text{there exist $d_0 > 0$, and a neighbourhood $U$ in $\Omega$
of $\partial\Omega$, such that}\\
\qquad\qquad\dsize\Mint_B |u - p|\geq d_0\quad\forall B\subset U,
B\in\Cal D.\endcases
\tag 2.2
$$
In particular, (2.2) holds if $|u - p|\geq d_0$ a.e. in some
neighbourhood $U$ of $\partial\Omega$.  Clearly for $u\in C^0(\overline
\Omega)$, (2.1) and (2.2) are equivalent.
\medskip
\noindent
{\bf Notation.}  For $\varepsilon > 0$, set
$$
\Omega_\varepsilon = \{x\in\Omega;\;\text{dist}(x,\partial\Omega) >
\varepsilon\}.
$$
\medskip
\noindent
{\bf Definition of degree for $u\in\text{VMO}$ satisfying (2.2)}:
\medskip
Given $u\in\text{VMO}(\Omega)$, we choose $\varepsilon_0 > 0$ so that
for all $x\in\Omega$,
$$
\Mint_{B_\vare(x)} |u - \overline{u}_\vare(x)|\leq d_0/2
\tag 2.3
$$
for all $\varepsilon\leq \varepsilon_0$ and $\varepsilon\leq
\dfrac12\text{dist}(x,\partial\Omega)$.  This is possible in view of
(1.4).  We may also take $\varepsilon_0$ to satisfy 
$$
\{x\in\Omega;\;\text{dist}(x,\partial\Omega)\leq 3\varepsilon_0\}\subset U,
$$
with $U$ as in (2.2).
\medskip

Combining (2.2) and (2.3) we have
$$
|\overline u_\vare(x) - p|\geq d_0/2\quad\text{if
$x\in\partial \Omega_{2\vare}$ and $\vare\leq \vare_0$}.
\tag 2.4
$$
Hence
$$
\deg(\overline u_\vare, \Omega_{2\vare},p)\quad\text{is defined for
every  $\vare\leq \vare_0$}.
$$
\medskip
\noindent
{\bf Claim:}  This degree is independent of $\vare$ for $0 < \vare\leq
\vare_0$.
\medskip
  We then define
$$
\deg(u,\Omega,p) = \deg(\overline u_\vare, \Omega_{2\vare},
p)\quad\text{for $\vare\leq \vare_0$}.
$$
\medskip
\noindent
{\it Proof of Claim:}  We may suppose $p = 0$.
\medskip
  We shall prove that for any given $\vare$ in $(0,\vare_0]$,
\; there exists $\delta$ depending on $\vare$ such that
$$
\deg(\overline u_t, \Omega_{2t},0) = \deg(\overline
u_\vare,\Omega_{2\vare},0)\quad\text{for $|t - \vare| < \delta$}.
\tag 2.5
$$
This yields the claim.
\bigskip

The map $\overline u_t$ is continuous in $x$ and $t$ where it is
defined.  Using (2.4) we see that there exists $\delta > 0$ such that
$$
|\overline u_t(x)|\geq \frac{d_0}{4}\quad\text{if $|t - \vare| <
\delta$ and dist$(x,\partial \Omega_{2\vare}) < \delta$}.
\tag 2.6
$$
Therefore
$$
\deg (\overline u_t, \Omega_{2\vare}, 0)\quad\text{is defined for $|t
- \vare| < \delta$}.
$$
By homotopy invariance and (2.6), this degree is independent of $t$,
and so
$$
\deg(\overline u_t, \Omega_{2\vare}, 0) = \deg(\overline u_\vare,
\Omega_{2\vare},0)\quad\text{for $|t - \vare| < \delta$}.
$$
Finally, by excision, and (2.6) again,
$$
\deg(\overline u_\vare, \Omega_{2\vare},0) = \deg(\overline u_t,
\Omega_{2t},0),
$$
and the claim is proved.
\medskip

Consequently, $\deg(u,\Omega,p)$ is defined. It is clear that if $u\in
C^0(\overline\Omega)$ then the degree just defined agrees with the usual degree.
\medskip
We verify now some of the standard properties of degree:
\medskip
\noindent
{\bf Property 1.}  If $u\in\text{VMO}(\Omega,\Bbb R^n)$ satisfies
(2.2) and
$$
\deg(u,\Omega,p)\not=0,
$$
then
$$
p\in \text{ess} R(u).
$$
(The essential range of a map $u$, ess$R(u)$, is defined in \S I.4
of [BNI]).  In fact
$$
B_{d_0} (p)\subset\text{ ess }R(u).
$$
\medskip
The proof follows that of Property 1 in \S I.4 of [BNI].
\medskip
\noindent
{\bf Property 2.  (Stability of degree in the BMO topology)}.
\medskip
Let $(u_j)$ and $u$ belong to VMO$(\Omega)$ and satisfy
$$
u_j\longrightarrow u\quad\text{in BMO$(\Omega)\cap
L^1_{\text{loc}}(\Omega)$}
\tag 2.7
$$
and
$$
\cases  \text{for some $p\in\Bbb R^n$, there exist a $d_0 > 0$ and a
neighbourhood $U$ of $\partial\Omega$ in $\Omega$},\\
\text{such that $\dsize\Mint_B |u_j - p|\geq d_0\quad\forall j,\quad\forall
B\subset U, B\in \Cal D$},
\endcases
\tag 2.8
$$
(in view of (2.7), the same holds for $u$).
\medskip
Then
$$
\deg(u_j,\Omega,p) = \deg(u,\Omega,p)
$$
for all $j$ sufficiently large.
\medskip

\demo{Proof}  We may take $p = 0$.  As in Lemma 4 of I.1 of
[BNI] we see that
$$
\lim\Sb |B|\rightarrow0\\B\in \Cal C\endSb\; \Mint_B |u_j - \overline
u_{j,B}| = 0\text{\qquad  uniformly in $j$}. 
\tag 2.9
$$
(Recall that $B\in\Cal C$ means that if $B = B_r(x)$, then $r\leq
\dfrac12$ dist$(x,\partial\Omega)$.)  It is here that we use the assumption that $u_j\rightarrow u$ in
BMO$(\Omega)$.  (2.8) and (2.9) imply that there exists $\vare_0$ such
that for all $\vare\in (0,\vare_0)$,
$$
\bigg\vert\Mint_{B_\vare(x)} u_j\bigg\vert \geq
\frac{d_0}{2}\qquad\forall j,\quad\forall x\in\partial\Omega_{2\vare}.
$$
\enddemo
\medskip
{\it Fix} some $\vare \in (0,\vare_0)$. Since $u_j\rightarrow u$ in
$L^1_{\text{loc}} (\Omega)$, we have
$$
\overline u_{j,\vare}\longrightarrow \overline
u_\vare\qquad\text{uniformly in $\overline\Omega_{2\vare}$}.
$$
Thus
$$
\deg(\overline u_{j,\vare}, \Omega_{2\vare},0) = \deg(\overline u_\vare, \Omega_{2\vare}, 0)
$$
for $j$ sufficiently large.  By our definition of degree we obtain
the desired result.
\medskip
\remark{\bf Remark 4}  In the argument above it is essential that
(2.8) holds {\it uniformly} in $j$.  Here is an illuminating example in
which uniformity in (2.8) is dropped and the conclusion fails.  Let
$\Omega = (0,1)$, and set
$$
u_j(x) = f_j(x) - \frac12
$$
where $f_j$ is the sequence defined in Example 6 of \S I.2 in [BNI].  Since
$u_j(0) = \dfrac12$ and\newline $u_j(1) = -\dfrac12, \deg(u_j,\Omega,0) = -1$. 
\endremark
\medskip

On the other hand, $u_j\rightarrow u\equiv - \dfrac12$ in BMO and in
$L^1$, and $\deg(u,\Omega,0) = 0$.
\medskip
An immediate corollary of the above is the invariance under suitable
homotopy:
\medskip
\proclaim{Corollary 1}  Let $H_t(\cdot)$ be a one-parameter family of
VMO maps from $\Omega$ to $\Bbb R^n$, depending continuously---in the
BMO$\cap L^1_{\text{loc}}$ topology---on the parameter $t$.  Assume in
addition, that (2.8) holds uniformly in $t$, i.e., the same $d_0$ and
$U$ for all $H_t$.  Then
$$
\deg (H_t,\Omega,p)\quad\text{is independent of $t$}.
$$
\endproclaim
\medskip
\proclaim{Corollary 2}  Suppose $u,v$ are VMO maps from $\Omega$ into
$\Bbb R^n$ both satisfying (2.2).  Suppose that for some $d_1 < d_0$,
$$
\Mint_B |u - v|\leq d_1,\quad\forall B\subset U,\;B\in\Cal D.
$$
Then
$$
\deg(v,\Omega,p) = \deg(u,\Omega,p).
$$
\endproclaim
To prove this, just use the homotopy $H_t = t v + (1-t)u,\quad0\leq
t\leq 1$, and apply the preceding corollary.
\medskip
\noindent
{\bf Property 3 (Borsuk)}.  Suppose $u\in\text{VMO}(\Omega,\Bbb R^n)$
and (2.2) holds with $p = 0$.  If $0\in\Omega$, $\Omega$ is
symmetric about the origin and $u$ is odd near $\partial\Omega$, then
$$
\deg(u,\Omega,0)\quad\text{is odd}.
$$

This is an immediate consequence of our definition of degree---via
Borsuk's theorem for continuous maps.
\medskip
\noindent
\remark{\bf Remark 5}  The definition of degree extends in a
straightforward way to VMO maps from a domain $\Omega$, with compact
closure, in a smooth oriented Riemannian manifold $X$, with values in
another oriented smooth manifold $Y$, $\dim Y =\dim X$.  Namely for
$u\in\text{VMO}(\Omega,Y)$, and for $p\in Y$ such that (2.2) holds,
where $|u(z) - p|$ is replaced by dist$(u(z),p)$, one defines
$$
\deg(u,\Omega,p)
$$
as in the Euclidean case.
\endremark
\bigskip
\noindent
{\bf II.3.  VMO functions having a VMO trace; VMO$_\bold\varphi$}
\medskip
In general, VMO functions on a domain $\Omega$ do not have a well
defined trace on $\partial\Omega$---even if $\partial\Omega$ is
smooth.  An example for $\Omega = (0,1)$ is the function
$\cos(\log|\log x|)$.  It is in VMO---even in $H^{1/2}$---but has no
trace at $0$.
\medskip
It is useful to introduce a subclass which {\it does} admit a trace on
$\partial\Omega$ belonging to VMO$(\partial\Omega)$.  As usual,
$\Omega$ is a bounded domain in $\Bbb R^n$.
\medskip
\definition{Definition of VMO$_{\bold 0}$}  A function
$f\in\text{VMO}(\Omega)$ belongs to VMO$_0(\Omega)$ if its extension $g$ outside $\Omega$ as identically zero, belongs to VMO$(B)$, where $B$
is an open ball containing $\overline\Omega$.
\enddefinition
\medskip
\remark{\bf Remark 6}  A function $f\in\text{VMO}(\Omega)$ which is
identically zero near $\partial\Omega$ belongs to VMO$_0(\Omega)$.
Indeed its extension $g$, by zero outside $\Omega$ lies in BMO$(B)$,
as is clear by Remark 1.  That it lies in VMO$(B)$ is a consequence of
Theorem 1.
\endremark
\medskip
A simple characterization in case $\partial\Omega$ is {\it smooth} is given
by
\medskip
\proclaim{Theorem 2}  $f\in\text{VMO}(\Omega)$ belongs to
VMO$_0(\Omega)$ iff
$$
\lim\Sb |B|\rightarrow 0\\B\in\Cal D\endSb\; \Mint_B |f| = 0. \tag 3.1
$$
\endproclaim

Condition (3.1) means that the average of $|f|$ over balls
$B_\vare(x)$ tends to zero as $x\rightarrow\partial\Omega$ provided
$\vare = \dfrac12$ dist$(x,\partial\Omega)$.

\demo{Proof}  
\medskip
\noindent
{\bf 1.}  Proof that $f\in \text{VMO}_0(\Omega)\Rightarrow (3.1)$ if
$\partial\Omega$ is smooth:  To see this, consider a ball
$B_\vare(x)\in \Cal D$, i.e., $\vare = \dfrac12$
dist$(x,\partial\Omega)$.  Let $z$ be a closest point on
$\partial\Omega$ to $x$.  Since $\partial\Omega$ is smooth, there is
some $\alpha > 0$, and some $\vare_0 > 0$ such that
$$
|B_{3\vare}(y)\cap\Omega^c|\geq \alpha |B_{3\vare}(y)|\quad\forall
y\in\partial\Omega,\quad\forall \vare\leq \vare_0. \tag 3.2
$$

Since $g\in\text{VMO}(B)$, given $\delta > 0$, there exists $\vare_1 >
0$ such that for $\vare < \vare_1$,
$$
\Mint_{B_{3\vare}(z)} |g - \bar g_{3\vare}(z)| < \delta. \tag 3.3
$$
It follows that
$$
|\bar g_{3\vare}(z)|
\frac{|B_{3\vare}(z)\cap\Omega^c|}{|B_{3\vare}(z)|} < \delta,
$$
so that
$$
|\bar g_{3\vare}(z)|\leq \frac\delta\alpha\quad\forall \vare <
\vare_1. \tag 3.4
$$
\medskip
\enddemo

By Lemma A.4 in [BNI]
$$
|\bar g_\vare(x) - \bar g_{3\vare}(z)|\leq 3^n \Mint_{B_{3\vare}(z)}
|g - \bar g_{3\vare}(z)|\leq 3^n\delta.
$$
Using (3.4) we find that
$$
|\bar g_\vare(x)|\leq (3^n + \frac{1}{\alpha})\delta. \tag 3.5
$$

Since $g$ is in VMO$(B)$, there is $\vare_2 < \vare_1$ such that
$$
\Mint_{B_\vare(x)} |g - \bar g_\vare (x)|\leq \delta\quad\text{for
$\vare < \vare_2$}.
$$
Combining this with (3.5) we obtain the desired result.
\bigskip
It is clear from the proof that what is required of $\Omega$ is simply
(3.2) rather than regularity.  Thus $\partial\Omega$ might merely be a
Lipschitz boundary.  However, some regularity of $\partial\Omega$ is
necessary.  For example if $\Omega = \text{unit disc in $\Bbb R^2$}$
minus the origin, and $f$ is smooth in $\Omega$ with $f = 1$ in $0 <
|x| < \frac12$ and $f = 0$ for $|x| > 3/4$, then $f \in
\text{VMO}_0(\Omega)$ but does not satisfy (3.1).
\medskip
One also observes from the proof above that $f\in\text{VMO}_0(\Omega)$
implies
$$
\operatornamewithlimits{\lim}_{\varepsilon\rightarrow 0}
\Mint_{B_\varepsilon(x)} |f| = 0
$$
where $\varepsilon = \text{dist}(x,\partial\Omega)$.
\bigskip



\noindent
{\bf 2.}  Proof that $(3.1)\Rightarrow f\in \text{VMO}_0(\Omega)$.
This is true for any bounded domain $\Omega$. 
\medskip
We have to show that given any $\delta > 0$ there is some $\vare_0 >
0$ such that, for $\vare < \vare_0$,
$$
\operatornamewithlimits{\Mint}_{B_\vare(x)}\;\operatornamewithlimits{\Mint}_{B_\vare(x)}
|g(y) - g(z)| < 2\delta
$$
where $B_\vare(x)$ is any ball in $\Bbb R^n$.  If $B_\vare(x)$ is in
$\Omega^c$ or if $B_\vare(x)$ is ``well inside'' $\Omega$, this is
clear.  Thus we may assume that
$$
B_\vare (x) \cap \Omega\not=\emptyset\quad\text{and $B_{2\vare} (x)
\cap \Omega^c \not=\emptyset$},
$$
and in particular dist$(x,\partial\Omega) \leq 2\vare$.
\medskip
Set $A = B_\vare(x)\cap\Omega$.  It suffices to prove that, for $\vare
<$ some $\vare_0$,
$$
\frac{1}{|B_\vare(x)|} \int_A |f| < \delta.
$$
\medskip
Consider a maximal family of disjoint open balls $B_{\vare_{i/3}}
(x_i)$ with centres $x_i\in A$ and $\vare_i =\dfrac12$
dist$(x_i,\partial\Omega)$.  Since $x_i\in B_\vare(x)$ we have
$$
\vare_i \leq \frac12 (\text{dist}(x_i,x) +
\text{dist}(x,\partial\Omega)) \leq \frac32\; \vare\quad\forall i.
$$
\medskip
\noindent
{\bf Claim:}  $G = \operatornamewithlimits{\bigcup}_i
B_{\vare_i}(x_i)\;\;$ covers $A$.
\medskip
Suppose not.  Suppose some $y\in A$, $y\notin G$.  Set $\gamma =
\dfrac12 \text{dist}(y, \partial\Omega)$; by maximality there exists
some $i$ such that
$$
B_{\vare_i/3} (x_i)\quad\text{intersects $B_{\gamma/3} (y)$}.
$$
Then
$$
\vare_i \leq \text{dist}(y, x_i) \leq \frac13 (\gamma + \vare_i),
$$
so that
$$
2\vare_i \leq \gamma.
$$
But
$$
\align
2\gamma = \text{dist}(y,\partial\Omega)&\leq \text{dist}(y,x_i) +
\text{dist}(x_i,\partial\Omega)\\
      &\leq \frac13 (\gamma + \vare_i) + 2 \vare_i,
\endalign
$$
i.e., $5 \gamma \leq 7\vare_i \leq \dfrac72 \gamma$.  Impossible.
\bigskip
This proves the claim; we return to the proof of the theorem.  We have
$$
\int_A |f|\leq \sum_i \int_{B_{\vare_i}(x_i)} |f| = 3^n \sum_i
|B_{\vare_i/3} (x_i)
|\operatornamewithlimits{\Mint}_{B_{\vare_i}(x_i)} |f|.
\tag 3.6
$$
By (3.1) we may find $r_0 > 0$ such that, for every ball $B_r(a)$ with
$r = \dfrac12$ dist$(a,\partial\Omega) < r_0$,
$$
\operatornamewithlimits{\Mint}_{B_r(a)} |f| < \delta/6^n.
$$
We take $\vare_0 = \dfrac23 r_0$ and thus, for $\vare < \vare_0$, we
have
$$
\vare_i \leq \frac32 \vare \leq r_0\qquad\forall i
$$
and hence
$$
\operatornamewithlimits{\Mint}_{B_{\vare_i}(x_i)} |f| <
\delta/6^n\qquad\forall i.
$$
Consequently, by (3.6),
$$
\int_A |f|\leq \frac{\delta}{2^n} \sum_i |B_{\vare_i/3} (x_i)|.
$$
The balls $B_{\vare_i/3} (x_i)$ are disjoint and they are all
contained in $B_{2\vare}(x)$; it follows that
$$
\sum_i |B_{\vare_i/3} (x_i)\leq |B_{2\vare} (x)| = 2^n |B_\vare(x)|.
$$
We conclude that
$$
\frac{1}{|B_\vare(x)|} \int_A |f| < \delta.
$$
$\quad\hfill\qed$
\medskip


\remark{\bf Remark 7}  One may think that VMO$_0(\Omega)$ is a closed
subspace of VMO$(\Omega)$ but this is not true.  In fact, it is dense
in VMO$(\Omega)$; see Remark 3.
\endremark
\medskip
\remark{\bf Remark 8}  The space $W^{1,n}_0(\Omega)$ is contained in
VMO$_0(\Omega)$.  This is clear from the definition of VMO$_0$, for the
extension of $u\in W^{1,n}_0 (\Omega)$ as zero outside $\Omega$ is in
$W^{1,n}(B)\subset\text{VMO}(B)$---see Example 1.
\endremark
\bigskip
Next we are going to define a class VMO$_\varphi(\Omega)$ where
$\varphi$ is a given function in VMO$(\partial\Omega)$, assuming
$\partial\Omega$ is smooth.  VMO$_\varphi(\Omega)$ will consist of
functions having ``trace'' $\varphi$ on $\partial\Omega$.  First we
need
\medskip
\proclaim{Lemma 5}  Let $\Omega$ be a smooth bounded domain and let
$\varphi\in\text{VMO}(\partial\Omega)$.  There exists a function
$\widetilde\varphi$ defined on a neighbourhood $\widetilde\Omega$ of
$\overline\Omega$ such that
$\widetilde\varphi\in\text{VMO}(\widetilde\Omega)$, and for $x$ close to
$\partial\Omega$,
$$
\widetilde\varphi(x) = \varphi (P(x)) \tag 3.7
$$
where $P$ is the projection to the closest point in $\partial\Omega$.
\endproclaim

\demo{Proof}  We first define $\widetilde\varphi$ by (3.7) in a tubular
neighbourhood $U$ of $\partial\Omega$,
$$
U = \{x\in\Bbb R^n; \text{dist}(x,\partial\Omega) < \delta\}.
$$
\enddemo
\noindent
{\bf Claim:}  $\widetilde\varphi \in \text{VMO}(U)$.
\medskip
In view of Lemma A.10 in [BNI] it suffices to prove the claim when
the boundary is on $\{x_n = 0\}$, for $\widetilde U = \{x\in\Bbb R^n;
|x_n| < \delta\}$.  If $Q$ is a cube with edges parallel to the
axes, then it is clear that
$$
\Mint_Q |\widetilde\varphi - \Mint_Q \widetilde\varphi |\leq \|\varphi\|_{\text{BMO}(\partial\Omega)}.
$$
If $B$ is a ball in $\widetilde U$, then it lies in such a cube $Q$, with
side length $= \text{diam} B$, and then the inequality
$$
\Mint_B |\widetilde\varphi - \Mint_B \widetilde\varphi|\leq
C\|\varphi\|_{\text{BMO}(\partial\Omega)}
$$
follows with the aid of Lemma A.4 of [BNI].  We have proved that
$\widetilde\varphi\in \text{BMO}(U)$; that it is in VMO$(U)$ is proved
either by approximation or repeating the computation above, and
letting $|B|\rightarrow 0$.  The claim is proved.
\medskip

To complete the proof of the lemma we simply multiply $\widetilde\varphi$
by a smooth cutoff function; here we rely on Lemma B.8 of [BNI].

$\quad\hfill\qed$
\medskip
Now, the
\definition{Definition of VMO$_\varphi$}  Let $\Omega$ and $\varphi$
be as above, and let $f\in\text{VMO}(\Omega)$.  We say that $f$ has
trace $\varphi$ on $\partial\Omega$, i.e., $f\in\text{VMO}_\varphi$,
provided
$$
(f - \widetilde\varphi)\quad\text{is in VMO$_0(\Omega)$}.
$$
\enddefinition
\medskip
This definition also makes sense if $\Omega\subset X$, a Riemannian
manifold.
\medskip
\remark{\bf Remark 9}  Though $\widetilde\varphi$ is not quite unique---it
depends on the choice of cutoff---the notion of VMO$_\varphi$ is
independent of our choice.  This follows immediately with the aid of
Remark 6.  Furthermore, it is clear that
$f\in\text{VMO}_\varphi\Leftrightarrow$ the following function $\widetilde
f$ belongs to VMO$(\widetilde\Omega)$:
$$
\widetilde f = \cases  f&\quad\text{in $\Omega$}\\
        \widetilde\varphi&\quad\text{in $\widetilde\Omega\backslash\Omega$}.
\endcases
$$
\endremark
\medskip
\remark{\bf Remark 10}  It follows from Theorem 1 that for any fixed
$\varphi\in\text{VMO}(\partial\Omega)$, the space
VMO$_\varphi(\Omega)$ is dense in VMO$(\Omega)$ in the BMO topology.
\endremark
\medskip
The notion of VMO$_\varphi$ is invariant under diffeomorphisms.  In
particular, if $\Omega$ is a domain (with compact closure) in a smooth
manifold $X$, the notion of VMO$_\varphi$ is independent of the choice
of Riemannian metric on $X$.  We have namely
\medskip
\proclaim{Lemma 6}  Let $X_1,X_2$ be smooth Riemannian manifolds
without boundaries and let $\Omega_1,\Omega_2$ be subdomains,
respectively, with compact closures and smooth boundaries.  Let $H$ be
a $C^1$ diffeomorphism from $\overline\Omega_1$ onto
$\overline\Omega_2$; $H$ maps $\partial\Omega_1$ onto $\partial\Omega_2$
as a $C^1$ diffeomorphism.  Let
$\varphi\in\text{VMO}(\partial\Omega_2)$ and let
$f\in\text{VMO}_\varphi(\Omega_2)$.  Then
$$
f\circ H\quad\text{belongs to VMO$_{\varphi\circ H} (\Omega_1)$}.
$$
\endproclaim
\medskip
\demo{Proof}  For $i = 1,2,$ let $\widetilde\Omega_i$ be a neighbourhood of
$\overline\Omega_i$ so that for every
$x\in\widetilde\Omega_1\backslash \Omega_1$ there is a 
unique closest point $P(x)$ on $\partial\Omega_1$.  We define an
extension $\widetilde H$ of $H$ to $\widetilde\Omega_1$ as follows:  For
$x\in\widetilde\Omega_1\backslash\overline\Omega_1$, we set
$$
\widetilde H(x) = y\in\widetilde\Omega_2\backslash\Omega_2
$$
where $y$ is the unique point there with $P(y) = H(P(x))$, and
dist$(y,H(P(x)) = \text{dist}(x,\partial\Omega_1)$.  To define $y$ we
may have to shrink $\widetilde\Omega_1$.  Clearly $\widetilde H$ is a
bi-Lipschitz map of $\widetilde\Omega$ onto a neighbourhood
$\widetilde\Omega_2$ of $\overline\Omega_2$.
\enddemo

Turning to the function $f$, set, as in Remark 9,
$$
\widetilde f = \cases  f&\quad\text{in $\Omega_2$}\\
        \widetilde \varphi&\quad\text{in $\widetilde\Omega_2\backslash\Omega_2$},
\endcases
$$
so that $\widetilde f\in\text{VMO}(\widetilde\Omega_2)$.  Consider now
$\widetilde f\circ \widetilde H$; it is defined on $\widetilde\Omega_1$.
\medskip

\noindent
{\bf Claim:}  $\widetilde f\circ \widetilde H \in\text{VMO}(\omega)$ where
$\omega$ is any open set with compact closure in $\widetilde\Omega_1$.
\medskip

Once the claim is proved, we are through, for if
$x\in\widetilde\Omega_2\backslash\overline\Omega_2$, then $\widetilde
f\circ\widetilde H(x) =$\newline $(\varphi\circ H) (P(x))$.
\medskip
\noindent
{\it Proof of Claim:}  Let $B_\vare(x)$ be a ball in $\omega$ with
$\vare\leq \dfrac12$ dist$(x,\partial\omega)$.  Consider
$$
\align
I &= \Mint_{B_\vare(x)}\;\Mint_{B_\vare(x)} |\widetilde f\circ \widetilde H(y)
- \widetilde f\circ \widetilde H (z)|\\
   &\leq \frac{C}{|B_\vare(x)|^2} \int_{\widetilde H(B_\vare(x))}
\int_{\widetilde H(B_\vare(x))} |\widetilde f (\eta) - \widetilde f(\zeta)|
\endalign
$$
since $(\widetilde H)^{-1}$ is Lipschitz.  Hence
$$
I \leq \frac{C}{|B_\vare(x)|^2} \int_{B_{\vare K}(\widetilde H(x))}
\int_{B_{\vare K}(\widetilde H(x))} |\widetilde f(\eta) - \widetilde f(\zeta)|
$$
since $\widetilde H$ is Lipschitz with Lipschitz constant $K$.  We also
require that
$$
\vare < \frac{1}{2K}\text{dist}(\widetilde H(\omega),
\partial\widetilde\Omega_2) =: r_0.
$$
Clearly $I\leq C \|\widetilde f\|_{\text{BMO}}$.  By Remark 1 we see that 
$$
\|\widetilde f\circ\widetilde H\|_{\text{BMO}(\omega)} \leq C \|\widetilde
f\|_{\text{BMO}(\widetilde\Omega_2)}.
$$
By density we conclude that $\widetilde f\circ \widetilde H$ is in
VMO$(\omega)$.

$\quad\hfill\qed$
\bigskip

Next, we present some examples of functions in VMO$_\varphi$.


\medskip
\noindent
{\bf Example 1.}  If $f\in C(\overline\Omega)$, and $\varphi =
f_{|\partial\Omega}$, then $f\in\text{VMO}_\varphi(\Omega)$.
\bigskip
\noindent
\medskip
\noindent
{\bf Example 2.}  If $f\in W^{1,n}(\Omega)$ and $\varphi =
f_{|\partial\Omega}$ then $f\in\text{VMO}_\varphi(\Omega)$.  Recall
that $f\in\text{VMO}(\Omega)$ and $\varphi = f_{|\partial\Omega} \in
W^{1-\frac1n,n}(\partial\Omega)$ also lies in VMO$(\partial\Omega)$,
by Example 2 in \S I.1 in [BNI].
\medskip
\demo{Proof} Since both conditions $f\in W^{1,n}$ and
$f\in\text{VMO}_\varphi$ are invariant under diffeomorphisms, we may
locally flatten the boundary $\partial\Omega$.  In addition we may
suppose that the metric is locally Euclidean near the flat portion of
boundary.  Near the origin in the flat boundary, we may use
coordinates $(x',x_n)$, $x'\in\Bbb R^{n-1}$, with $x_n > 0$ in
$\Omega, x_n = 0$ on $\partial\Omega$.  In view of Theorem 2 it
suffices to show that
$$
\lim\Sb |B|\rightarrow 0\\ B\in\Cal D\endSb\; \Mint_B |f(x',x_n) -
f(x',0)| dx = 0.
$$
For $B\in\Cal D$, let $Q = Q' \times (\vare,3\vare)$ be the smallest
cube with edges parallel to the axes containing $B$.  Then
$$
\align
\Mint_Q |f(x',x_n) &- f(x',0)|\leq \frac{2\vare}{(2\vare)^n}
\int_{Q'\times(0,3\vare)} |f_{x_n}|\\
     &\leq C(\int_{Q'\times (0,3\vare)} |f_{x_n}|^n)^{1/n}\rightarrow 0
\endalign
$$
as $\vare\rightarrow 0$.
\enddemo
\medskip
\noindent
{\bf Example 3.}  Consider, as usual, a domain $\Omega$ having compact
closure in a smooth Riemannian manifold $X$ without boundary; $\partial\Omega$ is smooth.  Let $\varphi$
belong to VMO$(\partial\Omega)$.  The following particular extension
$f$ of $\varphi$ inside $\Omega$ belongs to VMO$_\varphi(\Omega)$.
Let $U = \{x\in\Omega; \text{dist}(x,\partial\Omega) < \delta\}$ with
$\delta$ so small that any point $x$ in $U$ has a unique closest point
$P(x)$ on $\partial\Omega$.  The geodesics starting on
$\partial\Omega$ and orthogonal to $\partial\Omega$ cover $U$ simply.
Denote dist$(x,\partial\Omega)$ by $d(x)$.  For $x$ in $U$, define
$$
f(x) = \overline\varphi_{d(x)} (P(x))
$$
i.e., $f(x)$ is the average of $\varphi$ on a ball on $\partial\Omega$
centred at $P(x)$, having radius $d(x)$.  We extend $f$ to all of
$\Omega$ by multiplying it by a smooth cutoff function with support in
$U$ and which is identically one near $\partial\Omega$, and we
continue to denote by $f$ the extension to all of $\Omega$.
\medskip
\proclaim{Lemma 7}  $f$ belongs to VMO$_\varphi(\Omega)$.
\endproclaim
\medskip
\demo{Proof}  By Lemma 6, the property of belonging to VMO$_\varphi$
is independent of the particular metric on $X$.  It is convenient to
replace the given Riemannian metric on $\overline\Omega$ by a
different one.  We describe the new metric just in $U$; it is easily
extended to $\Omega$.  The new metric preserves all geodesics starting
on $\partial\Omega$ and orthogonal to $\partial\Omega$, and preserves
arc length on them.  But it is a product metric.  Namely, if $x' = (x_1,\ldots,x_{n-1})$ are local coordinates near a point $\bar y$ on
$\partial\Omega$, with $x' = 0$, $t = 0$ at $\bar y$, and $t > 0$ in
$U$, the lines $x' = \text{constant}$,\newline $0 < t < \delta$,
correspond to our special geodesics orthogonal to $\partial\Omega$.
The new metric has the form
$$
\widetilde{ds}^2 = dt^2 + ds^{'2} \tag 3.8
$$
where $ds^{'2} = ds^2_{\big\vert\partial\Omega}$.
\enddemo
\medskip

The function $f$ is continuous in $\Omega$. Therefore, to prove the
lemma we need only consider  balls $B_\vare(x)$ in $U$ belonging to
our family $\Cal C$.  We have to show that
$$
\Mint_{B_\vare(x)}\;\Mint_{B_\vare(x)} |f(y) - f(z)|\leq C
\|\varphi\|_{\text{BMO}}, \tag 3.9
$$
with $C$ a fixed constant independent of the ball; by density this
proves that $f$ is in VMO$(\Omega)$.  To verify that $f$ is in
VMO$_\varphi$ we have to show that
$$
\Mint_{B_\vare(x)} |f(y) - \varphi(P(y))|\quad\text{is small for $\vare
= \frac12 d(x)$ small}. \tag 3.10
$$
\medskip
We may use the local coordinates $(x',t)$ described above, and
suppose that $B_\vare(x)$ is the ball
$$
B_\vare(x) = B_\vare(0,\tau)\quad\text{with $2\vare\leq \tau\leq
\delta$}.
$$
Denote the ball in $\partial\Omega$, i.e., on $t = 0$, centred at
$P(x)$, which in our local coordinates is the origin, and having radius
$\vare$ by $B' = B'_\vare(0)$.  Now $B_\vare(0,\tau)$ lies in the
cylinder
$$
D = B'_\vare \times (\tau - \vare, \tau + \vare),
$$
and since $|D|\leq C|B_\vare(x)|$, to prove (3.9) it suffices to prove
that
$$
I = \Mint_D\;\Mint_D |f(y) - f(z)|\leq C\|\varphi\|_{\text{BMO}}.
$$
Now if $B'$ is the ball in $\Bbb R^{n-1}$ with centre $0$ and radius
$\vare$ (measured in our metric $ds'$), we have
$$
I = \operatornamewithlimits{\Mint}\Sb y'\in B'\\\tau - \vare < s< \tau + \vare\endSb\;
\operatornamewithlimits{\Mint}\Sb z'\in B'\\\tau - \vare < t < \tau + \vare\endSb
|\overline\varphi_s(y') - \overline\varphi_t(z')|.
$$
If $B'_s(y')$ is the ball (in our metric $ds'$) about $y'$ of
radius $s$ then
$$
B'_t (z'), B'_s(y')\subset B'_{\tau+2\vare}(0),\qquad\text{if
$\tau - \vare < s, t < \tau+\vare$}.
$$
Since
$$
\frac{\tau + 2\vare}{t}, \frac{\tau + 2\vare}{s} \leq \frac{\tau +
2\vare}{\tau/2}\leq C\quad\text{independent of $\tau$ and $\vare\leq
\frac12 \tau$},
$$
we see with the aid of Lemma A.4 in [BNI] that
$$
|\overline\varphi_s(y') - \overline\varphi_{\tau + 2\vare}(0)|,\;
|\overline\varphi_t(z') - \overline\varphi_{\tau + 2\vare} (0) |\leq
C\|\varphi\|_{\text{BMO}}
$$
if $\tau$ is small; thus
$$
|\overline\varphi_s(y') - \overline\varphi_t(z')|\leq
C\|\varphi\|_{\text{BMO}}.
$$
Inserting this in $I$ above we obtain (3.9).
\medskip

Turning to the proof of (3.10), we consider again the cylinder $D$,
with now, $\tau = 2\vare$.  It suffices to prove that
$$
\Mint_D |f(y) - \varphi(P(y))|\quad\text{is small,}
$$
i.e., that for $\vare$ small,
$$
J: = \operatornamewithlimits{\Mint}\Sb x'\in B'\\\vare < t < 3\vare\endSb
|\overline\varphi_t(x') - \varphi(x')|\quad\text{is small}.
$$
\noindent
Since $\varphi$ is in VMO,
$$
\Mint_{B'} |\varphi - \overline\varphi_\vare (0)|\quad\text{is small
for $\vare$ small}. \tag 3.11
$$
With the aid of Lemma A.4 in [BNI], we see, as above, that for $\vare$
small,
$$
|\overline\varphi_t (x') - \overline\varphi_\vare(0)|\quad\text{is
small if $x'\in B'$ and $\vare\leq t\leq 3\vare$}. \tag 3.12
$$
Thus
$$
J \leq \operatornamewithlimits{\Mint}\Sb x\in B'\\\vare < t < 3\vare\endSb
|\overline\varphi_t(x') - \overline\varphi_\vare(0)| + \Mint_{B'}
|\overline\varphi_\vare (0) - \varphi(x')|.
$$
The first term on the right is small by (3.12), and the second, by
(3.11).  

 $\quad\hfill\qed$
\bigskip
\noindent
{\bf Example 4.}  Consider $\Omega,X$ and $\varphi$ as in Example 3,
$\varphi\in \text{VMO}(\partial\Omega)$.
\medskip
\proclaim{Lemma 8}  The harmonic function in $\Omega$, which equals
$\varphi$ on $\partial\Omega$, belongs to VMO$_\varphi (\Omega)$.
\endproclaim

The proof is given in Appendix 3, see Theorem A3.1.
\bigskip
\noindent
{\bf Example 5.}  Consider $u = (\log |x|) \ast f$ in $\Bbb R^n$,
$n\geq 2$, where $f\in L^1(\Bbb R^n)$ with compact support (for
simplicity).  Let $\Omega\subset\Bbb R^n$ be a smooth bounded domain.
Clearly, $u\in W^{1,p}(\Omega)\quad\forall p < n$, but it need not
belong to $W^{1,n}(\Omega)$.  Hence $u$ has a trace on
$\partial\Omega$, say $\varphi$.
\medskip
\proclaim{Lemma 9}  $\varphi$ belongs to VMO$(\partial\Omega)$ and $u$
belongs to VMO$_\varphi(\Omega)$.
\endproclaim
\demo{Proof}  First, note that $u\in\text{VMO}(\Omega)$.  Indeed, by
density, this follows from the fact that
$$
\|u\|_{\text{BMO}(\Omega)} \leq C \|f\|_{L^1}.
$$
Next, that $\varphi$ belongs to VMO$(\partial\Omega)$ follows from the
estimate
$$
\|\varphi\|_{\text{BMO}(\partial\Omega)} \leq C \|f\|_{L^1}.
$$
This is derived in turn from the inequality
$$
\|\log |x - a|\|_{\text{BMO}(\partial\Omega)} \leq C\quad\forall
a\in\Bbb R^n
$$
where $C$ depends only on $\Omega$.  To prove the last inequality we
need only establish for $\varepsilon$ small,
$$
J:=\Mint_{B'_\varepsilon(x)} \;\Mint_{B'_\varepsilon(x)} \big\vert\log|y - a|
- \log |z - a|\big\vert d\sigma(y)d\sigma(z)\leq C\quad\forall
a\in\Bbb R^n,
\tag 3.13
$$
where $C$ depends only on $\Omega$.  Here $x\in\partial\Omega$ and
$B'_\varepsilon(x)$ is the geodesic ball on $\partial\Omega$ centred
at $x$.  We consider two cases:
\medskip
(i)\quad $\,|x - a|\geq 6\varepsilon$,
\smallskip
(ii)\quad$|x - a| < 6\varepsilon$.
\medskip

Case (i) is obvious, since for $\varepsilon$ small, if $y, z\in
B'_\varepsilon(x)$,
$$
|x - y| < \varepsilon,\quad |x - z| < \varepsilon
$$
and thus
$$
\frac12 \leq \frac{|y - a|}{|z - a|} \leq 2.
$$
\medskip

In Case (ii) we have
$$
J\leq 2\Mint_{B'_\varepsilon(x)} |\log \frac{|y - a|}{\varepsilon}|
d\sigma(y) \leq C(\Omega). \tag 3.14
$$
\medskip
Finally, we prove that $u\in\text{VMO}_\varphi(\Omega)$.  By Theorem 2
it suffices to show that
$$
\operatornamewithlimits{\lim}_{\varepsilon\rightarrow 0}
\Mint_{B_\varepsilon(a)} | u - \widetilde\varphi| = 0
$$
where $\varepsilon=\dfrac12$dist$(a,\partial\Omega)$ and
$\widetilde\varphi$ is as in (3.7).  By density (as in the proofs of
Theorem A3.1 and A3.2) it suffices to establish that
$$
\Mint_{B_\varepsilon(a)} |u - \widetilde\varphi|\leq C \|f\|_{L^1} \tag 3.15
$$
for $\varepsilon$ small, where $C$ depends only on $\Omega$.
\medskip

Inequality (3.15) follows from
$$
\Mint_{B_\varepsilon(a)} \big\vert\log|x - y| - \log |P(x) - y|\big\vert dx\leq
C(\Omega) \tag 3.16
$$
for every $y\in\Bbb R^n$ and for every $\varepsilon <$ some
$\varepsilon_0$.  To prove (3.16) we consider, as before, two cases:
\medskip
(i)\quad $|y - a|\geq 6\varepsilon$,
\smallskip
(ii)\quad$|y - a| < 6\varepsilon$.
\medskip
Case (i) is obvious since, for $x\in B_\varepsilon(a)$,
$$
\frac13 \leq \frac{|x - y|}{|P(x) - y|} \leq 3.
$$
In Case (ii) one shows, in fact, that
$$
\Mint_{B_\varepsilon(a)} \big\vert\log \frac{|x -
y|}{\varepsilon}\big\vert dx \leq C_n \tag 3.17
$$
and
$$
J :=\Mint_{B_\varepsilon(a)} \big\vert\log\frac{|P(x) -
y|}{\varepsilon}\big\vert dx \leq C(\Omega). \tag 3.18
$$
Inequality (3.17) is clear.  To verify (3.18) one has, first, as in
(3.14), that for $\varepsilon$ small,
$$
J\leq C \Mint_{B'_{2\varepsilon} (P(a))} \;|\log
\frac{|\xi-y|}{\varepsilon}| d\sigma(\xi)
$$
where $B'_{2\varepsilon} (P(a))$ is the geodesic ball on
$\partial\Omega$ centred at $P(a)$.  Now, for $\xi\in
B'_{2\varepsilon} (P(a))$,
$$
|\xi - y|\leq 1 0 \varepsilon.
$$
Furthermore, for $\varepsilon$ small, one sees that for such $\xi$,
$$
|\xi - y|\geq \frac12 |\xi - P(y)|.
$$
Hence
$$
\align
J&\leq C + C \Mint_{B'_{2\varepsilon} (P(a))} \big\vert\log\frac{|\xi -
P(y)|}{20\varepsilon} \big\vert d\sigma(\xi)\\
  &\leq C + C \Mint_{B'_{2\varepsilon}(P(a))} \big\vert\log
\frac{|\xi - P(y)|}{20 \varepsilon}\big\vert d\sigma (\xi) \leq
C(\Omega)
\endalign
$$
since the last integral is bounded by a constant depending only on
$\Omega$. $\quad\hfill\qed$
\enddemo
\medskip

We conclude this section with an answer to a question raised by
H.~Amann.  Let $X,Y$ be smooth $n$-dimensional compact oriented
manifolds without boundaries; $Y$ is smoothly embedded in some $\Bbb
R^N$.  Consider two maps $\varphi,\psi\in\text{VMO}(X,Y)$; by [BNI] the
degrees are well defined.  Suppose $\varphi$ and $\psi$ are connected
by some homotopy $H(x,t), 0\leq t\leq 1$.  In Corollary 6 of [BNI] it
was shown that if $H$ is continuous in $[0,1]$ with values in
VMO$(X,Y)$ then $\deg \varphi = \deg\psi$.  Amann's question was whether
the same conclusion holds in case
$$
H\in \text{VMO}(X\times (0,1),Y). \tag 3.19
$$
The answer is yes, provided one makes a slightly stronger assumption
on $H$ for $t$ near $0$ and $1$.  In fact, under condition (3.19) it is
not clear what is meant by saying that $H(\cdot\,,0) = \varphi,
H(\cdot\,,1) = \psi$.
\medskip
\proclaim{Corollary 3}  Assume in addition to (3.19) that
$$
\aligned
&\operatornamewithlimits{\Mint}^h_0 \; \operatornamewithlimits{\Mint}_{B_h(x)} |H(y,t) - \varphi(y)|
d\sigma(y) dt\rightarrow 0\quad\text{as $h\rightarrow 0$, uniformly in
$x\in X$}\\
&\operatornamewithlimits{\Mint}^1_{1-h}\;\operatornamewithlimits{\Mint}_{B_h(x)}|H(y,t)
- \psi(y)| d\sigma(y) dt\rightarrow 0\quad\text{as $h\rightarrow 0$,
uniformly in $x\in X$}.
\endaligned
\tag 3.20
$$
Then
$$
\deg(\varphi,X,Y) = \deg (\psi,X,Y).
$$
\endproclaim

\demo{Proof}  Consider the manifold $\widetilde X = X\times \Bbb R$ with
the product metric, and set $\Omega = X\times (-1,2)$ in $\widetilde
X$,
$$
\widetilde H(x,t) = \cases  \varphi(x)&\quad\text{for $t\leq 0$}\\
                   H(x,t)&\quad\text{for $0 < t < 1$}\\
                   \psi(x)&\quad\text{for $t \geq 1$}.
\endcases
\tag 3.21
$$
\medskip
By Theorem 2, conditions (3.20) imply that $\widetilde
H\in\text{VMO}(\Omega,Y)$. (It is easy to see that (3.20) is, in fact,
equivalent to the property that $\widetilde H\in\text{VMO}(\Omega)$.) As in [BNI] we now define
$$
\widetilde H_\vare (x,t) = P
\operatornamewithlimits{\Mint}_{B_\vare(x,t)} \widetilde H
$$
where $P$ is the projection to the closest point in $Y$.  In view of Lemma
A.4 of [BNI] we may also work with
$$
G_\vare(x,t) = P\;\operatornamewithlimits{\Mint}_{Q_\vare(x,t)} \widetilde H
$$
where $Q_\vare(x,t)$ is the cylinder $B_\vare(x)\times
(t-\vare,t +\vare)$, for by Lemma A.4 of [BNI],
$$
\operatornamewithlimits{\sup}\Sb x\in X\\ t\in\Bbb R\endSb |\widetilde
H_\vare (x,t) - G_\vare(x,t)|\rightarrow 0\quad\text{as
$\vare\rightarrow 0$}.
$$
Clearly for $t < -\vare$, $G_\vare (x,t) = \varphi_\vare (x) =
P\overline\varphi_\vare(x)$, and for $t > 1 + \vare$, $G_\vare(x,t) =
\psi_\vare (x) = P\overline\psi_\vare(x)$.  By standard homotopy
$$
\deg (G_\vare (\cdot\,,t), X,Y) \qquad\text{is independent of $t$}.
$$
Thus, for $\vare$ small, $\deg(\varphi,X,Y) = \deg(\varphi_\vare,X,Y)
= \deg(\psi_\vare,X,Y) = \deg(\psi,X,Y)$.
\enddemo
$\quad\hfill\qed$
\bigskip
\remark{\bf Remark 11}  In connection with (3.19), a word of warning:  If
$f\in C([-1,1]$, VMO$(X)\cap L^1(X))$ one might think that 
$f$ is in VMO$(X\times [- \dfrac12,\dfrac12])$.  This need not be the case;
here is an example.  Take $X = [-1,1]$ in $\Bbb R$.  For $t > 0$
consider
$$
f(x,t) = \cases 1&\qquad\text{if $|x|\leq t$}\\
    -1+2\dfrac{\log|x|}{\log t}&\qquad\text{if $t < |x| < \sqrt
t$}\\
               0&\qquad\text{if $|x| \geq \sqrt t$}
\endcases
$$
and for $t < 0$, $f(x,t)\equiv 0$.  By Example 6 in \S I.2 of
[BNI], $f\in C([-1,1], \text{ VMO}(X))$.  Continuity with values in
$L^1$ is clear.  But $f$ does not belong to VMO$(X\times
[-\dfrac12,\dfrac12])$, for
$$
\operatornamewithlimits{\Mint}_{Q_h}\;
\operatornamewithlimits{\Mint}_{Q_h} |f(x,t) - f(\xi,\tau)| dxdt
d\xi d\tau \geq \frac14,
$$
where $Q_h = [-h,+h]\times[-h,+h]$.
\endremark
%Section II.4-9/26/95
\bigskip
\noindent
{\bf II.4.  For $u\in\text{VMO}_\varphi, \deg(u,\Omega,p) = $ a
boundary degree}
\medskip
Recall the standard result that for a {\it continuous} map $u :
\overline\Omega\rightarrow \Bbb R^n$, with $u_{\vert\partial\Omega} =
\varphi$, and with $\varphi\not=p$ everywhere on $\partial\Omega$ for some
point $p\in\Bbb R^n$,
$$
\deg(u,\Omega,p) = \deg\left(\frac{\varphi - p}{|\varphi - p|},
\partial\Omega, S^{n-1}\right). \tag 4.1
$$
\bigskip
Here we extend this result to maps $u\in\text{VMO}_\varphi$ provided
$|\varphi - p|\geq d_0 > 0$ a.e. on $\partial\Omega$.
\medskip
\proclaim{Theorem 3}  Assume the above, with $\varphi\in\text{VMO}(\partial\Omega)$.
Then there is a neighbourhood $U$ of $\partial\Omega$ in $\Omega$ such
that
$$
\Mint_B |u - p|\geq \frac{d_0}{2}\quad\forall B\subset U, B\in\Cal D
$$
---so that $\deg(u,\Omega,p)$ is defined.  Furthermore, (4.1) holds.
\endproclaim

\demo{Proof}  We may take $p = 0$.  Set $\widetilde\varphi(x)=
\varphi(Px)$ where $P$ is the nearest point projection on
$\partial\Omega, \widetilde\varphi$ is defined in a neighbourhood $U$
of $\partial\Omega$.  Let $\zeta$ be a cutoff function with support
in $U$, and $\zeta \equiv 1$ near $\partial\Omega$.  Set
$\overline\varphi(x) = \zeta(x)\widetilde\varphi(x)$, so that
$\overline\varphi\in\text{VMO}_\varphi(\Omega)$; since $u\in
\text{VMO}_\varphi$,
$$
\operatornamewithlimits{\lim}\Sb |B|\rightarrow 0\\B\in\Cal D\endSb
\operatornamewithlimits{\Mint}_B |u - \overline\varphi| = 0.
$$
But
$$
\align
\operatornamewithlimits{\Mint}_B |u - \overline\varphi|&\geq
\operatornamewithlimits{\Mint}_B |\overline\varphi| -
\operatornamewithlimits{\Mint}_B |u|\\
   &\geq d_0 - \operatornamewithlimits{\Mint}_B |u|
\endalign
$$
for $|B|$ small, since $|\overline\varphi|\geq d_0$ near
$\partial\Omega$.  Hence there exists a neighbourhood $U$ of $\partial\Omega$ such
that
$$
\operatornamewithlimits{\Mint}_B |u|\geq \frac{d_0}{2}\quad \forall
B\subset U, B\in\Cal D.
$$
\enddemo

We have proved the first assertion of the theorem.  To verify (4.1)
we make use of
\medskip
\proclaim{Lemma 10}  Assume $\psi\in\text{VMO}(\partial\Omega,\Bbb
R^n)$ and $|\psi| = 1$ a.e. on $\partial\Omega$.  For $x\in\Omega$, let
$\overline{\psi}(x) = \zeta(x)\psi(Px)$ as above.  Then
$$
\deg(\overline{\psi},\Omega,0) = \deg(\psi,\partial\Omega,S^{n-1})
$$
\endproclaim

\demo{Proof}  We know (see Corollary 5 in [BNI]) that there exists a
sequence $\psi_j\in C^\infty (\partial\Omega, S^{n-1})$ such that
$\psi_j\rightarrow \psi$ in BMO and a.e.  By (4.1) for continuous maps,
$$
\deg(\psi_j,\partial\Omega,S^{n-1}) = \deg(\overline\psi_j,\Omega,0)
$$
where 
$$
\overline\psi_j(x) = \zeta(x) \psi_j(Px).
$$
As $j\rightarrow\infty$,
$\deg(\psi_j,\partial\Omega,S^{n-1})\rightarrow
\deg(\psi,\partial\Omega,S^{n-1})$ (by Theorem 1 in [BNI]).
\medskip

On the other hand we claim that
$\overline\psi_j\rightarrow\overline\psi$ in both $L^1(\Omega)$ and
BMO$(\Omega)$.  Indeed, convergence in $L^1$ follows from dominated
convergence.  Convergence in BMO uses the easily verified fact that
$\psi_j(Px)\rightarrow \psi(Px)$ in BMO$(U)$, and the estimate for
products, namely Lemma B.8 in [BNI].  Moreover
$$
|\overline\psi_j(x)|\equiv 1\quad\text{in some {\it fixed} (uniform)
neighbourhood of $\partial\Omega$}.
$$
Hence, by the stability of degree in the BMO topology (Property 2 in
II.2), we see that
$$
\deg(\overline\psi_j,\Omega,0)\rightarrow\deg(\overline\psi,\Omega,0).
$$
$\qquad\hfill\qed$
\enddemo

\noindent
{\it Proof of Theorem 3.}  Set $\psi(x) = \dfrac{\varphi(x)}{|\varphi(x)|}, x\in \partial\Omega$,
so that $\psi\in \text{VMO}$ (by Lemma A.7 in [BNI], on compositions
of VMO maps with Lipschitz maps).  Thus, by the previous lemma, with $\overline\psi$ as defined there,
$$
\deg(\overline\psi,\Omega,0) = \deg(\frac{\varphi}{|\varphi|},
\partial\Omega,S^{n-1}).
$$
\medskip
Next we have
$$
\deg(\overline\psi,\Omega,0) = \deg(\overline\varphi,\Omega,0) \tag
4.2
$$
where $\overline\varphi(x) = \zeta(x)\varphi(Px)$.  Indeed we may
consider the homotopy
$$
\align
H(x,t) &= t\overline\psi(x) + (1 - t) \overline\varphi(x) = \zeta(x)\varphi(Px) \left[\frac{t}{|\varphi(Px)|}+(1-t)\right]
\endalign
$$
and note that for every $x$ in some fixed neighbourhood of
$\partial\Omega$,
$$
|H(x,t)|\geq \big[ t + (1-t) d_0\big] \geq
\min(d_0,1)\qquad\forall t\in [0,1].
$$
Applying Property 2 once more, we obtain (4.2).
\medskip
Finally, it remains to prove that
$$
\deg(\overline\varphi,\Omega,0) = \deg(u,\Omega,0). \tag 4.3
$$
Recall that since $u\in\text{VMO}_\varphi$ we have
$$
\operatornamewithlimits{\lim}\Sb |B|\rightarrow 0\\B\in\Cal D\endSb
\operatornamewithlimits{\Mint}_B |u - \overline\varphi| = 0.
$$
\medskip
Assertion (4.3) then follows from
\medskip
\proclaim{Lemma 11}  Assume $u,v\in\text{VMO}(\Omega,\Bbb R^n)$ and
$$
\operatornamewithlimits{\lim}\Sb |B|\rightarrow 0\\B\in\Cal D\endSb
\operatornamewithlimits{\Mint}_B |u- v| = 0.
$$
Assume that, for some neighbourhood $U$ of $\partial\Omega$ in
$\Omega$,
$$
\operatornamewithlimits{\Mint}_B |u| \geq d_0 > 0\qquad \forall
B\subset U, B \in\Cal D
$$
so that $\deg(u,\Omega,0)$ is defined.  Then there is a neighbourhood
$U'$ of $\partial\Omega$ in $\Omega$ such that
$$
\operatornamewithlimits{\Mint}_B |v|\geq \frac{d_0}{2}\qquad\forall
B\subset U', B\in \Cal D.
$$
Moreover
$$
\deg(v,\Omega,0) = \deg(u,\Omega,0).
$$
\endproclaim

\demo{Proof}  The existence of $U'$ is clear.  Recall that, by
definition (see \S II.2),
$$
\deg(u,\Omega,0) = \deg(\overline u_\vare,\Omega_{2\vare},0)
$$
and
$$
\deg(v,\Omega,0) = \deg(\overline v_\vare, \Omega_{2\vare},0)
$$
for $\vare < \vare_0$.
\medskip
But we may fix $\vare$ so small that (see (2.4))
$$
|\overline u_\vare(x)|\geq \frac{d_0}{2},\quad|\overline v_\vare
(x)|\geq \frac{d_0}{2}\quad\forall x\in\partial\Omega_{2\vare}
$$
and, similarly,
$$
|\overline u_\vare(x) - \overline v_\vare (x)|\leq
\frac{d_0}{4}\qquad\forall x\in\partial\Omega_{2\vare}
$$
(since $\dsize\operatornamewithlimits{\lim}\Sb |B|\rightarrow 0\\B\in\Cal
D\endSb \;\dsize\operatornamewithlimits{\Mint}_B |u - v| = 0$).  Hence, by linear homotopy for the continuous maps,
$$
\deg(\overline u_\vare, \Omega_{2\vare},0) = \deg(\overline v_\vare,
\Omega_{2\vare}, 0).
$$
\medskip
This proves Lemma 11 and completes the proof of Theorem 3.
\enddemo
\medskip
\noindent
{\bf An application.}  Consider the equation
$$
\alignat2
\Delta u &= f     &&\quad\text{in $\Omega$},\\
       u &=\varphi&&\quad\text{on $\partial\Omega$},
\endalignat
$$
where $\Omega\subset\Bbb R^n$ is a smooth bounded domain with $n\geq
2$.  Assume
$$
\align
&f\in L^{n/2}(\Omega,\Bbb R^n),\tag 4.4\\
&\varphi\in\text{VMO}(\partial\Omega,S^{n-1}), \tag 4.5\\
\intertext{with}
&\deg(\varphi,\partial\Omega, S^{n-1})\not=0.\tag 4.6
\endalign
$$
\medskip
\proclaim{Corollary 4}  Under the conditions above
$$
\text{\rm ess}R(u)\supset B_1(0).
$$
\endproclaim

For the definition of ess$R(u)$, see \S I.4 in [BNI].
\medskip

\demo{Proof}  We claim that
$$
u\in\text{VMO}_\varphi(\Omega). \tag 4.7
$$
The assertion in the corollary then follows from Theorem 3 and
Property 1 in II.2.  To prove (4.7) we distinguish two cases:
\medskip
\noindent
{\bf Case (i)}:\quad $n\geq 3$,
\medskip
\noindent
{\bf Case (ii)}:\quad$n = 2$.
\medskip
In Case (i) we write $u = v + w$ where $v$ is the solution of
$$
\align
\Delta v &=f\qquad\text{in $\Omega$}\\
       v &=0\qquad\text{on $\partial\Omega$}
\endalign
$$
and $w$ the harmonic extension of $\varphi$ in $\Omega$.  Since $v\in
W^{2,n/2} (\Omega)$, then $v\in W^{1,n}(\Omega)$, and in fact in
$W_0^{1,n}(\Omega)$.  Thus $v\in\text{VMO}_0(\Omega)$ by Example 2 in
\S II.3.  On the other hand $w\in\text{VMO}_\varphi(\Omega)$ by
Theorem A3.1 in Appendix 3.  Thus $u = v + w \in\text{VMO}_\varphi(\Omega)$.
\medskip

In Case (ii), we use the same decomposition $u = v + w$.  But here we
cannot assert that $v\in W^{1,2}$.  Set
$$
\widetilde v = c(\log |x|) \ast f
$$
(here $f$ is extended as $0$ outside $\Omega$) so that $\Delta
\widetilde v = f$.
\medskip
By Lemma 9, $\tilde v\in\text{VMO}_\psi(\Omega)$ where $\psi = \tilde
v_{\vert\partial\Omega} \in\text{VMO}(\partial\Omega)$.  We have
$$
\alignat2
\Delta (\tilde v - v)&= 0&&\quad\text{in $\Omega$}\\
          \tilde v - v&=\psi&&\quad\text{on $\partial\Omega$}.
\endalignat
$$
Hence $\tilde v - v\in\text{VMO}_\psi(\Omega)$ by Theorem A3.1.  Thus
$u = v + w = (v - \tilde v) + \tilde v + w
\in\text{VMO}_\varphi(\Omega)$.
\medskip
$\quad\hfill\qed$
\enddemo
\medskip
\remark{\bf Remark 12}  If $n\geq 3$, condition (4.4) is sharp in the
sense that if $f\in L^{(n/2) - \varepsilon}$ (any $\varepsilon > 0$),
the conclusion of Corollary 4 need not hold.  This may be easily seen
on $\Omega = B_1(0)$; the function $u(x) = x/|x|$ satisfies (4.4) with
$f\in L^p(\Omega)$, for all $p < n/2$, but not with $p = n/2$.
\endremark

\bigskip
%Appendix 1 for 2nd Selecta paper, 10/4/95.
%\pageno=23
\def\Mint{\diagup\hskip-.50truecm\int}
\noindent
{\bf Appendix 1.  Some properties of BMO and VMO in domains}
\bigskip
We present the proofs of a number of results in \S II.1.  In
particular, the equivalence of various notions of BMO is
established---for general bounded open sets $\Omega$.  In addition we
show that $C_0^\infty(\Omega)$ is dense in BMO$(\Omega)$.  These
results are due to Peter~Jones and some are implicit in P.~Jones [1].
\medskip
We start with an easy result; we use the definitions of \S II.1 and do
not repeat them here.
\medskip
\proclaim{Lemma A1.1} Consider $0 < k_1 < k_2 < 1$.  Then
$$
\|\;\;\|_{\text{BMO},k_1}\leq \|\;\;\|_{\text{BMO},k_2}\leq C
\|\;\;\|_{\text{BMO},k_1} \tag A1.1
$$
where $C$ may depend on $n,k_1$, and $k_2$.
\endproclaim
\medskip
A deeper result, Theorem A1.1, implies that the constant $C$ depends
only on $k_1$.
\medskip
\noindent
{\it Proof of Lemma A1.1.} Throughout the proof, $C$ denotes various
constants depending on $n, k_1, k_2$.  Fix a ball $B_r(x)$ in $\Omega$
with
$$
r \leq k_2\text{ dist}(x,\partial\Omega).
$$
\medskip
Our aim is to show that
$$
I =
\operatornamewithlimits{\Mint}_{B_r(x)}\;\operatornamewithlimits{\Mint}_{B_r(x)}|f(y) - f(z)|\leq C\|f\|_{\text{BMO},k_1}. \tag A1.2
$$
\medskip
We use a covering argument similar to one in the proof of Lemma A.14 in
[BNI].
\medskip
Consider a maximal family of disjoint open balls $B_\rho(x_i)$, with
centres $x_i\in B_r(x)$, and radius
$$
\rho = Ar\qquad\text{  with $A = \dfrac{k_1(1 - k_2)}{2 k_2} < 1$}. \tag
A1.3
$$
Each ball of double radius, $B_{2\rho}(x_i)$, belongs to the class
$\Cal C_{k_1}$.  Indeed
$$
\align
r\leq k_2\text{ dist}(x,\partial\Omega) &\leq k_2(|x - x_i| +
\text{ dist}(x_i,\partial\Omega))\\
    &\leq k_2 r + k_2\text{ dist}(x_i,\partial\Omega),
\endalign
$$
so that
$$
r\leq \frac{k_2}{1 - k_2}\text{ dist}(x_i,\partial\Omega)
$$
and
$$
2\rho = 2A r \leq \frac{2Ak_2}{1 - k_2}\text{dist}(x_i,\partial\Omega)
      = k_1\text{ dist}(x_i,\partial\Omega).
$$
\medskip
\noindent
Furthermore, clearly,
$$
B_r (x) \subset \bigcup_i B_{2\rho}(x_i).
$$
Thus
$$
I\leq \frac{C}{r^{2n}} \bigg[\sum_i
\int_{B_{2\rho}(x_i)}\;\int_{B_{2\rho}(x_i)} |f(y) - f(z)| +
\sum_{i\not=j} \int_{B_{2\rho}(x_i)} \int_{B_{2\rho}(x_j)} |f(y) -
f(z)|\bigg].
\tag A1.4
$$
\medskip
The first sum is bounded by
$$
\align
2\|f\|_{\text{BMO},k_1} \sum_i |B_{2\rho}(x_i)|^2&\leq
C\|f\|_{\text{BMO},k_1} \sum |B_\rho(x_i)|^2\\
    &\leq C\|f\|_{\text{BMO},k_1} |B_{r+\rho}(x)|^2\tag A1.5\\
    &\leq C \|f\|_{\text{BMO},k_1} r^{2n}.
\endalign
$$
\medskip
To estimate the second sum in (A1.4) we have
$$
\align
J &= \sum_{i\not=j} \int_{B_{2\rho}(x_i)} \int_{B_{2\rho}(x_j)} |f(y) -
f(z)|\\
  &\leq \sum_{i\not=j} \int_{B_{2\rho}(x_i)}\, \int_{B_{2\rho}(x_j)} \big[|f(y) - \overline
f_{2\rho}(x_i)| + |\overline f_{2\rho}(x_i) - \overline f_{2\rho}
(x_j)\big] + [\overline f_{2\rho}(x_j) - f(z)]\\
   &\leq C \|f\|_{\text{BMO},k_1} \sum_{i\not=j}
|B_\rho(x_i)|\,|B_\rho(x_j)| + C\sum_{i\not=j}
|B_\rho(x_i)|\,|B_\rho (x_j)|\,|\overline f_{2\rho}(x_i) - \overline
f_{2\rho}(x_j)|.
\endalign
$$
\medskip
We now claim that for $i\not=j$
$$
|\overline f_{2\rho}(x_i) - \overline f_{2\rho}(x_j)|\leq C
\|f\|_{\text{BMO},k_1}. \tag A1.6
$$
\medskip
Assuming (A1.6) we see that
$$
\align
J&\leq C \|f\|_{\text{BMO},k_1} (\sum |B_\rho (x_i)|)^2\\
   &\leq C \|f\|_{\text{BMO},k_1} |B_{r+\rho} (x) |^2\tag A1.7\\
   &\leq C r^{2n} \|f\|_{\text{BMO}, k_1}.
\endalign
$$
If we combine this with (A1.5) and (A1.4) we obtain (A1.2).
\bigskip
\demo{Proof of (A1.6)}  This is done as in [BNI] (proof of inequality
(A.12)).  Namely, for any two points $y,z$ in $B_r(x)$,
$$
|\overline f_{2\rho} (y) - \overline f_{2\rho} (z)| \leq C
\frac{r}{\rho}\;\;\|f\|_{\text{BMO},k_1} \tag A1.8
$$
In view of (A1.3), we then obtain (A1.6).  To verify (A1.8) consider a
chain of points $y,y_1,\ldots,y_{\ell-1},z$ in $B_r(x)$ such that the
distance between any two successive ones is bounded by $\rho$, and
with $\ell \leq C r/p$.  For any two successive points of the chain,
say $y_i, y_{i+1}$, we see, using Lemma A.4 of [BNI], that
$$
\align
\big\vert\overline f_{2\rho} (y_i) - \overline f_{2\rho} (y_{i+1})\big\vert
&\leq C \operatornamewithlimits{\Mint}_{B_{3\rho}(y_i)} |f - \overline
f_{3\rho} (y_i)|\\
    &\leq C \|f\|_{\text{BMO},k_1}.
\endalign
$$
Consequently, adding these inequalities for all successive points we
obtain (A1.8).  $\qquad\hfill\qed$
\enddemo
\bigskip
\remark{\bf Remark A1.1}  The definition of $\|\;\;\|_{\text{BMO},k}$  involves balls in
$\Bbb R^n$, and we have only spoken of Euclidean balls.  The reader may
verify that Lemma A1.1 holds if we replace the Euclidean metric by any
norm on $\Bbb R^n$.
\endremark
\bigskip

Using Lemma A1.1 it is easy to give the

\demo{Proof of Lemma 2}  Consider a ball $B_r(x)$ in $\Omega_1$ with
$r < k\text{ dist}(x,\partial\Omega_1)$, $k$ to be chosen.  We wish to
estimate
$$
I
=\operatornamewithlimits{\Mint}_{B_r(x)}\;\operatornamewithlimits{\Mint}_{B_r(x)}|f (H(y)) - f(H(z))|.
$$
\medskip

In view of Lemma A1.1 it suffices to consider any $k$ in $(0,1)$.  We
have
$$
I\leq \frac{C}{r^{2n}} \int_{H(B_r(x))} \int_{H(B_r(x))} |f(\eta) -
f(\zeta)|;
$$
$C$ depends on a bound for the Jacobian of $H^{-1}$.  Note that
$$
H(B_r(x))\subset B_{\alpha r}(H(x))
$$
for suitable $\alpha$ depending on the Lipschitz constant of $H$.  Furthermore,
$$
\text{dist}(H(x), \partial\Omega_2)\geq \delta \text{
dist}(x,\partial\Omega_1)
$$
where $\delta$ depends on the Lipschitz constant of $H^{-1}$.  Thus
$$
\alpha r < \alpha k\text{ dist}(x,\partial\Omega_1) < \frac{\alpha
k}{\delta} \text{ dist}(H(x),\partial\Omega_2).
$$
We now fix $k$ so that, for example, $\alpha k/\delta = 1/2$.  Then we find
$$
I\leq C \|f\|_{\text{BMO}(\Omega_2)}.
$$

$\quad\hfill\qed$
\enddemo
\bigskip

We now come to one of the main results in this Appendix, the
equivalence, due to Peter Jones, of the various notions of BMO, i.e.,
using all balls or just balls well inside the domain.  In fact the
balls need not be Euclidean ones.  They may be balls in any norm on
$\Bbb R^n$.  In the statement of Theorem A1.1, and in the proof, the
balls and distance may be measured in any given norm.   
\medskip  

\proclaim{Theorem A1.1}  Let $\Omega$ be an open bounded set in $\Bbb
R^n$.  For any real function $f\in L^1_{\text{loc}} (\Omega)$,
consider two (semi) norms
$$
\|f\| = \|f\|_{\text{BMO}} = \sup\Sb x\in\Omega\\\vare\leq
\frac12\text{dist}(x,\partial\Omega)\endSb \Mint_{B_\vare (x)} |f -
\Mint_{B_\vare(x)} f|,
$$
$$
\|f\|' = \|f\|'_{\text{BMO}} = \sup\Sb x\in\Omega\\\vare <
\text{dist}(x,\partial\Omega)\endSb \operatornamewithlimits{\Mint}_{B_\vare(x)} |f -\operatornamewithlimits{\Mint}_{B_\vare(x)} f|.
$$
\medskip
There is a constant $\overline C$ depending only on $n$ and the choice
of norm on $\Bbb R^n$, such that
$$
\|f\|_{\text{BMO}} \leq \|f\|'_{\text{BMO}} \leq \overline C
\|f\|_{\text{BMO}}. \tag A1.9
$$
\endproclaim
\bigskip
The proof of Theorem A1.1 relies on the following two lemmas.
\medskip
\proclaim{Lemma A1.2}  There is a covering of $B = B_1(0)$ by balls
$B_i = B_{r_i} (x_i)$ with\newline $r_i = \dfrac12 (1 - |x_i|) > 0$ such that
for every $\gamma > (n-1) / n$,
$$
\sum_i |B_i|^\gamma = c_\gamma < \infty.
$$
In particular,
$$
\sum_i |B_i|\;|\log|B_i|| < \infty. \tag A1.10
$$
\endproclaim

\demo{Proof}  Let $0 < b < 1$ and set, for $j = 1,2,\ldots,$
$$
A_j = \{x\in B;\;\; 1 - b^{j-1} \leq |x|\leq 1 - b^j\}.
$$
Note that
$$
B = \bigcup^\infty_{j=1} A_j.
$$
For each fixed $j$, consider a maximal family $F_j$ of disjoint balls
$B_\rho(x_i)$ with $x_i\in A_j\quad\forall i$,\,  and $\rho = \dfrac14
b^j$.  Clearly, the family $B_{2\rho}(x_i)$ covers $A_j$.  The
corresponding family $B_i = B_{r_i} (x_i)$ with $r_i = \dfrac12 (1 -
|x_i|) \geq 2\rho$ also covers $A_j$.  Moreover
$$
\bigcup_{i\in F_j} B_\rho (x_i)\subset A = \left\{x;\;\; 1 - b^{j-1} - \rho <
|x| < 1 - b^j + \rho\right\}
$$
and so
$$
\sum_{i\in F_j} |B_\rho(x_i)|\leq |A|\leq C b^j,
$$
where $C$ is independent of $j$.  It follows that
$$
\text{card}F_j\leq C b^{j(1-n)}.
$$
Thus we obtain, since $r_i \leq C b^j\quad\forall i$,
$$
\sum_{i\in F_j} |B_i|^\gamma \leq C b^{j(1-n)} b^{nj\gamma}\leq C
d^j,
$$
where $d = b^{n\gamma - n+1} < 1$.  Consequently
$$
\sum^\infty_{j=1} \sum_{i\in F_j} |B_i|^\gamma \leq C
\sum^\infty_{j=1} d^j < \infty.
$$
$\qquad\hfill\qed$
\enddemo
\medskip
\proclaim{Lemma A1.3}  There is a constant $C$ depending only on $n$
and the choice of norm on $\Bbb R^n$ such that 
$$
|\overline f_r(x) - \overline f_{1/2}(0)|\leq C\|f\|_{\text{BMO}}
\log (1/r)\quad\forall x\in B_1(0) \tag A1.11
$$
with
$$
r = \frac12 (1 - |x|).
$$
\endproclaim
\medskip
Assuming Lemma A1.3 it is easy to derive Theorem A1.1.
\medskip
\demo{Proof of Theorem A1.1}  It suffices to show that, for any ball
$B = B_\delta (x_0)\subset \overline{B_\delta(x_0)} \subset\Omega$,
$$
I: = \operatornamewithlimits{\Mint}_B |f  - f_0|\leq \overline C
\sup\Sb x\in B\\\vare\leq \frac12\text{dist}(x,\partial B)\endSb
\operatornamewithlimits{\Mint}_{B_\vare(x)} |f - \overline f_\vare
(x)|
$$
for some constant $f_0$ and some constant $\overline C$ depending only
on $n$ and the given norm on $\Bbb R^n$.  Without loss of generality
we may suppose $B = B_1(0)$ and $\|f\|_{\text{BMO}(B)} = 1$.
\medskip
Consider a covering $B_i = B_{r_i} (x_i)$ of $B$ as in Lemma A1.2.
Set
$$
f_0 = \overline f_{1/2} (0)
$$
and
$$
f_i = \overline f_{r_i}(x_i).
$$
We deduce from (A1.11) that, for all $i$,
$$
|f_i - f_0|\leq C \log\frac{1}{r_i}\leq C |\log|B_i|| + C. \tag A1.12
$$
Therefore
$$
\align
I\leq \frac{1}{|B|} \sum_i \int_{B_i} |f - f_0|&\leq \frac{1}{|B|}
\sum_i |B_i|\operatornamewithlimits{\Mint}_{B_i} |f - f_i| +
\frac{1}{|B|} \sum_i |B_i||f_i - f_0|\\
   &\leq C
\endalign
$$
by (A1.12) and Lemma A1.2.

$\qquad\hfill\qed$
\enddemo
\bigskip
We now return to the
\medskip
\demo{Proof of Lemma A1.3}  The line from $0$ to $x$ is identified
with $\Bbb R$ and we assume $0\leq x < 1$.  Consider the sequence
$B_{r_k}(x_k)$ of balls centred on that line with
$$
x_k = 1 - (1 - x) 2^{k-1}
$$  
and
$$
r_k = \frac12 (1 - x_k) = (1 - x) 2^{k-2}.
$$
Let $k_0$ be the largest integer such that $x_k \geq 0$.  We always
assume that $k\leq k_0$, so that $\overline{B_{r_k}(x_k)}\subset B_1(0)$.
\enddemo
\medskip
It is easy to check that
$$
B_{r_k/2} \left(x_k - \frac{r_k}{2}\right)\subset
B_{r_k}(x_k)\cap B_{r_{k+1}} \left(x_{k+1}\right),
$$
and thus, by Lemma A.4 of [BNI], we have
$$
|\overline f_{r_k} (x_k) - \overline f_{r_k/2} \left(x_k -
\frac{r_k}{2}\right)|\leq C \|f\|_{\text{BMO}}
$$
$$
|\overline f_{r_{k+1}} \left(x_{k+1}\right) - \overline
f_{r_k/2} \left(x_k - \frac{r_k}{2}\right)|\leq C
\|f\|_{\text{BMO}}.
$$
Set
$$
f_k = f_{r_k} (x_k);
$$
we infer that 
$$
|f_k - f_{k+1}|\leq C \|f\|_{\text{BMO}}\quad\forall k\leq k_0 - 1.
$$
Adding these inequalities we find
$$
|f_1 - f_{k_0}|\leq C \|f \|_{\text{BMO}} (k_0 - 1) \leq C
\|f\|_{\text{BMO}} \log \frac1r. \tag A1.13
$$
Note that
$$
f_1 = \overline f_r(x)\quad\text{with\quad $r = \dfrac12 (1 - x)$}.
$$
Finally we claim that
$$
|f_{k_0} - \overline f_{1/2} (0)| \leq C \|f\|_{\text{BMO}}. \tag
A1.14
$$
The desired conclusion (A1.11) then follows from (A1.13) and (A1.14).
\medskip
\demo{Proof of (A1.14)}  Since $x_{k_0 + 1}\leq 0$ we have $x_{k_0}
\leq \dfrac12$ and $r_{k_0} = \dfrac12(1 - x_{k_0})\geq \dfrac14$.  It
follows that $B_{1/2}(0)\cap B_{r_{k_0}} (x_{k_0})$ contains the ball
$\hat B = B_{1/8} (x_{k_0} - \dfrac18)$.  Applying Lemma A.4 of [BNI]
once more we obtain
$$
\align
|\overline f_{1/2}(0) &- \overline f_{\hat B} |\leq C
\left\|f\right\|_{\text{BMO}}\\
      |f_{k_0} &- \overline f_{\hat B} | \leq
C\left\|f\right\|_{\text{BMO}}
\endalign
$$
and thus (A1.14) is established.
\enddemo
\medskip
\remark{\bf Remark A1.2}  Theorem A1.1 holds for any open set
$\Omega$, with $\overline\Omega$ compact in a smooth open Riemannian
manifold $X_0$.  In the definitions of the norms $\|\;\;\|$ and
$\|\;\;\|'$, one also restricts the radii of the balls to be less than
the injectivity radius $r_0$ of $X_0$---assumed to be positive.  The
constant $\overline C$ in (A1.9) then depends on the Riemannian metric
on $X_0$.  The proof of this more general result proceeds as in the
proof above with minor modifications.
\endremark
\bigskip
Here are some consequences of the above results.
\medskip
\proclaim{Corollary A1.1}  Let $\Omega$ be an open bounded set in
$\Bbb R^n$.  Suppose $\|\;\;\|_1$ and $\|\;\;\|_2$ are two norms on
$\Bbb R^n$.  Associated with these are two notions of BMO$(\Omega)$:
$$
\left\|f\right\|_{\text{BMO}_i} =
\underset\overline{B_\vare^i(x)}\subset\Omega\to{\sup}\;
\operatornamewithlimits{\Mint}_{B_\vare^i(x)} |f -
\operatornamewithlimits{\Mint}_{B_\vare^i(x)} f|\qquad i = 1,2.
$$
Here the ball $B_\vare^i(x)$ is measured in the norm $\|\;\;\|_i$.
Then the two BMO norms are equivalent (and the equivalence constants
depend only on $n$ and the equivalence constants for $\|\;\;\|_1$ and
$\|\;\;\|_2$).
\endproclaim
\bigskip
Next we take up the
\medskip
\demo{Proof of Lemma 3 in \S II.1}  Consider the
function
$$
\varphi(x) = \log \frac{1}{\widetilde d(x,\partial\Omega)}
$$
where $\widetilde d$ is the distance measured in some metric equivalent to
the Riemannian one.  For any ball $B_\vare(x)$ in $\Omega$, with
$\vare\leq \dfrac12$ dist$(x,\partial\Omega)$---here dist refers
to our Riemannian metric---we have to estimate
$$
J = \operatornamewithlimits{\Mint}_{B_\vare(x)}
\;\operatornamewithlimits{\Mint}_{B_\vare(x)} |\varphi(y) -
\varphi(z)|.
$$
Clearly if $y\in B_\vare(x)$, dist$(y,\partial\Omega) > \vare$, and
thus $\widetilde d(y,\partial\Omega) > \alpha\vare$ for some constant
$\alpha$.  Hence for $y, z\in B_\vare(x)$,
$$
|\varphi(y) - \varphi(z)|\leq \frac{C}{\vare}\widetilde d(y,z)\leq
\frac{C}{\vare}\text{  dist}(y,z)\leq C.  \tag A1.15
$$
Consequently $J \leq C$.

$\qquad\hfill\qed$
\enddemo
\medskip

With the aid of Theorem 1 we also give the
\medskip

\demo{Proof of Lemma 4}  Consider a ball $B_\vare(x)$ in $\Omega$ with
$\vare\leq \dfrac12$dist$(x,\partial\Omega)$.  In view of Theorem 1 we
have to show that given $\delta > 0$, there exists $\vare_0 > 0$ such
that
$$
J = \operatornamewithlimits{\Mint}_{B_\vare(x)}\;\operatornamewithlimits{\Mint}_{B_\vare(x)} |\varphi^\alpha(y) -
\varphi^\alpha(z)| < \delta\quad\forall x\in\Omega,
$$
and for all $\vare \leq \min\{\vare_0,
\dfrac12$dist$(x,\partial\Omega)\}$.  Since $\varphi$ is continuous in
$\Omega$ we need only consider such balls with
dist$(x,\partial\Omega)$ small.  We have
$$
|\varphi^\alpha(y) - \varphi^\alpha(z)|\leq \alpha\frac{|\varphi(y) -
\varphi(z)|}{\min\{\varphi(y),\varphi(z)\}^{1 - \alpha}} ;
$$
by (A1.15),
$$
|\varphi^\alpha(y) - \varphi^\alpha(z)| \leq \frac{C}{\vare}\;\,
\frac{\text{dist}(y,z)}{\underset B_\vare(x)\to{|\min}\; \varphi|^{1-\alpha}}.
$$
Consequently
$$
J\leq C |\min_{B_\vare(x)} \varphi|^{\alpha - 1}
$$
with $C$ independent of $x$ and $\vare$.  Thus for
dist$(x,\partial\Omega)$ small, $\underset B_\vare(x)\to{\min}
\varphi$ is as large as wanted, so that $J$ is small.

$\quad\hfill\qed$
\enddemo
\bigskip



We turn finally to the proof of Theorem 1.  The proof we present is a
slight modification of one shown to us by Peter Jones.
\medskip
\demo{Proof of Theorem 1}  We need only prove that (1.6) implies
(1.7), namely, if $f\in\text{BMO}(\Omega)$ and satisfies
$$
\lim\Sb \vare\rightarrow 0\\\vare\leq
\frac12\text{dist}(x,\partial\Omega)\endSb
\operatornamewithlimits{\Mint}_{B_\vare(x)} |f - \overline f_\vare(x)|
= 0\quad\text{uniformly in $x$},
\tag A1.16
$$
then
$$
  \text{there exists a sequence $(f_j)$ in
$C^\infty_0(\Omega)$ converging to $f$ in BMO$(\Omega)\cap
L^1_{\text{loc}}(\Omega)$}.
\tag A1.17
$$
\enddemo
\medskip
 The proof makes use of the following simple
\medskip
\proclaim{Lemma A1.4}  Assume that $f$ is in BMO$(\Omega)$ and
satisfies (A1.16).  Then each truncation
$$
f^k(x)  = \cases  k&\quad\text{if $f(x) \geq k$}\\
          f(x)&\quad\text{if $- k < f(x) < k$}\\
          -k&\quad\text{if $f(x) \leq -k$}
\endcases
$$
also satisfies (A1.16) and moreover,
$$
f^k\rightarrow f\quad\text{\rm in BMO$\cap L^1_{\text{loc}}$ as
$k\rightarrow\infty$}.
\tag A1.18
$$
\endproclaim

The lemma is a variant of Lemma A.17 in [BNI] and is proved in the
same way.
\medskip
In view of Lemma A1.4 we may assume that our $f$ satisfying (A1.16) is
in $L^\infty$.  The main step is to show that $f$ may then be
approximated in BMO$\cap L^1$ by $L^\infty$ functions $F$ satisfying
(A1.16) and which, furthermore, have compact support in $\Omega$.
Once this is done it is easy to complete the proof of the theorem:  We
may think of $\Omega$ as lying in a compact manifold $X_0$, without
boundary and consider $F$ defined on $X_0$ to be zero outside
$\Omega$.  By Sarason's result (see Lemma 3 in [BNI]) $F$ belongs to
VMO$(X_0)$.  By Corollary 1 in [BNI], $\overline F_\vare$ is close to
$F$ in BMO$\cap L^1$, if $\vare$ is small.  But for $\vare$ small,
$\overline F_\vare$ also has compact support in $\Omega$.  Since
$\overline F_\vare$ is continuous, it may be approximated in the
$L^\infty$ norm---and hence in BMO$\cap L^1$---by smooth functions
with compact support in $\Omega$.  The proof of Theorem 1 would then
be complete.
\medskip
As usual, it is convenient to replace the BMO norm by an equivalent
one:
$$
\|f\|_\star = \operatornamewithlimits{\sup}\Sb x\\\vare < \frac12\text{dist}(x,\partial\Omega)\endSb\;
\operatornamewithlimits\Mint_{B_\vare(x)} \;\operatornamewithlimits\Mint_{B_\vare(x)} |f(y) - f(z)|,
$$
and to rewrite (A1.16) as
$$
\lim\Sb\vare\rightarrow 0\\\vare\leq
\frac12\text{dist}(x,\partial\Omega)\endSb \operatornamewithlimits\Mint_{B_\vare(x)}
\;\operatornamewithlimits\Mint_{B_\vare(x)} |f(y) - f(z)| = 0\quad\text{uniformly in $x$}.
\tag"{$(A1.16)'$}"
$$
\medskip
To carry out the main step, consider $f$ satisfying $(A1.16)'$ with
$|f|\leq k$.  Using suitable cutoff functions we will construct the
approximating functions $F_j$.
\medskip

Recalling the function of Lemma 3,
$$
\varphi(x) = \log \frac{1}{\text{dist}(x,\partial\Omega)},
$$
without loss of generality, we may always assume that for all $x$,
dist$(x,\partial\Omega) \leq 1$, so that $\varphi(x) \geq 0$.  For $j
= 1,2,\ldots,$ set
$$
h_j(x) = (1 - \frac1j\varphi(x))^+
$$
and
$$
F_j = h_j f.
$$
\medskip
We claim that the $F_j$ have all the desired properties:
\medskip
\noindent
(i)\quad $F_j\in L^\infty$,
\medskip
\noindent
(ii)\quad each $F_j$ satisfies (A1.16),
\medskip
\noindent
(iii)\quad the $F_j$ have compact support,
\medskip
\noindent
(iv)\quad $F_j\rightarrow f$ in $L^1$,
\medskip
\noindent
(v)\quad $F_j\rightarrow f$ in BMO.
\medskip
\noindent
Clearly (i), (iii) and (iv) are trivial.
\medskip
\noindent
{\it Proof of\/} (ii).  Each function $h_j$ is Lipschitz on $\Omega$,
with Lipschitz constant $k_j$.  Then, for our usual balls $B_\vare(x)$,
$$
\align
\operatornamewithlimits\Mint_{B_\vare(x)}\; \operatornamewithlimits\Mint_{B_\vare(x)} |h_j (y) f(y) &- h_j (z) f(z)|\\
    &\leq \operatornamewithlimits\Mint_{B_\vare(x)} \; \operatornamewithlimits\Mint_{B_\vare(x)} |f(y) - f(z)| + k_j
\|f\|_{L^\infty} \operatornamewithlimits\Mint_{B_\vare(x)} \;\operatornamewithlimits\Mint_{B_\vare(x)}
\text{dist}(y,z)\\\openup 2\jot
    &\longrightarrow 0\text{\quad as \quad $\vare \longrightarrow 0$\qquad
by $(A1.16)'$}.
\endalign
$$

\medskip
\noindent
{\it Proof of\/} (v).  Given $\delta > 0$, there exists $\vare_0 > 0$
such that
$$
\operatornamewithlimits{\Mint}_{B_\vare(x)}\;\operatornamewithlimits{\Mint}_{B_\vare(x)} |f(y) - f(z)| <
\frac{\delta}{3}\quad\text{for $\vare < \vare_0, \vare\leq
\dfrac12$dist$(x,\partial\Omega)$}.
$$
Consider
$$
I = \operatornamewithlimits{\Mint}_{B_\vare(x)}\; \operatornamewithlimits{\Mint}_{B_\vare(x)} |h_j(y) f(y) - h_j(z) f(z) -
(f(y) - f(z))|.
$$
We will prove that $I < \delta$ for $j$ sufficiently large
(independent of $x$ and $\vare$).  As usual, we distinguish two cases.
\medskip
\noindent
(a)\quad If $\vare < \vare_0$ then
$$
\align
I &\leq 2 \operatornamewithlimits{\Mint}_{B_\vare(x)}\;\operatornamewithlimits{\Mint}_{B_\vare(x)} |f(y) - f(z)| +
\|f\|_{L^\infty} \operatornamewithlimits{\Mint}_{B_\vare(x)}\;\operatornamewithlimits{\Mint}_{B_\vare(x)} |h_j(y) -
h_j(z)|\\
    &< \frac{2\delta}{3} + \frac1j \|f\|_{L^\infty}
\operatornamewithlimits{\Mint}_{B_\vare(x)}\;\operatornamewithlimits{\Mint}_{B_\vare(x)} |\varphi(y) - \varphi(z)|\\
    &\leq \frac{2\delta}{3} + \frac{C}{j} \|f\|_{L^\infty} \qquad\text{by
Lemma 3},\\
    &< \delta\quad\text{for $j$ sufficiently large}.
\endalign
$$
\medskip
\noindent
(b)\quad  If $\vare \geq \vare_0$ then
$$
I \leq 2 \operatornamewithlimits{\Mint}_{B_\vare(x)} |h_j(y) - 1|\;|f(y)|\leq C \|f\|_{L^\infty} \int_\Omega (1 - h_j).
$$
This can be made less than $\delta$ for $h_j$ large, by dominated
convergence.

$\quad\hfill\qed$

\bigskip
%10/5/96 Appendix 2 for Selecta paper.
\noindent
{\bf Appendix 2 (with P. Mironescu). Toeplitz operators and VMO}
\bigskip
In this appendix we discuss Toeplitz operators on the circle $S^1$.
Let us first recall the classical Toeplitz operators.  Consider
complex valued $L^2$-functions on $S^1$ and the closed subspace
$$
\Cal H^2 = \left\{f\in L^2(S^1); \int_{S^1} e^{in\theta} f(\theta) d\theta
= 0,\quad n=1,2,\ldots\right\},
$$
and more generally, for $p$ in $[1,\infty]$,
$$
\Cal H^p = \left\{f\in L^p(S^1); \int_{S^1} e^{in\theta} f(\theta) d\theta
= 0,\quad n = 1,2,\ldots\right\}.
$$
Let $P$ be the orthogonal projection from $L^2$ onto $\Cal H^2$.
\medskip
Given a function $\varphi\in L^\infty(S^1,\Bbb C)$ we denote by $
M_\varphi$ the operation on $L^2$ of multiplication by $\varphi$.  The
associated Toeplitz operator (with symbol $\varphi$), is
$$
T_\varphi = PM_\varphi P; \tag A2.1
$$
the associated Hankel operator is
$$
H_\varphi = (I - P) M_\varphi P. \tag A2.2
$$
\medskip
$T_\varphi$ is often considered as an operator from $\Cal H^2$ to
$\Cal H^2$.
\medskip
A classical result is that if $\varphi$ is continuous and nowhere
zero, then $T_\varphi$ is a Fredholm operator and
$$
\text{index $(T_\varphi) = - \deg\left(\frac{\varphi}{|\varphi|}, S^1,
S^1\right)$}.
\tag A2.3
$$
\medskip
See, for example, R.~G.~Douglas [1], Theorem 7.26 and R.~G.~Douglas~[2]; further references and
history may be found there.  A number of authors have extended this
result to other classes of functions $\varphi$, not necessarily
continuous.  See for example Theorem 7.36 in R.~G.~Douglas~[1] and D.~Sarason
[1],[2],[3],[4], and the recent book by I.~Gohberg and N.~Krupnik~[1].
\medskip

Since the right hand side of (A2.3) makes sense for
$\varphi\in\text{VMO}(S^1)$ with $|\varphi| \geq a > 0$---by [BNI], it is
natural to extend the classical result above to functions $\varphi$
satisfying 
$$
\varphi\in\text{VMO}(S^1)\cap L^\infty (S^1), \quad|\varphi|\geq a > 0.
\tag A2.4
$$
We present such a result
\medskip
\proclaim{Theorem A2.1}  Let $\varphi$ satisfy (A2.4).  Then
$T_\varphi$ is Fredholm and (A2.3) holds.
\endproclaim

This follows, in fact, from Theorem 7.36 in R.~G.~Douglas~[1].  His result is more
general:  it asserts that if $\varphi$ is in $\Cal H^\infty + C^0$ and if
$\hat\varphi$, the harmonic extension of $\varphi$ to the unit disc
$D$, satisfies
$$
|\hat\varphi (r e^{i\theta})|\geq \alpha > 0\quad\text{for $1 - \delta
< r < 1$}, \tag A2.5
$$
then $T_\varphi$ is Fredholm.  Moreover,
$$
\text{index $(T_\varphi) = - \deg\left(\frac{\hat\varphi(r
e^{i\theta})}{|\hat\varphi(r e^{i\theta}|} , S^1, S^1
\right)$\quad for every  $r$ in $(1 - \delta,1)$}. \tag A2.6
$$
To derive Theorem A2.1 from Douglas' result one uses two facts:
\medskip
(i)\quad If $\varphi\in\text{VMO}\cap L^\infty$, then $\varphi\in
\Cal H^\infty+ C^0$.  More precisely,
$$
\varphi\in \text{VMO $\cap
L^\infty\Longleftrightarrow \varphi$ and $\overline\varphi$ belong to
$\Cal H^\infty + C^0$}.
$$
This result is due to D.~Sarason [1].  The space
VMO$\cap L^\infty$ is sometimes called QC (quasi continuous);
\medskip
(ii)\quad  If $\varphi\in\text{VMO}$ and $|\varphi|\geq a > 0$ then
its harmonic extension $\hat\varphi$ satisfies (A2.5); see Lemma 5 in
D.~Sarason~[3], and also Theorem A3.2 in Appendix 3 here.
\bigskip
It seems worthwhile to present here a different proof which is more or
less self contained.  It is elementary except for the Fefferman
inequality (see (A2.10) below).
\medskip
We derive Theorem A2.1 from the classical case---for $\varphi$ continuous---by
approximation.  The convergence of the right hand side of (A2.3), in
the approximation, holds by stability of degree in VMO, see Theorem 1
in [BNI]. The convergence of the left hand side is more subtle since $T_\varphi$
does {\it not} depend continuously in the operator norm on the BMO
norm of $\varphi$; see Remark A2.1.  It turns out that $H_\varphi$ has
that property:
\medskip
\proclaim{Lemma A2.1}  There is a constant $C$ such that
$$
\|H_\varphi\|\leq C \|\varphi\|_{\text{BMO}} \qquad\forall \varphi\in
L^\infty(S^1). \tag A2.7
$$
\endproclaim

\demo{Proof}  Clearly $H_\psi = 0$ if $\psi\in \Cal H^\infty$.  Thus for any
$\psi\in \Cal H^\infty$,
$$
\|H_\varphi\| = \|(I - P) M_{\varphi - \psi} P\|\leq \|M_{\varphi -
\psi} \| \leq \|\varphi - \psi\|_{L^\infty}.
$$
Hence
$$
\|H_\varphi\|\leq \operatornamewithlimits{inf}_{\psi\in \Cal H^\infty}
\|\varphi - \psi\|_{L^\infty} = \text{dist$(\varphi, \Cal H^\infty)$ in
$L^\infty$}.
$$
(In fact equality holds by Nehari's theorem; see D.~Sarason~[4], page 100.)
\medskip
The assertion of the lemma follows from the
\medskip
\noindent
{\bf Claim}:
$$
\text{dist$(\varphi, \Cal H^\infty) \leq C \|\varphi\|_{\text{BMO}}$\quad for
$\varphi\in L^\infty$}. \tag A2.8
$$
\enddemo
\medskip
\demo{Proof of Claim}  Recall that if $X$ is a real Banach space, and
$M$ is a linear subspace of $X$ then for any $f\in X^\star$,
$$
\operatornamewithlimits{\sup}\Sb u\in M\\\|u\|\leq 1\endSb \langle
f,u\rangle = \text{dist}(f,M^\perp), \tag A2.9
$$
where $M^\perp$ is the set of points in $X^\star$ which annihilate $M$.  We
take $X = L^1 (S^1,\Bbb C)\simeq L^1(S^1,\Bbb R^2)$, $M = $the set of
finite linear combinations (over $\Bbb C$) of $e^{-in\theta}, n =
1,2,\ldots$\,.  Feffermans' inequality (see C.~Feffermann [1]; see also
C.~Fefferman and E.~Stein [1], and E.~Stein~[1]) implies that for $u\in M$,
$$
\left\vert\int_{S^1} f u \right\vert\leq C \|f\|_{\text{BMO}} \|u\|_{L^1} \tag A2.10
$$
\medskip
By definition, $M^\perp = \Cal H^\infty$, and (A2.7) then follows from
(A2.9) and (A2.10).

$\quad\hfill\qed$
\enddemo
\medskip
\remark{\bf Remark A2.1}  There is no estimate of the form
$$
\|T_\varphi\|\leq C\left(\|\varphi\|_{\text{BMO}} +
\|\varphi\|_{L^1}\right)\qquad\forall \varphi\in L^\infty. \tag A2.11
$$
\endremark
\medskip
\demo{Proof}  Write $f\in L^2$ as
$$
f = Pf + (I - P) f = Pf + \overline{P\overline f} - \Mint f. \tag
A2.12
$$
Since $H_\varphi + T_\varphi = M_\varphi P$ we may write, for any
$f\in L^2$,
$$
\align
M_\varphi f &= M_\varphi (Pf) + M_\varphi ((I - P)f)\\
    &= M_\varphi (Pf) + M_\varphi (\overline{P\overline f}) - (\Mint
f)\varphi\\
    &= M_\varphi(Pf) + \overline{M_{\overline\varphi} P\overline f} - (\Mint f)
\varphi\\
   &= H_\varphi(f) + T_\varphi(f) +
\overline{H_{\overline\varphi}(\overline f)} +
\overline{T_{\overline\varphi}(\overline f)} - (\Mint f)\varphi.
\endalign
$$
Thus, if (A2.11) were to hold, by (A2.11) and Lemma A2.1,
$$
\|M_\varphi f\|_{L^2} \leq C(\|\varphi\|_{\text{BMO}} +
\|\varphi\|_{L^1}) \|f\|_{L^2} + |\Mint f| \|\varphi\|_{L^2}.
$$
In particular,
$$
\|M_\varphi \| \leq C(\|\varphi\|_{\text{BMO}} + \|\varphi\|_{L^2}).
$$
But $\|M_\varphi\| = \|\varphi\|_{L^\infty}$.  This yields a
contradiction if we choose for $\varphi$ the truncations of a function in
BMO which is not in $L^\infty$.

$\quad\hfill\qed$
\medskip
\proclaim{Lemma A2.2}  For $\varphi \in\text{VMO} \cap L^\infty$,
$$
H_\varphi\text{  is compact from $L^2$ into itself}.
$$
\endproclaim

\demo{Proof}  There is a sequence $(\varphi_j)$ of functions in $C^0$
such that $\varphi_j\rightarrow\varphi$ in BMO; see D.~Sarason~[1].  By
Lemma A2.1
$$
\|H_{\varphi_j} - H_\varphi \|\leq C \|\varphi_j -
\varphi\|_{\text{BMO}} \rightarrow 0. \tag A2.13
$$
On the other hand, for every continuous $\psi$, $H_\psi$ is compact.
This fact is classical and is easily verified by noting that for every
$\psi$ of the form
$$
\psi(\theta) = \sum^{+N}_{n = -N} a_n e^{in\theta}
$$
$H_\psi$ is a finite rank operator.

$\quad\hfill\qed$
\enddemo
\medskip
\proclaim{Corollary A2.1}  For $\varphi\in L^\infty$ and
$\psi\in\text{VMO}\cap L^\infty$
$$
T_\varphi T_\psi - T_{\varphi\psi}\quad\text{is compact}
$$
\endproclaim

\demo{Proof}  Just write
$$
T_\varphi T_\psi - T_{\varphi\psi} = - P M_\varphi H_\psi \tag A2.14
$$
and apply Lemma A2.2.
\enddemo
\medskip
\proclaim{Lemma A2.3}  Assume (A2.4), then $T_\varphi$ is Fredholm in
$\Cal H^2$.
\endproclaim

\demo{Proof}  By Lemma $2'$ in [BNI] we know that
$\varphi^{-1}\in\text{VMO}\cap L^\infty$ and so, by Corollary A2.1, we
have, on $\Cal H^2$
$$
T_\varphi T_{\varphi^{- 1}} = I + K,\quad K\text{  compact}.
$$
Similarly, we have, on $\Cal H^2$,
$$
T_{\varphi^{-1}} T_\varphi = I + K',\quad K'\text{  compact}.
$$
It follows that (see e.g. S.~Lang[1]) $T_\varphi$ is Fredholm.
\medskip
Before continuing with the proof of Theorem A2.1, it is convenient to
introduce the class
$$
A = \{\varphi\in\text{VMO};\;\; \varphi\in L^\infty\text{  and  }
\varphi^{-1}\in L^\infty\}.
$$
Note that if $\varphi\in A$, then $\varphi^{-1}\in\text{VMO}$; see
Lemma $2'$ in [BNI].
\enddemo
\medskip
\proclaim{Lemma A2.4}  Let $(\psi_j)$ be a sequence in $A$ such that
$\|\psi_j\|_{L^\infty}\leq C$, $\|\psi_j^{-1} \|_{L^\infty}\leq C$ and
$\|\psi_j\|_{\text{BMO}} \rightarrow 0$.  Then $T_{\psi_j}$ is
invertible in $\Cal H^2$ for $j$ sufficiently large.
\endproclaim

\demo{Proof}  By (A2.14) we have, in $\Cal H^2$,
$$
T_{\psi_j} T_{\psi_j^{-1}} = I - P M_{\psi_j} H_{\psi_j^{-1}} \tag A2.15
$$
and
$$
T_{\psi_j^{-1}} T_{\psi_j} = I - P M_{\psi_j^{-1}} H_{\psi_j}. \tag A2.16
$$
Passing to a subsequence, we may always assume (by Lemma A.1 in [BNI])
that $\psi_j\rightarrow c$, for some constant $c$, in $L^1$.  It
follows (by Lemma A.7 in [BNI]) that $\psi_j^{-1}\rightarrow 0$ in
BMO.  Applying Lemma A2.1 we conclude that
$$
\|P M_{\psi_j} H_{\psi_j^{-1}}\|\rightarrow 0\text{  and  } \|P
M_{\psi_j^{-1}} H_{\psi_j}\|\rightarrow 0.
$$
Hence $I - PM_{\psi_j} H_{\psi_j^{-1}}$ and $I - PM_{\psi_j^{-1}}
H_{\psi_j}$ are invertible for $j$ sufficiently large;
the conclusion of the lemma follows easily from (A2.15) and (A2.16).
\enddemo
\medskip
Next, a useful lemma about the product of functions in BMO.
\medskip
\proclaim{Lemma A2.5}  Let $g\in\text{VMO} \cap L^\infty$.  Then for
every $\delta > 0$ there exists a constant $C_\delta$ (depending on
$\delta $ {\bf and} $g$) such that
$$
\|fg\|_{\text{BMO}} \leq \delta \|f\|_{L^\infty} + C_\delta
(\|f\|_{\text{BMO}} + \|f\|_{L^1})\quad\forall f\in L^\infty.
$$
\endproclaim

\demo{Proof}  Recall (see $(1'')$ in [BNI]) that
$$
\left\|fg\right\|_{\text{BMO}} \leq \operatornamewithlimits{\sup}_{\vare,x}
\operatornamewithlimits{\Mint}_{B_\vare(x)}
\;\operatornamewithlimits{\Mint}_{B_\vare(x)} |f(y) g(y) - f(z) g(z)|.
$$
But
$$
\align
\operatornamewithlimits{\Mint}_{B_\vare(x)}\;
\operatornamewithlimits{\Mint}_{B_\vare(x)} |f(y)g(y) - f(z)g(z)|&\leq
L +
\operatornamewithlimits{\Mint}_{B_\vare(x)}\;\operatornamewithlimits{\Mint}_{B_\vare(x)}
|f(y) - f(z)|\, |g(z)|\\
      &\leq L + 2 \|g\|_{L^\infty} \|f\|_{\text{BMO}},
\endalign
$$
where
$$
L = \operatornamewithlimits{\Mint}_{B_\vare(x)}\;\operatornamewithlimits{\Mint}_{B_\vare(x)}|f(y) (g(y) - g(z))|.
$$
Clearly, two estimates hold for $L$:
$$
L\leq 2\|g\|_{L^\infty} \operatornamewithlimits{\Mint}_{B_\vare(x)}
|f|,\;\;\text{  and  $L\leq \|f\|_{L^\infty}$}
\operatornamewithlimits{\Mint}_{B_\vare(x)}\;\operatornamewithlimits{\Mint}_{B_\vare(x)}
|g(y) - g(z)|. \tag A2.17
$$
\medskip
Since $g\in \text{VMO}$, there exists $\vare_0$ depending only on $g$
such that
$$
\operatornamewithlimits{\Mint}_{B_\vare(x)}\;\operatornamewithlimits{\Mint}_{B_\vare(x)}
|g(y) - g(z)|\leq \delta\qquad\text{if $\vare\leq \vare_0$},
$$
and thus $L\leq \delta \|f\|_{L^\infty}$ by the second estimate in
(A2.17).  For $\vare > \vare_0$ we use the first estimate in (A2.17),
namely
$$
L\leq 2 \frac{\|g\|_{L^\infty}}{\vare_0} \int_{B_\vare(x)} |f|\leq C
\|f\|_{L^1}
$$
and the conclusion of the lemma follows.
\enddemo
\medskip
\proclaim{Lemma A2.6}  Let $\varphi \in A$ and $(\varphi_j)$ be a
sequence in $A$ such that $\|\varphi_j\|_{L^\infty} \leq C$,
$\|\varphi_j^{-1}\|_{L^\infty}\leq C$ and
$\varphi_j\rightarrow\varphi$ in BMO$\cap L^1$.  Then
$$
\text{index  $(T_{\varphi_j}) = $ index $(T_\varphi)$ for $j$ sufficiently
large}.
$$
\endproclaim

\demo{Proof}  Lemma A2.5 (applied to $f = \varphi_j - \varphi$ and $g
= \varphi^{-1}$) implies that
$$
\left\|\frac{\varphi_j}{\varphi} \right\|_{\text{BMO}} \longrightarrow
0\quad\text{as $j\rightarrow \infty$}.
$$
We deduce from Lemma A2.4 that $T_{\varphi_j/\varphi}$ is invertible
in $\Cal H^2$ for $j$ sufficiently large.
\enddemo

\medskip
By Corollary A2.1 we have, in $\Cal H^2$,
$$
\align
T_{\varphi_j/\varphi} &= T_{\varphi_j} T_{1/\varphi} + K\\
   T_{1/\varphi} T_\varphi &= I + K'
\endalign
$$
where $K$ and $K'$ are compact.  Applying the standard properties of
the index (see e.g. S.~Lang [1]) we conclude that, for $j$
sufficiently large
$$
\align
0 &= \text{index$(T_{\varphi_j/\varphi}) = $index$(T_{\varphi_j}
T_{1/\varphi}) = $}\\
    &= \text{index$(T_{\varphi_j}) + $ index$(T_{1/\varphi}) = $index$(T_{\varphi_j}) - $ index$(T_\varphi)$}.
\endalign
$$ 
\enddemo
\medskip
We may now prove Theorem A2.1 by approximation using (A2.3) for
continuous $\varphi$.
\medskip
\demo{Proof of Theorem A2.1}  Given $\varphi\in A$ there is a sequence
$(\varphi_j)$ of continuous functions such that
$\|\varphi_j\|_{L^\infty}\leq C$, $\|\varphi_j^{-1}\|_{L^\infty} \leq
C$ and $\varphi_j\rightarrow\varphi$ in BMO$\cap L^1$; see
e.g. Corollary 4 in [BNI].  We have
$$
\text{index}T_{\varphi_j} = - \deg(\varphi_j/|\varphi_j|).
$$
For $j$ sufficiently large, the left hand side equals index
$T_\varphi$ (by Lemma A2.6) and the right hand side equals
$\deg(\varphi/|\varphi|)$ by Theorem 1 in [BNI].  $\qquad\hfill\qed$
\enddemo

\medskip
Here is an alternative proof of Theorem A2.1 which does not make use
of Lemmas A2.4, A2.5 and A2.6.  It is slightly shorter, but it relies
on an additional ingredient:  the lifting property for maps in
VMO$(S^1,S^1)$ with degree zero (see Theorem 3 in Section I.6 of
[BNI]).  On the other hand this proof is totally self contained---it
does not rely on the classical case ($\varphi$ continuous).  The key
observation is the following:
\medskip
\proclaim{Lemma A2.7}  Consider a map $m$:  $A\rightarrow \Bbb Z$
satisfying
$$
m(\varphi\psi)  = m(\varphi) + m(\psi)\quad\forall \varphi,\psi\in A.
$$
Then there is an integer $k$ such that
$$
m(\varphi) = k \deg \left(\frac{\varphi}{|\varphi|},
S^1,S^1\right)\quad\forall \varphi\in A. \tag A2.18
$$
\endproclaim

\remark{\bf Remark A2.2}  Surprisingly, in Lemma A2.7, no continuity
is required of $m$.  The condition on $m$ is purely algebraic.
\endremark
\medskip
\demo{Proof}  We first claim that
$$
m(\psi) = 0\quad\forall \psi\in A\text{\quad with $\deg(\frac{\psi}{|\psi|}) =
0$}. \tag A2.19
$$
Indeed we may write, by Theorem 3 in \S I.6 of [BNI],
$$
\psi = |\psi| e^{i\sigma}
$$
for some function $\sigma\in\text{VMO}(S^1,\Bbb R)$.  For every
integer $n$, let
$$
\psi_n = |\psi|^{\frac1n} e^{i\sigma/n} \in A,
$$
so that 
$$
m(\psi) = m(\psi_n^n) = n m(\psi_n).
$$
Thus, if $m(\psi) \ne 0$, $|m(\psi)|\geq n\quad\forall n$---impossible;
(A2.19) is proved.
\medskip
\enddemo
For $\varphi\in A$ let
$$
d = \deg \left(\frac{\varphi}{|\varphi|}, S^1, S^1\right)
$$
and write
$$
\varphi = e^{di\theta} |\varphi| e^{i\eta}, \eta\in
\text{VMO}(S^1,\Bbb R)
$$
(see Remark 10 in \S I.6 of [BNI]).  Then
$$
m(\varphi) = m(e^{di\theta}) + m (|\varphi| e^{i\eta}) = d m
(e^{i\theta})
$$
by (A2.19).  This proves (A2.18) with $k = m(e^{i\theta})$.
\medskip
\demo{Proof of Theorem A2.1}  For every $\varphi\in A$ we know that
$T_\varphi$ is Fredholm by Lemma A2.3.  Set
$$
m(\varphi) = \text{index}(T_\varphi)
$$
We have, by Corollary A2.1, for some compact operator $K$,
$$
m(\varphi\psi) = \text{index}(T_{\varphi\psi}) =
\text{index}(T_\varphi T_\psi + K) = \text{index}(T_\varphi T_\psi) =
m(\varphi) + m(\psi).
$$
by standard properties of Fredholm operators.  Applying Lemma A2.4 we conclude that
$$
m(\varphi) = k \deg\left(\frac{\varphi}{|\varphi|}, S^1,S^1\right)
$$
for some integer $k$.  Choosing $\varphi(\theta) = e^{i\theta}$ we see
that $k = - 1$.

$\qquad\hfill\qed$

%Append App3 onto Allpart2.tex
%\def\Mint{\diagup\hskip-.50truecm\int}
\bigskip
\noindent
{\bf Appendix 3.  The harmonic extension of VMO maps}
\medskip
In this appendix we discuss properties of the harmonic extension $u$
of a BMO \hbox{( or VMO)} map $\varphi$ defined on the boundary
$\partial\Omega$ of a domain $\Omega\subset\Bbb R^n$; throughout we
assume that $\Omega$ is smooth and bounded.
\medskip
The two main properties which are related to the core of our paper are
the following:
\medskip
\proclaim{Theorem A3.1}  Assume $\varphi$ is a function in
VMO$(\partial\Omega)$.  Then its harmonic extension $u$ belongs to
VMO$_\varphi(\Omega)$.
\endproclaim

\proclaim{Theorem A3.2}  Assume $\varphi\in
\text{VMO}(\partial\Omega,\Bbb R^N)$ and $\varphi(x)\in\Sigma$ a.e. on
$\partial\Omega$, where $\Sigma$ is a closed set in $\Bbb R^N$.  Then,
for any $\delta > 0$ there is a neighbourhood $U$ of $\partial\Omega$
in $\Omega$ such that
$$
\text{\rm dist}(u(x),\Sigma)\leq \delta\qquad\forall x\in U. \tag A3.1
$$
\endproclaim
\medskip
\noindent
{\bf Remark A3.1.}  The two theorems above hold in the general setting
where $\Omega$ is a domain on a manifold; the proofs carry over.
\bigskip

First some notation.  Fix a neighbourhood $V$ of $\partial\Omega$ in
$\Omega$ such that every point $x\in V$ has a unique projection $P(x)$
on $\partial\Omega$.  Set
$$
d(x) = \text{dist}(x,\partial\Omega).
$$
Clearly, there is a constant $C$ such that
$$
C^{-1} (d^2(x) + |P (x) - \xi|^2)\leq |x - \xi|^2\leq C(d^2(x) +
|P(x) - \xi|^2)\quad\forall x\in
V,\quad\forall\xi\in\partial\Omega.  \tag A3.2
$$
Given a function $\varphi$ defined on $\partial\Omega$, consider (as
in \S II.3, Example 3), for $x\in V$,
$$
\overline u(x) = \overline\varphi_{d(x)} (P(x)) = \Mint_{B_{d(x)}
(P(x))} \varphi.
$$
\medskip

The next result provides a useful connection between the harmonic
extension $u$ of $\varphi$ and the function $\overline u$; it will
allow us to derive, easily, Theorems A3.1 and A3.2 from the
corresponding properties of $\overline u$.
\medskip
\proclaim{Lemma A3.1}  There is a constant $C$ such that
$$
\|u - \overline u\|_{L^\infty(V)} \leq C
\|\varphi\|_{\text{BMO}(\partial\Omega)}. \tag A3.3
$$
\endproclaim
\medskip
The proof of Lemma A3.1 relies on the following two lemmas; the first
one is a variant of an observation due to C.~Fefferman and
E.~Stein~[1]:
\medskip
\proclaim{Lemma A3.2}  There is a constant $C$, depending only on $n$,
such that
$$
\int_{y\in B_R} \frac{t|\psi(y) - \overline\psi_t(a)|}{(t^2 + |a -
y|^2)^{n/2}} dy \leq C \|\psi\|_{\text{BMO}(B_{2R})}  \tag A3.4
$$
where $B_R = \{y\in\Bbb R^{n-1}; |y|\leq R\}, a\in B_{R/2}, 0 < t <
R/2$ and
$$
\overline\psi_t(a) = \Mint_{B_t(a)} \psi.
$$
\endproclaim
\medskip

\proclaim{Lemma A3.3}  Let $H$ be a smooth diffeomorphism from $B_R$ onto a
subset of $\partial\Omega$.  Then there are constants $C$ and $t_0$
such that
$$
|(\overline{\varphi\circ H})_t(y) - \overline\varphi_t(H(y))|\leq
C\|\varphi\|_{\text{BMO}(\partial\Omega)} \tag A3.5
$$
for all $\varphi\in\text{BMO}(\partial\Omega)$, $|y| \leq R/2$ and $0 <
t < t_0$.
\endproclaim
\medskip
Assuming Lemmas A3.2 and A3.3 we present the
\medskip
\demo{Proof of Lemma A3.1}  We may suppose that
$\|\varphi\|_{\text{BMO}(\partial\Omega)} = 1$ and
$\int_{\partial\Omega} \varphi = 0$.  Let $P(x,\xi)$ be the Poisson
kernel so that
$$
u(x) = \int_{\partial\Omega} P(x,\xi)\varphi(\xi) d\xi.
$$
Recall (see e.g. M. Avellaneda and F.~H.~Lin~[1], Lemma 21) the
estimate
$$
0\leq P(x,\xi)\leq C \frac{\text{dist}(x,\partial\Omega)}{|x -
\xi|^n}\quad\forall x\in\Omega,\quad\forall
\xi\in\partial\Omega. \tag A3.6
$$
For every constant $c$ we have
$$
|u(x) - c| \leq \int_{\partial\Omega} P(x,\xi) |\varphi(\xi) - c|
d\xi. \tag A3.7
$$
We apply (A3.7) with $c = \overline u(x) = \overline\varphi_{d(x)}
(p(x))$ and set
$$
t = \text{dist}(x,\partial\Omega) = d(x).
$$
From the estimate (A3.6) we obtain
$$
|u(x) - \overline u(x)|\leq Ct\int_{\partial\Omega}
\frac{|\varphi(\xi) - \overline\varphi_t(P(x))|d\xi}{|x -
\xi|^n}.
\tag A3.8
$$
\medskip
Consider a finite family of smooth maps $H_i:
B_{2R}\rightarrow\partial\Omega$ such that each $H_i$ is a diffeomorphism
(onto its image) and
$$
\underset i\to\bigcup\; H_i (B_{R/2})\quad\text{covers
$\partial\Omega$}.
$$
For each $x\in V$ there is some $i$ such that
$$
P(x)\in H_i (B_{R/2}). \tag A3.9
$$
Thus we have
$$
|u(x) - \overline u(x)|\leq Ct\int_{H_i(B_R)} \big[\;\;\;\big] + Ct
\int_{H_i(B_R)^c} \big[\;\;\;\big] = I_1 + I_2, \tag A3.10
$$
where
$$
\big[\;\;\;\big] = \frac{|\varphi(\xi) - \overline\varphi_t(P(x))|}{|x
- \xi|^n}.
$$
To estimate $I_2$ note that, by (A3.2),
$$
|x - \xi|\geq C^{-1/2}|P(x) - \xi|\geq \alpha > 0,
$$
since $\xi\in H_i(B_R)^c$ and $p(x)\in H_i(B_{R/2})$.  Therefore
$$
I_2\leq Ct\left(\|\varphi\|_{L^1(\partial\Omega)} +
|\overline\varphi_t(P(x))|\right)\leq C  \tag A3.11
$$
by Lemmas A.1 and B.7 in [BNI].  We recall that Lemma B.7 implies that\newline
$\|\overline\varphi_t\|_{L^\infty}\leq C(1 + |\log t|)$; the proof of
this fact uses the John-Nirenberg inequality.
\medskip

To estimate $I_1$, use the change of variables $\xi = H_i(y)$, so
that by (A3.2),
$$
I_1 \leq Ct \int_{B_R} \frac{|\varphi(H_i(y)) -
\overline\varphi_t(P(x))|}{(t^2 + |P(x) - H_i(y)|^2)^{n/2}} dy,
$$
and thus
$$
I_1\leq Ct \int_{B_R} \frac{|\psi(y) - \overline\varphi_t(P(x))|}{(t^2
+ |a - y|^2)^{n/2}} dy, \tag A3.12
$$
where $\psi = \varphi\circ H_i$ and $a = H^{-1}_i(P(x))$.
\bigskip
From (A3.12) we deduce that
$$
\spreadlines{4pt}
\aligned
I_1&\leq Ct \int_{B_R} \frac{|\psi(y) - \overline\psi_t(a)| +
|\overline\psi_t(a) - \overline\varphi_t(P(x))|}{(t^2 +
|a-y|^2)^{n/2}} dy\\\vspace{1\jot}
   &\leq C \|\psi\|_{\text{BMO}(B_R)} +
C\|\varphi\|_{\text{BMO}(\partial\Omega)},
\endaligned
\tag A3.13
$$
by Lemmas A3.2 and A3.3.  Note that $|a|\leq R/2$ by (A3.9), and that
we may choose a neighbourhood $V'$ of $\partial\Omega$, $V'\subset V$,
so that, for every $x\in V'$, $t = d(x)\leq \min\{t_0, R/2\}$; here
$t_0$ is defined in Lemma A3.3.

\medskip
In view of Lemma 2 in \S II.1 we obtain
$$
I_1\leq C. \tag A3.14
$$
Combining (A3.11) and (A3.14) we conclude that
$$
|u(x) - \overline u(x)|\leq C\qquad\forall x\in V'.
$$
\medskip
If $x\in V\backslash V'$ we have
$$
|u(x)|\leq C \|\varphi\|_{L^1(\partial\Omega)}\leq C
$$
(since $u$ is harmonic), and clearly
$$
|\overline u(x)|\leq C \|\varphi\|_{L^1(\partial\Omega)}\leq C.
$$
Hence, in all cases,
$$
|u(x) - \overline u(x)|\leq C\qquad\forall x\in V.
$$

$\quad\hfill\qed$

We now return to the
\medskip
\demo{Proof of Lemma A3.2}  By scaling we may assume that $R = 1$.  We
may also suppose that
$$
\|\psi\|_{\text{BMO}(B_{2R})} = 1\quad\text{and that $\int_{B_R} \psi =
0$}.
$$
\medskip
Consider the sequence of balls in $\Bbb R^{n-1}$,
$$
B_R = B_{2^kt}(a)\qquad k = 0,1,2,\ldots
$$
and set
$$
A_k = B_k\backslash B_{k-1}\qquad k = 1,2,3,\ldots
$$
Let $k_0$ be the largest integer $k$ such that
$$
2^kt + |a|\leq 1,
$$
and set
$$
b_k =  \Mint_{B_k} \psi\qquad\text{for $0 \leq k\leq k_0$}.
$$
Note that
$$
b_0 = \overline\psi_t(a).
$$
By Lemma A.4 in [BNI]---recall our definition of BMO$(B_{2R})$--- we have
$$
|b_{k+1} - b_k|\leq C\quad\text{for $0\leq k\leq k_0 - 1$}.
$$
Adding these inequalities yields
$$
|b_k - b_0|\leq Ck\quad\text{for $0\leq k\leq k_0$}. \tag A3.15
$$
On the other hand, note that
$$
\frac14 \leq 2^{k_0} t \leq 1. \tag A3.16
$$
By Lemma A.4 in [BNI] we have
$$
|b_{k_0} - \Mint_{|y|\leq 1} \psi |\leq C
$$
and thus
$$
|b_{k_0}|\leq C
$$
since $\dsize\int_{|y|\leq 1} \psi = 0$.  It follows from (A3.15) and
(A3.16) that
$$
|b_0|\leq C k_0\leq C \log (1/t). \tag A3.17
$$
\medskip
We have to estimate
$$
I = t\int_{|y|\leq 1} \frac{|\psi(y) - b_0|}{(t^2 + |a-y|^2)^{n/2}}
dy.
$$
We write
$$
I = I_1 + I_2 + I_3
$$
where
$$
\spreadlines{3pt}
\align
I_1 &= t\int_{B_0} \frac{|\psi(y) - b_0|}{(t^2 + |a - y|^2)^{n/2}},\\\vspace{1\jot}
I_2 &= \sum^{k_0}_{k=1} t \int_{A_k} \frac{|\psi(y) - b_0|}{(t^2 + |a
- y|^2)^{n/2}} = \sum^{k_0}_{k=1} J_k\\
\intertext{and}
I_3 &= t \operatornamewithlimits{\int}\Sb |y|\leq 1\\y\notin B_{k_0}\endSb  \frac{|\psi(y) -
b_0|}{(t^2 + |a - y|^2)^{n/2}}.
\endalign
$$
Clearly
$$
I_1 \leq \frac{1}{t^{n-1}} \int_{B_0} |\psi(y) - b_0|\leq C
\Mint_{B_0} |\psi(y) - b_0|\leq C. \tag A3.18
$$
\medskip

Next, we estimate $I_3$; observe that if $y\notin B_{k_0}$, $|a -
y|\geq 2^{k_0} t \geq 1/4$, and thus
$$
I_3 \leq C t\int_{|y|\leq 1} |\psi(y) - b_0|\leq Ct \left(\|\psi\|_{\text{BMO}(B_{2R})} + |b_0|\right).
$$
Therefore, by (A3.17),
$$
I_3\leq C t \left(1 + \log(1/t)\right)\leq C. \tag A3.19
$$
\medskip
Finally, we estimate $J_k$.  On $A_k$ we have $|a - y|\geq 2^{(k-1)}
t$ and thus
$$
J_k \leq \frac{t}{(t^2 + 2^{2(k-1)} t^2)^{n/2}} \int_{B_k} |\psi(y) -
b_0|.
$$
Consequently
$$
\align
J_k&\leq \frac{1}{t^{n-1}2^{n(k-1)}} \int_{B_k} (|\psi - b_k| + |b_k -
b_0|)\\
   &\leq \frac{C}{t^{n-1}2^{nk}} |B_k| \left(\Mint_{B_k} |\psi - b_k|
+ |b_k - b_0|\right)\\
   &\leq \frac{C}{2^k} (1 + k)\quad\text{by (A3.15)}.
\endalign
$$
It follows that
$$
I_2 = \sum^{k_0}_{k=1} J_k \leq C. \tag A3.20
$$
Combining (A3.18) - (A3.20) we obtain the desired estimate (A3.4).

$\quad\hfill\qed$
%texed and corrected 11/13
\medskip
Next, we give the
\medskip
\demo{Proof of Lemma A3.3}  For any constant $c$ we have
$$
\align
\big\vert \Mint_{B_t(y)} \varphi(H(\xi)) d\xi - c\big\vert&\leq
\Mint_{B_t(y)} |\varphi(H(\xi)) - c| d\xi\\
                &\leq \frac{C}{|B_t(y)|} \int_{H(B_t(y))} |\varphi (\eta) -
c| d\eta.
\endalign
$$
Choosing
$$
c = \Mint_{H(B_t(y))} \varphi (\zeta) d\zeta,
$$
we find
$$
\spreadlines{3pt}
\aligned
&\big\vert\Mint_{B_t(y)} \varphi (H(\xi))d\xi - \Mint_{H(B_t(y))}
\varphi (\zeta)d\zeta\big\vert\leq\\\vspace{1\jot}
&\leq \frac{C}{|B_t(y)|\,|H(B_t(y))|}
\int\int_{H(B_t(y))} |\varphi(\eta) - \varphi(\zeta)| d\eta
d\zeta. 
\endaligned
\tag A3.21
$$
There are constants $t_0 > 0$ and $K > 1$ such that
$$
B_{t/K}(H(y)) \subset H(B_t(y)) \subset B_{tK}(H(y))\quad\forall t <
t_0,\; |y|\leq R/2. \tag A3.22
$$
We deduce from (A3.21) and (A3.22) that
$$
\spreadlines{4pt}
\aligned
\big\vert\Mint_{B_t(y)} \varphi(H(\xi)) d\xi - \Mint_{H(B_t(y))}
\varphi(\zeta)d\zeta\big\vert&\leq C\Mint\Mint_{B_{tK}(H(y))}
|\varphi(\eta) - \varphi(\zeta)|d\eta d\zeta\\\vspace{1\jot}
             &\leq C \|\varphi\|_{\text{BMO}}.  
\endaligned
\tag A3.23
$$
On the other hand, by Lemma A.4 in [BNI], we have
$$
\big\vert\Mint_{H(B_t(y))} \varphi - \Mint_{B_{tK}(H(y))}
\varphi\big\vert\leq C \|\varphi\|_{\text{BMO}} \tag A3.24
$$
and
$$
\big\vert\Mint_{B_t(H(y))} \varphi - \Mint_{B_{tK}(H(y))}
\varphi\big\vert\leq C \|\varphi\|_{\text{BMO}}. \tag A3.25
$$
\medskip

Combining (A3.23), (A3.24) and (A3.25) we are led to the desired
conclusion
$$
\big\vert\Mint_{B_t(y)} (\varphi\circ H) - \Mint_{B_t(H(y))} \varphi
\big\vert\leq C \|\varphi\|_{\text{BMO}}.
$$
\enddemo
$\quad\hfill\qed$
\medskip

Finally, we turn to the
\medskip
\demo{Proof of Theorem A3.1}  Observe first that if
$\varphi\in\text{BMO}(\partial\Omega)$, then its harmonic extension
$u$ belongs to BMO$(\Omega)$ and
$$
\|u\|_{\text{BMO}(\Omega)} \leq C
\|\varphi\|_{\text{BMO}(\partial\Omega)}. \tag A3.26
$$
In proving (A3.26) we may assume, as usual, that
$\|\varphi\|_{\text{BMO}(\partial\Omega)} = 1$ and that
$\dsize\int_{\partial\Omega} \varphi = 0$.  Let $\zeta$ be a smooth
cutoff function with support in a small neighbourhood of
$\partial\Omega$ and such that $\zeta\equiv 1$ near $\partial\Omega$.
By Lemma A3.1 we have
$$
\|\zeta u - \zeta\overline u \|_{L^\infty(\Omega)} \leq C
$$
and, in particular,
$$
\|\zeta u - \zeta \overline u \|_{\text{BMO}(\Omega)} \leq C.
$$
On the other hand, by (3.8) in Lemma 7 of \S II.3 we have
$$
\|\zeta \overline u\|_{\text{BMO}(\Omega)} \leq C
$$
and therefore
$$
\|\zeta u\|_{\text{BMO}(\Omega)} \leq C.
$$
Since we clearly have
$$
\|(1 - \zeta) u \|_{L^\infty(\Omega)} \leq C,
$$
it follows that (A3.26) holds.  The fact that $u\in\text{VMO}(\Omega)$ whenever
$\varphi\in\text{VMO}(\partial\Omega)$ is derived from (A3.26) by a
standard density argument.
\enddemo
\medskip

Next we prove that if $\varphi\in\text{VMO}(\partial\Omega)$, then
$u\in\text{VMO}_\varphi(\Omega)$.  Since we already know that
$\zeta\overline u\in\text{VMO}_\varphi(\Omega)$ (see Lemma 7 in \S
II.3) it suffices to verify that
$$
(u - \zeta\overline u)\in\text{VMO}_0(\Omega).
$$
Given $\delta > 0$ we have to check (see Theorem 2 in \S II.3) that
$$
\Mint_{B_\varepsilon(x)} |u - \overline u| < \delta\quad\text{for
$\varepsilon = \frac12 d(x)$ small}. \tag A3.27
$$
Let $\psi$ be a continuous function on $\partial\Omega$.  Let $v$ be
its harmonic extension in $\Omega$ and let $\overline v(x) =
\overline\psi_{d(x)} (P(x))$ for $x\in V$.
\medskip
Write
$$
u - \overline u = \big[(u - v) - (\overline u - \overline v)\big] + (v
- \overline v).
$$
Application of Lemma A3.1 to $(\varphi - \psi)$ yields
$$
\|u - \overline u\|_{L^\infty(B_\varepsilon(x))} \leq C \|\varphi -
\psi\|_{\text{BMO}(\partial\Omega)} + \|v - \overline
v\|_{L^\infty(B_\varepsilon(x))} \tag A3.28
$$
provided $\varepsilon < \varepsilon_0$ with $\varepsilon_0$
sufficiently small such that $B_{\varepsilon_0} (x) \subset V$.
\medskip
Choose $\psi\in C^0(\partial\Omega)$ with
$$
C \|\varphi - \psi \|_{\text{BMO}(\partial\Omega)} < \delta/2 \tag
A3.29
$$
and then choose $\varepsilon_1 < \varepsilon_0$ sufficiently small so
that
$$
\|v - \overline v\|_{L^\infty(B_\varepsilon(x))} <
\delta/2\quad\text{for $\varepsilon < \varepsilon_1$}. \tag A3.30
$$
This is clearly possible since $v$ and $\overline v$ are continuous on
$\overline\Omega$ and $v = \overline v = \psi$ on $\partial\Omega$.
Together, (A3.28) - (A3.30) yield
$$
\|u - \overline u \|_{L^\infty(B_\varepsilon(x))} <
\delta\quad\text{for $\varepsilon = \frac12 d(x) < \varepsilon_1$}.
$$
The desired conclusion (A3.27) follows.  $\hfill\qed$
\bigskip
A similar procedure furnishes the
\medskip
\demo{Proof of Theorem A3.2}  As in the proof of Theorem A3.1 we write
$$
u = \overline u + [(u - v) - (\overline u - \overline v)] + (v -
\overline v). \tag A3.31
$$
Recall that, by Lemma A3.1,
$$
\|(u - v) - (\overline u - \overline v) \|_{L^\infty(V)} \leq C
\|\varphi - \psi \|_{\text{BMO}(\partial\Omega)}. \tag A3.32
$$
Fix $\varepsilon_0 > 0$ such that $d(x) < \varepsilon_0$ implies $x\in
V$.  Choose $\psi \in C^0 (\partial\Omega)$ such that
$$
C \|\varphi - \psi\|_{\text{BMO}(\partial\Omega)} < \delta / 3. \tag
A3.33
$$
Next, let $\varepsilon_1 < \varepsilon_0$ be so small that
$$
|v(x) - \overline v(x)| < \delta/3\quad\text{if $d(x) <
\varepsilon_1$}.
$$
Finally, we may find $\varepsilon_2 < \varepsilon_1$ such that
$$
\text{dist}(\overline u(x), \Sigma) < \delta\quad\text{if $d(x) <
\varepsilon_2$};  \tag A3.34
$$
this can be achieved since $\varphi\in\text{VMO}(\partial\Omega)$ (see
(7) and Remark 3 in [BNI]).
\medskip
Combining (A3.31) - (A3.34) we obtain the desired estimate
$$
\text{dist}(u(x), \Sigma) < \delta\qquad\text{if $d(x) <
\varepsilon_2$}.
$$
\enddemo
\enddemo
$\quad\hfill\qed$
\medskip
\noindent
{\bf Remark A3.2.}  Theorem A3.2 asserts that if $\varphi$ takes its
values into some closed set $\Sigma$, the harmonic extension $u$ has
the properties that, close to $\partial\Omega$, the values of $u$ lie
near $\Sigma$.  This need not be true for arbitrary extensions of
$\varphi$ in Sobolev spaces.  For example, with $n = 2$ and
$\varphi\equiv 0$:  If $u\in H^1_0(\Omega)$, near the boundary, $u$
need not be small.

\medskip
Here is such a function $u$ defined on $\Omega = \Bbb R^2_+ =
\{(x_1,x_2), x_2 > 0\}$.  Consider any decreasing sequence
$(\varepsilon_j)$ of positive numbers such that
$$
\sum^\infty_{j=1} \varepsilon_j < \infty
$$
and
$$
\sum^\infty_{j=1} \frac{1}{|\log \varepsilon_j|} < \infty;
$$
for example $\varepsilon_j = e^{-j^2}$ does it.  Let $(a_k)$ be the
sequence of points on the $x_2$-axis defined by
$$
a_k = (0, 2 \sum^\infty_{j=k} \varepsilon_j).
$$
Set
$$
u(x) = \sum^\infty_{j=1} \psi_j \left( |x - a_j|\right)
$$
where $\psi_j(r) = \log |\log r| - \log |\log \varepsilon_j|$ if $r <
\varepsilon_j$ and $\psi_j(r) = 0$ if $r\geq \varepsilon_j$. Note that $\operatorname{supp} u$ is contained in the set
$$
\bigcup^\infty_{j=1} B(a_j,\varepsilon_j),
$$
and that $u\in H^1(\Bbb R^2)$ since
$$
\int_{B(a_j,\varepsilon_j)} |\nabla u|^2 = 2\pi \int^{\varepsilon_j}_0
\frac{r dr}{r^2 |\log r|^2} = \frac{2\pi}{|\log \varepsilon_j|}.
$$
Clearly, $u(a_k) = +\infty\quad\forall k$ and $a_k \rightarrow 0$ as
$k\rightarrow \infty$.  $\quad\hfill\qed$
\enddemo
\bigskip
\noindent
{\bf Acknowledgment}.  The second author was partly supported by
grants ARO-DAAL-03-92-G-0143 and NSF-DMS-94-00912.
\bigskip
\Refs
\bigskip
\ref
\by  M. Avellaneda and F. H. Lin [1]
\paper  Compactness methods in the theory of homogenization
\jour  Comm. Pure Appl. Math.
\vol  40
\yr  1987
\pages  803--847
\endref
\medskip
\ref
\by  H. Brezis and L. Nirenberg [1]
\paper  $(=[BNI])$, Degree theory and BMO; Part I: Compact Manifolds
without Boundaries
\jour  Selecta Mathematica, New Series
\vol  1
\yr  1995
\pages  197--263
\endref
\medskip
\ref
\by  R. G. Douglas [1]
\paper Banach Algebra Techniques in Operator Theory
\jour  Acad. Press
\yr  1972
\endref
\medskip
\ref
\bysame  [2]
\paper  Banach Algebra Techniques in the Theory of Toeplitz Operators
\jour  C.B.M.S., Vol. 15, Amer. Math. Soc. (1973)
\endref
\medskip
\ref
\by  C. Fefferman [1]
\paper  Characterizations of bounded mean oscillation
\jour  Bull. Amer. Math. Soc.
\vol  77
\yr  1971
\pages  587-588
\endref
\medskip
\ref
\by  C. Fefferman and E. Stein [1]
\paper  $H^p$ spaces of several variables
\jour  Acta Math.
\vol  129
\yr  1972
\pages 137--193
\endref
\medskip
\ref
\by  I. Gohberg and N. Krupnik [1]
\book  One Dimensional Linear Singular Integral Equations
\publ Vol. II, Birkh\"auser (1992)
\endref
\medskip
\ref
\by L.~Greco, T.~Iwaniec, C.~Sbordone and B.~Stroffolini~[1]
\paper  Degree formulas for maps with nonintegrable Jacobian
\toappear
\endref
\medskip
\ref
\by  P. Jones [1]
\paper  Extension Theorems for BMO
\jour  Indiana Univ. Math. J.
\vol  29
\yr  1980
\pages  41--66
\endref
\medskip
\ref
\by  S. Lang [1]
\book  Real and Functional Analysis
\publ  Springer (third edition, 1993)
\endref
\medskip
\ref
\by  D. Sarason [1]
\paper  Functions of vanishing mean oscillation
\jour  Trans. AMS
\vol  207
\yr  1975
\pages  391--405
\endref
\medskip
\ref
\bysame [2]
\paper  Toeplitz operators with semi-almost periodic symbols
\jour  Duke Math. J.
\vol  44
\yr  1977
\pages  357--364
\endref
\medskip
\ref
\bysame  [3]
\paper  Toeplitz operators with piecewise quasicontinuous symbols
\jour  Indiana Univ. Math. J.
\vol  26
\yr  1977
\pages  817--838
\endref
\medskip
\ref
\bysame  [4]
\paper  Function Theory on the Unit Circle
\jour  Lecture notes, Virginia Polytech. Inst.
\yr  1978
\endref
\medskip
\ref
\by  E. Stein [1]
\paper  Harmonic Analysis:  Real Variable Methods, Orthogonality and
Oscillatory Integrals
\jour  Princeton Univ. Press
\yr  1993
\endref
\endRefs
\enddocument
\enddocument














