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\centerline{\bf Degree theory:  old and new}
\bigskip
\centerline{Ha\"{\i}m Brezis}
\smallskip
\centerline{Universit\'e P. et M. Curie, Paris, France}
\centerline{and}
\centerline{Rutgers University, New Brunswick, N.J., USA}
\bigskip
\noindent
{\bf 1.  Introduction}
\medskip
The theory of degree has a long history which is a cascade of
successive generalizations.  Presumably, the oldest notion is the
degree of a (smooth) map $u$ from $S^1$ into $S^1$ ($S^1 = $the unit
circle).  The degree of $u$, also called winding number, counts ``how
many times $u$ covers its range taking into account the algebraic
multiplicity.''  More generally, a smooth (say $C^1$) map $u$ from
$S^1$ into $\Bbb C$, such that $u\not= 0$ on $S^1$ has a degree which
may be computed through the very classical integral formula
$$
\deg u = \frac{1}{2\pi i} \int_{S^1} \frac{\dot u}{u} \tag 1.1
$$
which measures the ``algebraic change of phase'' of $u$ as the
variable goes around $S^1$ once.  Similarly, if $\Gamma$ is a simple
curve in $\Bbb R^2$ and $u$ is a smooth map from $\Gamma$ into $S^1$,
then its degree can be computed as
$$
\deg (u,\Gamma) = \frac{1}{2\pi} \int_\Gamma u \times u_\tau, \tag 1.2
$$
where $\times$ denotes the cross product of vectors in $\Bbb R^2$ (here
$S^1$ is viewed as a subset of $\Bbb R^2$, not $\Bbb C$) and $u_\tau$
denotes the tangential derivative of $u$ along $\Gamma$. 
\medskip
Starting at the end of the 19th century people realized that the notion
of degree also makes sense in higher dimensions.  To simplify the
presentation I will consider only (smooth) maps $u$ from $S^n$ into
$S^n$ ($= n$ dimensional unit sphere in $\Bbb R^{n+1}$), but the
theory extends to maps $u : X\rightarrow Y$ where $X$ and $Y$ are
smooth $n$-dimensional oriented manifolds without boundary.
\medskip
Here is the precise definition of degree.  Fix any $y\in S^n$ which is
a regular value of $u$, i.e.,
$$
\varphi^{-1} (y) = \{x_1, x_2, \ldots, x_p\}
$$
is a finite set and, for
each $i$, the Jacobian determinant $\text{det} J_u(x_i)\not= 0$.
(Recall that, by Sard's theorem, almost every $y$ is a regular value.)
The degree of $u$ is by definition the number of solutions of the
equation $u(x) = y$ taking into account their algebraic multiplicity:
$$
\deg u = \sum^p_{i=1} \text{sign det} J_u (x_i). \tag 1.3
$$
In principle this number depends on the choice of the regular value
$y$.  A {\bf remarkable property} is that $\deg u$ defined above is
{\bf independent of $y$}, so that one may talk about $\deg u$ without
specifying $y$.
\medskip
A very important representation formula, which is the $n$-dimensional
analogue of (1.1) or (1.2) allows to compute the degree as an integral
of the Jacobian determinant:
$$
\deg u = \frac{1}{|S^n|} \int_{S^n} \text{det} J_u \tag 1.4
$$
where the Jacobian determinant is computed using geodesic normal
coordinates (both in the domain and the range) and $|S^n|$ denotes the
measure (length, area, volume, etc...) of $S^n$.  For the proof of
(1.4) see e.g. L.~Nirenberg~[1] or H.~Brezis and L.~Nirenberg~[3].  It
is sometimes convenient to observe that
$$
\det J_u = \det (u,u_{x_1}, u_{x_2}, \ldots, u_{x_n}) \tag 1.5
$$
(recall that $u$ takes its values in $\Bbb R^{n+1}$ and $\det$ on the
righthand side of (1.5) refers to the determinant of an $(n+1) \times
(n+1)$ matrix); this follows easily from the fact that $|u|^2\equiv 1$
and thus $u\cdot u_{x_i} \equiv 0$ for every $i$.  (For example (1.2)
corresponds to the form (1.5)).
\medskip
There is a ``cousin'' of formula (1.4) where the ``surface'' integral
in (1.4) is replaced by a ``{\bf volume}'' integral in the unit ball
$B^{n+1}$ of $\Bbb R^{n+1}$.  Let $\widetilde u$ be any {\bf
extension} of $u$ to $B^{n+1}$ with values into $\Bbb R^{n+1}$, then
$$
\deg u = \frac{1}{|B^{n+1}|} \int_{B^{n+1}} \det J_{\widetilde
u}. \tag 1.6
$$
(Note that, in (1.6), $\det$ refers to the determinant of an
$(n+1)\times (n+1)$ matrix).  It is easy to pass from (1.5) to (1.6)
by writing $\det J_{\widetilde u}$ in a divergence form and then
integrating by parts (see e.g. H.~Brezis and L.~Nirenberg~[3]).  For
example, when $n = 1$ write
$$
\det J_{\widetilde u} = \widetilde u_x \times \widetilde u_y =
\frac12 \big[ (\widetilde u\times \widetilde u_y)_{x} + (\widetilde
u_x \times \widetilde u)_y\big].
$$
Green's formula allows us to replace the integral over $B^2$ by an
integral over $S^1 $ which coincides with (1.2).
\medskip
The next major step in degree theory came at the beginning of the 20th
century, especially through the work of Brouwer.  It was then realized
that the $C^1$ assumption about $u$ used either in (1.3) or (1.4) is
{\bf not} necessary to define a degree.  Continuity suffices.  The key
observation is the following
\bigskip
\proclaim{Lemma 1}  Assume $u, v\in C^1 (S^n, S^n)$ satisfy
$$
\|u - v\|_{L^\infty} < 1,
$$
then
$$
\deg u = \deg v.
$$
\endproclaim

Using Lemma 1 we may now define $\deg u$ for a general map $u\in
C^0(S^n, S^n)$.  Clearly, there is a sequence $(u_j)$ of $C^1$ maps
from $S^n$ to $S^n$ such that $u_j\rightarrow u$ uniformly.  Hence
$\|u_j - u_k\|_{L^\infty} < 1\quad\forall j, k\geq N$.  By definition
we let
$$
\deg u = \deg u_j\quad\text{for  $j \geq N$}.
$$
\medskip
In this manner every map $u\in C^0(S^n, S^n)$ has a well defined
degree which belongs to $\Bbb Z$.  Moreover the degree is {\bf stable}
(i.e., unchanged) under small $C^0$ perturbation:
\medskip
\noindent
{\bf Property 1.}  If $u, v\in C^0(S^n,S^n)$ are such that
$$
\| u - v\|_{L^\infty} < 1
$$
then
$$
\deg u = \deg v.
$$
\medskip
As a consequence the degree is constant under homotopy, i.e., if
$H(x,t)\in C^0(S^n \times [0,1], S^n)$ then
$$
\deg (H(\cdot\;,0)) = \deg (H(\cdot\;, 1)).
$$
\medskip
Let us summarize the main properties of the degree (they were
discovered during the first part of this century):
\medskip
\noindent
{\bf Property 2.}  If $u\in C^0(S^n, S^n)$ is such that
$$
\deg u \not= 0
$$
then
$$
u \quad\text{  maps  }\quad S^n\quad\text{onto}\quad S^n.
$$
\medskip
\noindent
{\bf Property 3 (Borsuk).}  If $u\in C^0(S^n,S^n)$ is odd, then
$$
\deg u\qquad\text{  is odd}.
$$
\medskip
\noindent
{\bf Property 4 (Hopf).}  If $u,v\in C^0(S^n,S^n)$ are such that
$$
\deg u = \deg v
$$
then there is a homotopy $H(x,t)$ (as in Property 1) connecting $u$
and $v$.
\medskip

A variant of the above degree theory has also been developed for maps
$u : \overline{\Omega} \rightarrow \Bbb R^n$ where $\Omega$ is a
bounded domain in $\Bbb R^n$.  Given a point $y\in\Bbb R^n$ such that
$$
y\notin u(\partial\Omega) \tag 1.7
$$
one defines $\deg (u,\Omega, y)$ provided $u\in C^0
(\overline\Omega,\Bbb R^n)$.
\medskip

The strategy is the same as above; namely one starts with a {\bf
smooth map} $u$ and a {\bf regular value} $y$.  The $\deg (u,\Omega,
y)$ is independent of $y$ provided $y$ stays in a connected component
of $\Bbb R^n\backslash u(\partial\Omega)$.  This allows to define
$\deg (u,\Omega,y)$ for any $y$ satisfying (1.7).  Next one defines
$\deg (u,\Omega,y)$ for any continuous map $u$ by a variant of Lemma
1.  The two notions above are closely connected; for example if
$\Omega = B^n$ the unit ball of $\Bbb R^n$ and $y\notin
u(\partial\Omega) = u(S^{n-1})$, then
$$
\deg (u,\Omega, y) = \deg \left(\frac{u - y}{|u - y|},\; S^{n-1}\right)
\tag 1.8
$$
where the degree on the righthand side of (1.8) refers to the degree
of a map from $S^{n-1}$ to $S^{n-1}$; see e.g. H.~Brezis and
L.~Nirenberg~[3].
\bigskip

A very important extension of degree theory to {\bf infinite
dimensional spaces} was discovered by J.~Leray and J.~Schauder~[1] in
the thirties.  It requires continuity and some kind of compactness.
It has many applications, in particular, in the study of nonlinear
partial differential equations (see e.g. H.~Brezis and
L.~Nirenberg~[3]).
\medskip
In what follows I propose to describe some recent extensions of degree
theory to a class of maps in finite dimensional spaces, which are
possibly {\bf discontinuous}.  This was first done for Sobolev maps,
in some limiting cases of the Sobolev imbedding.
\bigskip
\noindent
{\bf 2.  Degree theory for maps in the Sobolev class $H^1(S^2,S^2)$}
\medskip
In 1982 I was working with J.~M.~Coron on a problem raised by
M.~Giaquinta and S.~Hildebrandt~[1] concerning harmonic maps.  Let
$\Omega$ be unit disc in $\Bbb R^2$ and consider maps $u :
\Omega\rightarrow \Bbb R^3$ satisfying the system
$$
\cases  - \Delta u = u |\nabla u|^2&\qquad\text{in $\Omega$}\\
             \hphantom{-\delta } |u| = 1&\qquad\text{in $\Omega$}\\
                \hphantom{-\Delta }         u = g&\qquad\text{on $\partial\Omega$}. \endcases
\tag 2.1
$$
\medskip

Solutions of (2.1) correspond to critical points of the functional
$\int_\Omega |\nabla u|^2$ subject to the constraint
$$
u\in H^1_g (\Omega,S^2) = \{u \in H^1(\Omega,\Bbb R^3), |u| =
1\quad\text{a.e. in $\Omega$ and $u = g$ on $\partial\Omega$}\}
$$
where $H^1$ refers to the usual Sobolev space, $g :
\partial\Omega\rightarrow S^2$ is a given (smooth) map.  It is easy to
see that (2.1) has at least one solution, namely by considering
$$
\operatornamewithlimits{Min}_{u\in H^1_g(\Omega, S^2)} \;\int |\nabla
u|^2.
$$
\medskip
\noindent
Call such a minimizer $u_0$.  If $g = C$ is a constant then $u_0 = C$
is the only solution of (2.1).  The question of M.~Giaquinta and
S.~Hildebrandt was whether (2.1) has at least 2 solutions whenever
$g\not\equiv$ Const.  In support of their conjecture they considered
the special case
$$
g(x,y) = (Rx, Ry, (1 - R^2)^{1/2})\quad\text{with $0 < R < 1$}. \tag
2.2
$$
In this case one may write down explicitly two solutions of (2.1),
namely
$$
\underline{u} (x,y) = \frac{2\lambda}{\lambda^2 + r^2} (x,y,\lambda) +
(0,0, -1)
$$
and
$$
\overline u (x,y) = \frac{2\mu}{\mu^2 + r^2} (x,y, - \mu) + (0,0,1)
$$
where $r^2 = x^2 + y^2$, $\lambda = \dfrac{1}{R} + (\dfrac{1}{R^2} -
1)^{1/ 2}$ and $\mu = \dfrac1R - (\dfrac{1}{R^2} -
1)^{1/ 2}$.
\medskip
(Note that $\underline{u}$ and $\overline u$ are simply rescaled
stereographic projections from the north and south pole respectively.)
\medskip
We did answer positively the question of M.~Giaquinta and
S.~Hildebrandt:
\medskip
\proclaim{Theorem (see H.~Brezis and J.~M.~Coron~[1] and also
J.~Jost~[1])}  If $g\not\equiv$ Const, then the system (2.1) has at
least two solutions.
\endproclaim

The starting point in our proof is the observation that the space 
$H^1_g(\Omega, S^2)$ is {\bf not} connected.  In fact, it has
infinitely many connected components and they are classified using
degree theory.  Unfortunately, we {\bf cannot} use the classical degree
theory because maps in $H^1$ need {\bf not} be continuous in 2
dimensions.
\medskip
We were first led with J.~M.~Coron to investigate the class of maps
$\varphi\in H^1(S^2,S^2)$ and try to define their degree.  The natural
strategy is to consider the integral in (1.4) namely
$$
\frac{1}{4\pi} \int_{S^2} \det J_\varphi. \tag 2.3
$$
Note that the integral is well defined because $\varphi\in H^1$
implies $\det J_\varphi \in L^1$ (recall that $\det$ refers to the
determinant of a $2 \times 2$ matrix).  To have an interesting degree
we would like to know that the quantity in (2.3) is an integer (in
$\Bbb Z$).  This is a direct consequence of the following:
\medskip
\proclaim{Lemma 2 (R. Schoen and K. Uhlenbeck~[1])}  Given any
$\varphi\in H^1(S^2,S^2)$ there is a sequence $(\varphi_j)$ in
$C^1(S^2,S^2)$ such that $\varphi_j\rightarrow\varphi$ in $H^1$.
\endproclaim

I would like to sketch the proof, because it is quite interesting and
plays an important role in Section 4.  Consider a smoothing
process, by convolution or just by averaging for simplicity, say
$$
\overline\varphi_\varepsilon (x) = \frac{1}{|B_\varepsilon(x)|}
\int_{B_\varepsilon(x)} \varphi(y) dy
$$
where $B_\varepsilon(x)$ is a geodesic disc on $S^2$ of radius
$\varepsilon$ centered at $x$.  Clearly
$|\overline\varphi_\varepsilon|\leq 1$, but
$\overline\varphi_\varepsilon$, in general, does not take its values in
$S^2$.  {\bf If} we happen to know that $\varphi$ is {\bf also}
continuous then $\overline\varphi_\varepsilon\rightarrow\varphi$
uniformly, as $\varepsilon\rightarrow 0$, and then we may consider
$$
\varphi_\varepsilon = \overline\varphi_\varepsilon /
|\overline\varphi_\varepsilon| \tag 2.4
$$
which has all the required properties.  Unfortunately, if $\varphi\in
H^1$ only, then $\varphi$ need {\bf not} be continuous, and (2.4) does
not even make sense because (in principle)
$\overline\varphi_\varepsilon$ could vanish.  The key observation is
that $\overline\varphi_\varepsilon$ does {\bf not} vanish (for
$\varepsilon$ small) and in fact we have
$$
|\overline\varphi_\varepsilon(x)|\longrightarrow
1\qquad\text{\bf uniformly on $S^2$}. \tag 2.5
$$
The proof of (2.5) relies on Poincar\'e's inequality
$$
\int_{B_\varepsilon(x)} |\varphi(y) - \overline\varphi_\varepsilon(x)
| dy \leq C |B_\varepsilon(x)|^{1/ 2} \int_{B_\varepsilon(x)}
|\nabla\varphi| \tag 2.6
$$
and thus
$$
\frac{1}{|B_\varepsilon(x)|} \int_{B_\varepsilon(x)} |\varphi(y) -
\overline\varphi_\varepsilon(x)| dy \leq C\bigg[
\int_{B_\varepsilon(x)} |\nabla\varphi|^2\bigg] ^{1/ 2}. \tag 2.7
$$
Note that, for every $y$,
$$
|\varphi(y) - \overline\varphi_\varepsilon(x)|\geq
\text{dist}(\overline\varphi_\varepsilon(x), S^2) = 1 -
|\overline\varphi_\varepsilon(x)|
$$
and thus
$$
\text{dist}(\overline\varphi_\varepsilon(x), S^2) \leq C
\bigg[\int_{B_\varepsilon(x)} |\nabla\varphi|^2\bigg]^{1/2}\rightarrow
0\quad\text{uniformly as $\varepsilon\to 0$}.
$$
In particular, $|\overline\varphi_\varepsilon (x)|\rightarrow 1$
{\bf uniformly} and then it is not difficult to prove that
$\varphi_\varepsilon = \overline\varphi_\varepsilon /
|\overline\varphi_\varepsilon|$ converges to $\varphi$ in $H^1$.
\bigskip
\noindent
{\bf Remark 1.}  One may ask a more general question:  Is $C^1(M,N)$
dense in the Sobolev space $W^{1,p} (M,N)$ where $M$ and $N$ are
compact manifolds ($N$ has no boundary but $M$ may have a
boundary). If $p \geq \dim M$, the answer is positive (for every $N$ )
and the proof is the same as above.  If $p < \dim M$, a deep result of
F.~Bethuel~[1] asserts that there is density if and only if
$\Pi_{[p]}(N) = 0$.
\medskip
Let me now explain briefly how the $H^1$ degree is used to decompose
$H^1_g(\Omega, S^2)$ into its components.  Fix a ``reference'' map,
for example $u_0$ (the absolute minimizer).  Given any $u\in
H^1_g(\Omega,S^2)$, we ``glue'' the two maps $u$ and $u_0$ together
and define a map $\varphi\in H^1(S^2,S^2)$ by $\varphi = (u, u_0)$
(one copy of $\Omega$ is identified with $S^2_+$, the upper
half-hemisphere, and the other copy of $\Omega$ is identified with
$S^2_{-}$).  One may then write
$$
H^1_g(\Omega,S^2) = \bigcup^{+\infty}_{k = - \infty} \Cal E_k
$$
where $\Cal E_k = \{\varphi = (u,u_0);\; \deg\varphi = k\}$.  These
are the connected components of $H^1_g(\Omega, S^2)$.  It is then
natural to try to minimize the energy $\Omega$ on each $\Cal E_k$:
$$
\operatornamewithlimits{Inf}_{u\in\Cal E_k} \;\int_\Omega |\nabla
u|^2. \tag 2.8
$$
It is not clear at all that this infimum is achieved (because a
minimizing sequence converges weakly in $H^1$; the classes $\Cal E_k$
are closed for the strong $H^1$ topology, but {\bf not} for the weak
$H^1$ topology).  Using a delicate analysis we were able to prove that the
infimum in (2.8) is achieved at least when $k = + 1$ or $k = - 1$.  For
more details see H.~Brezis and J.~M.~Coron~[1], H.~Brezis~[1],[2].
\medskip

\noindent
{\bf Remark 2.}  In the case of the {\bf special} boundary condition
$g$ given by (2.2) one can prove that the infimum in (2.8) is achieved
only when $k = 0$ and $k = -1$ (and the corresponding minimizers are
given by $\underline u$ and $\overline u$).  It is a beautiful {\bf
open problem} to determine whether, for this $g$, $\underline u$ and
$\overline u$ are the {\bf only} solutions of (2.1).
\medskip
The method described above for $H^1(S^2,S^2)$ easily extends to
$W^{1,n}(S^n,S^n)$ and allows to define the degree of any map
$\varphi\in W^{1,n}(S^n,S^n)$.  It is given by the formula
$$
\deg \varphi = \frac{1}{|S^n|} \int_{S^n} \det J_\varphi.
$$
The fact that $\deg \varphi \in\Bbb Z$ is proved as above using the
property that $C^1(S^n,S^n)$ is dense in $W^{1,n}(S^n,S^n)$.  Recall
that $W^{1,n}(S^n,S^n)$ is {\bf not} contained in $C^0(S^n,S
^n)$.  This is again a limiting case for the Sobolev imbedding.
\bigskip

\noindent
{\bf 3.  Degree theory for maps in the Sobolev class
$H^{1/2}(S^1,S^1)$}
\medskip
In 1985 L. Boutet de Monvel and O. Gabber observed that maps in the
Sobolev class $H^{1/2}(S^1,S^1)$ have a well-defined degree.  (Note
that the space $H^{1/2}$ in one dimension is {\bf not} contained in
$C^0$.  Once more this is a limiting case for the Sobolev imbedding!)
Their motivation came from the Ginzburg-Landau model and their
argument is presented as an Appendix in A. Boutet de Monvel-Berthier,
V. Georgescu and R. Purice [1] (another application of the $H^{1/2}$
degree, also connected to the Ginzburg-Landau theory is presented in
F. Bethuel, H. Brezis and F. H\'elein [1]).
\medskip
The original argument of L. Boutet de Monvel and O. Gabber was the
following.  If $\varphi\in C^1(S^1,S^1)$, then
$$
\deg \varphi = \frac{1}{2\pi i} \int_{S^1} \frac{\dot\varphi}{\varphi}
= \frac{1}{2\pi i} \int_{S^1} \overline{\varphi} \dot\varphi. \tag 3.1
$$
However, this last integral makes sense if one merely assumes that
$\varphi\in H^{1/2}$ because $\dot\varphi\in H^{-1/2}$ and the
integral viewed as a scalar product in the duality between $H^{- 1/2}$ and
$H^{1/2}$ has a meaning.  One may wonder whether the resulting number
is an integer.  The answer is again positive because $C^1(S^1,S^1)$ is
dense in $H^{1/2} (S^1,S^1)$; the argument is essentially the same as
in the proof of Lemma 2, except that here one uses the fact that
$$
\int_{S^1}\,\int_{S^1} \frac{|\varphi(x) - \varphi(y)|^2}{|x - y|^2} dx dy <
\infty \tag 3.2
$$
to deduce that $|\overline\varphi_\varepsilon(x)|\rightarrow 1$ uniformly (see
Section 4).  Recall that (3.2) is one of the definitions of the space
$H^{1/2}$ (see e.g. R. A. Adams [1]).
\medskip
There are two other approaches which lead to the fact that maps in
$H^{1/2}(S^1,S^1)$ have a degree---each one with a different flavor.
\medskip
First, via the {\bf Fourier series}.  This grew out of a question of
I. M. Gelfand.  Given a complex-valued function $\varphi$ on $S^1$ let
$(a_j)$ denote its Fourier coefficients.  Then $\varphi\in
H^{1/2}(S^1,\Bbb C)$ if and only if
$$
\|\varphi\|^2_{H^{1/2}} = \sum^{+\infty}_{j=-\infty} |j|\;|a_j|^2 <
\infty. \tag 3.3
$$
On the other hand, if $\varphi\in C^1(S^1,S^1)$, its degree, given by
(3.1) takes the form
$$
\deg \varphi = \sum^{+\infty}_{j = -\infty} j |a_j|^2. \tag 3.4
$$
The fact that the sum of the series in (3.4) belongs to $\Bbb Z$ for
{\bf any} $\varphi\in H^{1/2}(S^1,S^1)$ is established, as above, via a
density argument. It would be good to have a direct and
simple proof of that property.  More precisely, if $(a_j)$ is a
sequence of complex numbers satisfying (3.3),
$$
\sum^{+\infty}_{j=-\infty} |a_j|^2 = 1 \tag 3.5
$$
and
$$
\sum^{+\infty}_{j=-\infty} a_j \overline{a}_{j+k} = 0\quad\forall
k\not= 0, \tag 3.6
$$
then
$$
\sum^{+\infty}_{j=-\infty} j |a_j|^2 \in \Bbb Z.
$$
Note that (3.5) and (3.6) correspond to the fact that the map
$$
\varphi(\theta) = \sum^{+\infty}_{j = -\infty} a_j e^{ij\theta}
$$
takes its values in $S^1$.  (Indeed $|\varphi|\equiv 1$ is equivalent to the
property that
$$
\int_{S^1} |\varphi|^2 e^{ik\theta} = 0\quad\forall k\not=0\text{   and
$\int_{S^1} |\varphi|^2 = 2\pi$},
$$
which may be written as (3.5) and (3.6).)
\medskip
In view of the formula (3.4) it is now quite natural that maps in
$H^{1/2}$ have a degree.  But it is far from obvious that maps in $C^0$
have a degree!  A general map $\varphi$ in $C^0(S^1,S^1)$ need not
belong to $H^{1/2}(S^1,S^1)$ and thus the series
$$
\sum^{+\infty}_{j = -\infty} |j|\;|a_j|^2
$$
may be divergent.  If this happens
$$
\deg\varphi = \sum^{+\infty}_{j= +1} j |a_j|^2 + \sum^{- 1}_{j=-\infty}
j |a_j|^2 = +\infty - \infty
$$
has no clear meaning.  Since $\deg\varphi$ makes sense for
$\varphi\in C^0(S^1,S^1)$, there must be some kind of cancellation of the two
infinite quantities, leaving us with a ``principal value''.  In
fact it would be very interesting to understand what summation process
(if any) may be used to compute
$$
\sum^{+\infty}_{j=-\infty} j |a_j|^2
$$
for a general $\varphi\in C^0(S^1,S^1)$.  In particular can one use
any of the standard methods, for example,
$$
\operatornamewithlimits{\lim}_{n\rightarrow\infty} \sum^{+n}_{j = - n}
j |a_j|^2
$$
or
$$
\operatornamewithlimits{\lim}_{r\uparrow 1} \sum^{+\infty}_{j =
-\infty} j |a_j|^2 r^{|j|}\quad ?
$$
\medskip
Here is still another approach which shows that maps in $H^{1/2}$ have
a degree.  It relies on the characterization of $H^{1/2}$ as {\bf
trace} space for $H^1$.  More precisely, given some $\varphi\in
H^{1/2}(S^1,S^1)$ we consider it as a map in $H^{1/2}(S^1,\Bbb C)$ and
then we may extend it to the unit disc $B^2$ in $\Bbb R^2$ by a map
$u\in H^1(B^2,\Bbb C)$.  We then recall the formula (1.6) which here
takes the form
$$
\deg\varphi = \frac{1}{\pi} \int_{B^2} \det J_u. \tag 3.7
$$
Formula (3.7) holds for smooth maps.  But, again, we observe that the
right hand side in (3.7) makes sense provided $u\in H^1$ and to have
such $u$ it suffices to assume that $\varphi\in H^{1/2}$.  Finally, a
density argument, as above, shows that the integral on the right hand
side of (3.7) belongs to $\Bbb Z$ for any $\varphi\in
H^{1/2}(S^1,S^1)$.
\medskip
This last approach also works in higher dimensions.  Suppose
$\varphi\in W^{s,p}(S^n,S^n)$ for some fractional Sobolev space.  We
may extend $\varphi$ inside the unit ball $B^{n+1}$ by some $u\in
W^{s+1/p,p}(B^{n+1}, \Bbb R^{n+1})$ and then use the formula (1.6)
$$
\deg \varphi = \frac{1}{|B^{n+1}|} \int_{B^{n+1}} \det J_u. \tag 3.8
$$
The integral on the right hand side of (3.8) makes sense when $u\in
W^{1,n+1}$ and so we may take $p = n+1\;\; s = 1 - \dfrac1p =
\dfrac{n}{n+1}$.  We now reach the conclusion that maps $\varphi\in
W^{\frac{n}{n+1}, n+1} (S^n, S^n)$ have a degree (in $\Bbb Z$).  This
class falls again in the category of the limiting Sobolev exponent and
it is slightly larger than $W^{1,n}(S^n,S^n)$ (via the fractional
Sobolev imbedding).  For example when $n=2$, there is a well-defined
degree for maps $\varphi\in W^{\frac23, 3} (S^2, S^2)$; this class is a
little bigger than the class $H^1(S^2,S^2)$ considered in Section 2.
\medskip
At this stage the situation was becoming rather confusing and we
decided, with Louis Nirenberg, to investigate a suggestion of
L. Boutet de Monvel and O. Gabber, namely, to define a degree for VMO
maps.  Such a degree is {\bf not defined via an integral formula but
rather via approximation} (in the same manner as one extends degree
theory from $C^1$ to $C^0$).  As we shall see in the next Section such
a class includes $C^0$ maps as well as {\bf all} {\bf Sobolev maps} in
the limiting case of the Sobolev exponent.
\bigskip
\noindent
{\bf 4.  Degree theory for maps in} VMO$(S^n,S^n)$
\medskip
Here, and throughout the rest of this paper we present our recent work
with Louis Nirenberg; see H.~Brezis and L.~Nirenberg~[1],[2].
\medskip
Let us first recall the definition of BMO; this is a celebrated space
introduced by F.~John and L.~Nirenberg~[1].
\medskip
An integrable function $f : S^n\rightarrow\Bbb R$ belongs to BMO if
$$
\|f\|_{\text{BMO}} = \operatornamewithlimits{Sup}_{B\subset S^n}
\Mint_B |f - \Mint_B f| < \infty \tag 4.1
$$
where the Sup is taken over all geodesic balls on $B$ on $S^n$, with radius
$r \leq 1$.  Formula (4.1) defines a semi-norm or a norm on BMO modulo
constants.  BMO is complete under this norm.  A {\bf very useful}
equivalent norm is given by
$$
\|f\|_\ast = \operatornamewithlimits{Sup}_{B\subset S^n} \Mint_B
\Mint_B |f(y) - f(z)| dy dz. \tag 4.2
$$
In fact its easy to check that
$$
\|f\|_{\text{BMO}} \leq \|f\|_\ast \leq 2 \|f\|_{\text{BMO}}. \tag 4.3
$$
Sometimes it is convenient to take the Sup in (4.1) (or (4.2)) over
all balls with radius $r \leq r_0$; this yields a norm which is
equivalent to the original BMO norm.
\medskip
Clearly,
$$
L^\infty \subset \text{BMO}
$$
and
$$
\|f\|_{\text{BMO}} \leq 2 \|f\|_{L^\infty}.
$$
But the converse is not true:  the well-known example is the log
function.  More precisely, fix a point $x_0$ on $S^n$ then
$$
f(x) = \zeta(x) \log |x - x_0|\in \text{BMO} \tag 4.4
$$
where $\zeta$ is a smooth cut-off function supported near $x_0$.  An
important inequality, due to F.~John and L.~Nirenberg~[1] asserts that
$$
\text{BMO} \subset L^p\quad\forall 1 \leq p < \infty.
$$
More precisely,
$$
\|f - \Mint_{S^n} f \|_{L^p}\leq C \|f\|_{\text{BMO}} \tag 4.5
$$
where $C$ depends only on $p$ and $n$.  In fact, a sharper form
asserts that $e^{c|f|}\in L^1$ whenever $\|f\|_{\text{BMO}} \leq 1$,
where $c$ depends only on $n$.
\medskip
It turns out that $C^0(S^n)$ is {\bf not} dense in BMO$(S^n)$.  Since
we plan to define the degree via approximation by smooth maps, it is
essential to deal with maps which can be regularized.  Hence we will
work with
$$
\text{VMO}(S^n) = \text{the closure of $C^0(S^n)$ in BMO$(S^n)$}
$$
i.e., a function $f\in\text{BMO}(S^n)$ belongs to VMO$(S^n)$ if there
is a sequence $(f_j)$ in $C^0(S^n)$ such that $\|f_j -
f\|_{\text{BMO}}\rightarrow 0$; without loss of generality, using
(4.5), we may also assume that $f_j\rightarrow f$ in $L^p\quad\forall
p < \infty$ and $f_j\rightarrow f$ a.e.
\medskip
The space VMO (for functions on $\Bbb R^n$) has been introduced by
D.~Sarason~[1] who also established a useful characterization.
\medskip
\proclaim{Lemma 3}  A function of $f\in\text{BMO}(S^n)$ belongs to
VMO$(S^n)$ if and only if
$$
\operatornamewithlimits{\lim}_{|B|\rightarrow 0} \Mint_B \big\vert f - \Mint_B
f\big\vert = 0 \tag 4.6
$$
or equivalently
$$
\operatornamewithlimits{\lim}_{|B|\rightarrow 0} \Mint_B \Mint_B |f(y)
- f(z)| dy dz = 0. \tag 4.7
$$
\endproclaim

The fact that $f\in\text{VMO}(S^n)\Rightarrow$ (4.6) is easy.  Indeed
we may write, for any function $g$,
$$
\Mint_B \big\vert f - \Mint_B f \big\vert \leq \|f - g\|_{\text{BMO}}
+ \Mint_B \big\vert g -
\Mint_B g\big\vert.
$$
Given $\varepsilon$ we may choose a continuous function $g$ such that
$\|f - g\|_{\text{BMO}} < \varepsilon$.  Then we may find a $\delta >
0$ such that
$$
\Mint_B \big\vert g - \Mint_B g\big\vert < \varepsilon\quad\forall B\text{ \rm with  } |B| < \delta.
$$
The converse (i.e., (4.6) $\Rightarrow f \in$ VMO) is more delicate
(see D.~Sarason~[1] or H.~Brezis and L.~Nirenberg~[1]).

\medskip
\noindent
{\bf Examples:}
\medskip
\noindent
1)  The function $f$ in (4.4) does {\bf not} belong to VMO$(S^n)$.
However $|f|^\alpha \in \text{VMO}(S^n)$ for any $0 < \alpha < 1$.
Hence, there are unbounded functions in VMO.  Similarly, the function
$\zeta(x) \log \big\vert\log|x - x_0|\big\vert$ belongs to VMO$(S^n)$.
\medskip
\noindent
2)  The Sobolev space $W^{1,n}(S^n)\subset\text{VMO}(S^n)$.  This
follows easily from the Poincar\'e inequality
$$
\int_B \big\vert f - \Mint_B f\big\vert \leq C |B|^{1/n} \int_B |\nabla f| \tag 4.8
$$
where $C$ depends only on $n$.  From (4.8) we deduce that
$$
\Mint_B \big\vert f - \Mint_B f \big\vert \leq C \left(\int_B |\nabla f|^n\right)^{1/n}
\tag 4.9
$$
and we may then apply Lemma 3 to infer that $f\in \text{VMO}$.
\medskip
\noindent
3)  More generally, functions in the fractional Sobolev space
$W^{s,p}(S^n)$ with $0 < s < n$ and $sp = n$ belong to VMO$(S^n)$
(note that the condition $sp = n$ is limiting for the Sobolev
imbedding).  To prove this, it suffices to consider the case $0 < s <
1$ (when $s\geq 1$, $W^{s,p}\subset W^{1,n}$ and we are reduced to the
previous example).
\medskip
Recall (see e.g. R.~Adams~[1]) that a function $f$ belongs to
$W^{s,p}$ provided
$$
\|f\|^p_{W^{s,p}} = \int_{S^n} \int_{S^n} \frac{|f(y) - f(z)|^p}{|y -
z|^{sp+n}} dy dz < \infty.
$$
We have, by H\"older,
$$
\align
\Mint_B \Mint_B |f(y) - f(z)|&\leq \frac{1}{|B|^2} \left(\int_B \int_B
|f(y) - f(z)|^p\right)^{1/p} |B|^{2/p'}\\
  &\leq \frac{1}{|B|^{2/p}} \left(\int_B \int_B \frac{|f(y) -
f(z)|^p}{|y - z|^{2n}} \right)^{1/p} (2r)^{2n/p}\\
    &\leq C \left(\int_B \int_B \frac{|f(y) - f(z)|^p}{|y -
z|^{sp+n}}\right)^{1/p},
\endalign
$$
(since $|y - z|\leq 2r$, for $y,z\in B =$ a ball of radius $r$).
Applying Lemma 3 once more we conclude that $f\in\text{VMO}(S^n)$.
\medskip
Finally, we say that a vector-valued function $u: S^n\rightarrow\Bbb
R^k$ belongs to VMO$(S^n,\Bbb R^k)$ if all its component are in VMO.
If $\Sigma$ is a closed subset of $\Bbb R^k$ we say that $u
\in\text{VMO}(S^n,\Sigma)$ provided $u\in\text{VMO}(S^n,\Bbb R^k)$ and
$u(x)\in\Sigma$ a.e. on $S^n$.
\medskip
Our main result is the following
\medskip
\proclaim{Theorem 1}  Any map $u\in\text{VMO}(S^n,S^n)$ has a well-defined degree in $\Bbb Z$.  If $u\in C^0(S^n,S^n)$ or if $u$ belongs
to one of the Sobolev classes described in Sections 2 and 3, the new
degree coincides with the degree previously defined.
\endproclaim

The properties of this new degree are very similar to the properties
of the standard degree:
\medskip
\noindent
{\bf Property 1.}  The degree is stable under small BMO perturbation,
i.e., if $u\in\text{VMO}(S^n,S^n)$ and $(u_j)$ is a sequence in
VMO$(S^n,S^n)$ such that
$$
\|u_j - u\|_{\text{BMO}} \rightarrow 0 \tag 4.10
$$
then
$$
\deg u_j = \deg u\quad\text{for $j$ sufficiently large}.
$$
As a consequence, the degree is constant under homotopy within VMO,
i.e., if $H(x,t)\in C([0,1]$, VMO$(S^n,S^n))$ then
$$
\deg(H(\cdot,\,0)) = \deg(H(\cdot,\,1)).
$$
\bigskip
\noindent
{\bf Property 2.}  If $u\in\text{VMO}(S^n,S^n)$ is such that
$$
\deg u \not=0
$$
then
$$
\text{ess} R(u) = S^n. \tag 4.11
$$
Note that since $u$ is only defined a.e. it does not make sense to
talk about the range of $u$.  Instead one considers the essential
range which is the smallest closed set $\Sigma\subset S^n$ such that
$u(x)\in\Sigma$ a.e. on $S^n$.  Property (4.11) says that there cannot
be a ``hole'' in the range of $u$, i.e., there is no open ball $B$ in
$S^n$ such that $u(x) \in S^n\backslash B$ a.e.
\medskip
\noindent
{\bf Property 3 (Borsuk).}  If $u\in\text{VMO}(S^n,S^n)$ is odd, then
$$
\deg u\qquad\text{is odd}
$$
\medskip
\noindent
{\bf Property 4 (Hopf).}  If $u, v\in\text{VMO}(S^n,S^n)$ are such
that
$$
\deg u = \deg v
$$
then there is a homotopy $H$ within VMO (as in Property 1) connecting
$u$ and $v$.
\bigskip
Our definition of degree for VMO maps is extremely simple.  Given $u$
in VMO$(S^n,S^n)$ set
$$
\overline{u}_\varepsilon(x) = \Mint_{B_\varepsilon(x)} \, u,\quad 0 <
\varepsilon < 1,
$$
where $B_\varepsilon(x)$ is the geodesic ball on $S^n$, with center
$x$ and radius $\varepsilon$.  Note that $\overline{u}_\varepsilon\in
C^0(S^n,B^{n+1})$ and also that
$$
1 - |\overline u_\varepsilon(x)| = \text{dist}(\overline
u_\varepsilon(x), S^n) \leq \Mint_{B_\varepsilon(x)} |u(y) - \overline
u_\varepsilon(x)| dy. \tag 4.12
$$
Since $u\in \text{VMO}$ the right hand side in (4.12) tends to $0$
{\bf uniformly} in $x$ as $\varepsilon\rightarrow 0$.  Therefore, as
$\varepsilon\rightarrow 0$,
$$
|\overline u_\varepsilon(x)|\rightarrow 1\quad\text{uniformly in $x$}
$$
and thus we may introduce, for $\varepsilon \leq \varepsilon_0$, the map
$$
u_\varepsilon(x) = \frac{\overline u_\varepsilon(x)}{|\overline
u_\varepsilon(x)|}.
$$
Since $u_\varepsilon\in C^0(S^n,S^n)$ we may consider
$$
\deg u_\varepsilon.
$$
This number is independent of $\varepsilon$ for $\varepsilon\leq
\varepsilon_0$ since we may use $\varepsilon$ as a homotopy parameter
to connect $u_\varepsilon$ and $u_{\varepsilon'}$.  By {\bf
definition} we let
$$
\deg u = \deg u_\varepsilon\qquad\text{for $\varepsilon\leq
\varepsilon_0$}.
$$
\medskip
For the proofs of all the above results we refer to H.~Brezis and
L.~Nirenberg~[1].  They are not very difficult, but, still, the VMO
degree theory is more subtle than the usual $C^0$ theory.  Here are
some delicate points:
\medskip
\noindent
{\bf Remark 3.}  In the $C^0$ case, Property 1 asserts that if $u,v\in
C^0(S^n,S^n)$ and
$$
\|u - v\|_{L^\infty} < 1
$$
then $\deg u = \deg v$.  In the VMO case such a statement does not
hold.  More precisely, there exists {\bf no uniform} $\delta > 0$ such that if
$u, v\in\text{VMO}(S^n,S^n)$ and
$$
\|u - v\|_{\text{BMO}} < \delta
$$
then $\deg u = \deg v$.  In fact we can construct (see Lemma 6 in
H. Brezis and L.~Nirenberg~[1]) sequences $(u_j)$ and $(v_j)$ in
$C^1(S^1,S^1)$ such that
$$
\|u_j - v_j\|_{\text{BMO}} \rightarrow 0\quad\text{as $j
\rightarrow\infty$}
$$
(even $\|u_j - v_j\|_{H^{1/2}}\rightarrow 0$ as $j\rightarrow\infty$)
and
$$
|\deg u_j - \deg v_j|\geq 1\quad\forall j.
$$
The exact formulation of Property 1 in the VMO case is:
\medskip
Given any $u\in\text{VMO}(S^n,S^n)$ there is some $\delta > 0$ {\bf
depending on} $u$ such that if $v\in\text{VMO}(S^n,S^n)$ and $\|v -
u\|_{\text{BMO}} < \delta$, then $\deg v = \deg u$.
\medskip
\noindent
{\bf Remark 4.}  As we have already pointed out the degree counts ``how
many times'' $\varphi$ covers its range (including algebraic
multiplicity).  If $\varphi\in C^0(S^1, S^1)$ then $\varphi$ covers
globally $S^1$ at most a {\bf finite} number of times; this follows
from the uniform continuity:  there is a $\delta > 0$ such that $|x -
y| < \delta\Rightarrow |\varphi(x) - \varphi(y)| < 1$ and so the
number of times that $\varphi$ may cover $S^1$ is at most of the order
of $1/\delta$.  When $\varphi\in\text{VMO}(S^1,S^1)$ it may cover
$S^1$ {\bf infinitely} many times.  Here is such an example.  Consider
a real-valued function $f(\theta)$ on $[0,2\pi]$ which is smooth on
$[0,2\pi]$ except at $\theta = \pi$, with
$\underset\theta\rightarrow\pi\to{\lim} f(\theta) = +\infty$ and $f(0)
= 0$, $f(2\pi) = 2\pi$.  We may construct such an $f$ which belongs to
VMO, choosing, for example,
$$
f(\theta) = \theta + \zeta(\theta) \big\vert\log|\theta - \pi|\big\vert
$$
where $\zeta$ is a smooth cut-off function supported near $\theta =
\pi$.  Set
$$
\varphi(\theta) = e^{if(\theta)}.
$$
Then $\varphi\in\text{VMO}(S^1,S^1)$ (it is easy to see, using (4.7)
that the composition $L\circ f$ of a Lipschitz map $L$ with a VMO map $f$
lies in VMO).  As $\theta\rightarrow \pi$, $\theta < \pi$,
$\varphi(\theta)$ turns around $S^1$ {\bf infinitely} many times in
the {\bf positive} direction.  As soon as $\theta$ crosses $\pi$,
$\varphi(\theta)$ turns around $S^1$ {\bf infinitely} many times in the {\bf
negative} direction!  Again, the degree seems to be a difference of
two infinite quantities and we encounter the same kind of {\bf
cancellation phenomenon} as in Section 3 with Fourier series.  Here
also the degree seems to be some sort of ``principal value''.  It
would be very interesting to clarify this point.
\bigskip
\noindent
{\bf 5.  Further properties of VMO maps in connection with Topology}
\medskip
We present here various remarks and additional results.
\medskip
\noindent\
{\bf A.  Homotopy classes and VMO}
\medskip
At this moment it is not clear whether VMO is the ``largest'' natural
class on which a degree can be defined.  What is certain is that
bigger classes such as $L^p(S^n,S^n)$, $1 \leq p\leq \infty$, or
BMO$(S^n,S^n)$ do {\bf not} have a degree.  The reason is that the
spaces $L^p(S^n,S^n)$, $1\leq p\leq \infty$, and BMO$(S^n, S^n)$ are
{\bf arcwise connected}; this is true even if $S^n$ is replaced in the
domain space by a manifold $X$ and in target space by a manifold $Y$;
see Section I.5 in H.~Brezis and L.~Nirenberg~[1].
\medskip
When dealing with continuous maps, topologists consider the {\bf
homotopy classes} say of $C^0(S^n,S^k)$.  These are the connected
components $\Cal C_i$ of $C^0(S^n,S^k)$.  One may ask what are the
connected components of VMO$(S^n,S^k)$?  It turns out that they are of
the same type as in the $C^0$ case.  More precisely, they are the
closures in BMO of the above $\Cal C_i$; see Section I.5 and Lemmas
A.18 - A.24 in H.~Brezis and L.~Nirenberg~[1].
\medskip
\noindent
{\bf B.  Lifting and VMO}
\medskip
Another topic of interest in Topology concerns {\bf lifting}.  For the
sake of simplicity let us consider maps from $S^1$ into $S^1$.  The
question is whether a map $\varphi: S^1\rightarrow S^1$ can be written
as
$$
\varphi = e^{if}
$$
for some $f: S^1\rightarrow \Bbb R$.  When $\varphi\in C^0(S^1,S^1)$,
a classical result asserts that there is such $f\in C^0(S^1,\Bbb R)$
if and only if $\deg\varphi = 0$.  Here is an extension to VMO:
\smallskip
\proclaim{Theorem 2}  Assume $\varphi\in\text{VMO}(S^1,S^1)$, then
$\varphi$ may be written as
$$
\varphi = e^{if}\quad\text{\rm for some $f\in\text{VMO}(S^1,\Bbb R)$}
$$
if and only if $\deg\varphi = 0$.
\endproclaim

The proof, which is much more elaborate than in the continuous case, is
presented in Section I.6  of H.~Brezis and L.~Nirenberg~[1].  It is
related to earlier work of R.~Coifman and Y.~Meyer~[1].  We have more
general results in the framework of 3 spaces $X,Y,Z$ and $F$ is a
continuous covering map of $Z$ to $Y$.  Under the natural topological
assumptions we prove that a map $\varphi\in\text{VMO}(X,Y)$ can be
lifted to $Z$, i.e., 
$$
 \varphi = F \circ f
$$ 
for some $f\in\text{VMO}(X,Z)$.
\medskip

\noindent
{\bf Remark 5.}  The question of lifting for {\bf Sobolev maps} is
more delicate than it seems and has been settled only recently.  Here
is the problem in a simple situation.  Let $\Omega\subset\Bbb R^n$ be
a smooth bounded domain and let $u\in W^{1,p}(\Omega,S^1)$ with $1\leq
p < \infty$.  Can one write
$$
u = e^{if}\quad\text{for some $f\in W^{1,p}(\Omega,\Bbb R)$}\; ?
$$
The answer is {\bf positive} if $p \geq 2$ and {\bf negative} if $1
\leq p < 2$.  This is a result of F.~Bethuel and X.~Zheng~[1]; a
simpler proof, due to P.~Mironescu is given in H.~Brezis~[3].
\medskip
\noindent
{\bf C.  Toeplitz operators and VMO}
\medskip
Let me recall briefly the notion of Toeplitz operators.  Consider the
Hilbert space $L^2(S^1,\Bbb C)$ and the closed subspace
$$
\Cal H^2 = \{f\in L^2(S^1,\Bbb C); \int_{S^1} f(\theta) e^{in\theta}
d\theta = 0\quad\forall n=1,2,\ldots\}.
$$
Let $P$ be the orthogonal projection from $L^2$ onto $\Cal H^2$.
Given a function $\varphi\in L^\infty(S^1,\Bbb C)$ consider the
multiplication operator, defined on $L^2$ by
$$
M_\varphi f = \varphi f.
$$
By definition the Toeplitz operator $T_\varphi$, with symbol $\varphi$,
is
$$
T = P M_\varphi
$$
considered as a bounded operator from $\Cal H^2$ into itself.
\medskip
A very classical result in the theory of Toeplitz operators (see
e.g. R.~Douglas~[1]) asserts that if $\varphi\in C^0(S^1,\Bbb C)$ and
$\varphi\not= 0$ on $S^1$, then $T_\varphi$ is Fredholm and
$$
\text{index}(T_\varphi) = -
\deg\left(\frac{\varphi}{|\varphi|}\right).
$$
Here is an extension to VMO.
\medskip
\proclaim{Theorem 3}  Assume $\varphi\in\text{VMO}(S^1,\Bbb C) \cap
L^\infty(S^1,\Bbb C)$ satisfies
$$
|\varphi|\geq \alpha > 0\quad\text{a.e. on $S^1$}.
$$
Then $T_\varphi$ is Fredholm and
$$
\text{\rm index}(T_\varphi) = - \deg
\left(\frac{\varphi}{|\varphi|}\right).
$$
\endproclaim

Of course, the degree is to be understood in the sense of degree for
VMO maps.  The proof of Theorem 3, which is joint with
P.~Mironescu, is presented in Appendix 2 of H.~Brezis and
L.~Nirenberg~[2].  It uses a deep result:  the Fefferman-Stein duality
of $\Cal H^1$ and BMO (see C.~Fefferman and E.~Stein~[1]).
\bigskip
\noindent
{\bf 6.  Degree theory for VMO maps on domains}
\medskip
As we have mentioned in the Introduction there is a classical notion
of degree for continuous maps on domains of $\Bbb R^n$.  Such a
concept can be extended to VMO maps.  We have first to define
precisely what is meant by BMO and VMO on domains.
\medskip
Let $\Omega\subset\Bbb R^n$ be an open (connected) bounded set.  A
function $f\in L^1_{\text{loc}}(\Omega)$ belongs to BMO$(\Omega)$
provided
$$
\|f\|_{\text{BMO}} = \operatornamewithlimits{Sup}_{\overline
B\subset\Omega} \Mint_B \big\vert f - \Mint_B f\big\vert < \infty, \tag 6.1
$$
where the Sup in (6.1) is taken over all balls $B$ whose closure is
contained in $\Omega$.  This notion depends (in principle) on the
choice of norm in $\Bbb R^n$---a different norm gives rise to a
different geometry of balls.  A deep result of P.~Jones asserts that
two different norms on $\Bbb R^n$ yield two {\bf equivalent} BMO
norms.  This is proved in H.~Brezis and L.~Nirenberg~[2], using the
methods of P.~Jones~[1].  The main idea is to show that if in (6.1) we
consider balls $B$ ``well-inside'' (i.e., $B = B_r(x)$ with $r\leq
\dfrac12$dist$(x,\partial\Omega)$) we obtain a smaller norm, which is
equivalent to the BMO norm.
\medskip
Now VMO$(\Omega)$ is the closure of $C^0(\overline\Omega)$ for the BMO
norm.  The analogue of Lemma 3 holds provided $\overline
B\subset\Omega$.  As above, the Sobolev space
$W^{s,p}(\Omega)\subset\text{VMO}(\Omega)$ when $0 < s < n$ and
$sp = n$.
\medskip
We wish to define
$$
\deg(u,\Omega,y)
$$
for a map $u\in\text{VMO}(\Omega,\Bbb R^{n+1})$ such that $y\notin
u(\partial\Omega)$.  This last condition does not make sense since VMO
maps do {\bf not}, in general, have a trace on the boundary.  We make
instead the following assumption:
$$
\cases  \text{there exist constants $\delta > 0$ and $r_0 > 0$ such
that}\\
  \dsize\Mint_{B_r(x)} |u(z) - y|\geq \delta\;\;\; \forall x\in\Omega\text{
with   $r = \dfrac12$dist$(x,\partial\Omega) \leq r_0$}.
\endcases
\tag 6.2
$$
Note that, if $u \in C^0(\overline\Omega,\Bbb R^{n+1})$, assumption
(6.2) is equivalent to the condition that\newline $y\notin u(\partial\Omega)$.
Of course, we could also have made the stronger assumption that
$$
|u - y|\geq \delta\text{  a.e. on some neighborhood of
$\partial\Omega$}. \tag 6.3
$$
However, such a condition would be too restrictive in our framework.
For example, if $u\in W^{1,n}$, let $\varphi = u_{\big\vert\partial\Omega}$ 
and assume that
$$
|\varphi - y|\geq \gamma > 0\qquad\text{a.e. on $\partial\Omega$},
$$
then (6.2) holds, but (6.3) does not hold.
\medskip
Our main result is
\medskip
\proclaim{Theorem 4}  Assume $u\in\text{VMO}(\Omega,\Bbb R^{n+1})$
satisfies (6.2) then
$$
\deg(u,\Omega,y)\quad\text{is well-defined (in $\Bbb Z$)}.
$$
\endproclaim

This new degree has all the properties that one expects for a degree.
Here are some:
\medskip
\noindent
{\bf Property 1.}  Assume $u\in\text{VMO}(\Omega,\Bbb R^{n+1})$ and
$(u_j)\subset\text{VMO}(\Omega,\Bbb R^{n+1})$ are such that
$u_j\rightarrow u$ in BMO and in $L^1_{\text{loc}}$, and (6.2) holds
for $u$ and for $u_j$ uniformly in $j$ (i.e., with the same $\delta$
and $r_0$).  Then
$$
\deg(u_j,\Omega,y) = \deg(u,\Omega,y)\quad\text{for $j$ large}.
$$
\medskip
\noindent
{\bf Property 2.}  Suppose $u\in\text{VMO}(\Omega,\Bbb R^{n+1})$
satisfies (6.2) and $\deg(u,\Omega,y)\not=0$.  Then
$y\in\text{ess}R(u)$ and more precisely
$$
B_\delta(y)\subset\text{ess} R(u).
$$
\medskip
In the classical theory, with $u\in C^0(\overline\Omega,\Bbb R^{n+1})$,
 assuming also $\Omega$ is smooth, we have
$$
\deg(u,\Omega,y) = \deg \left( \frac{u - y}{|u - y|},\;
\partial\Omega,\; S^{n+1}\right). \tag 6.4
$$
Formula (6.4) does not make sense for maps $u$ in VMO---again because
they do not have a trace on $\partial\Omega$.
\medskip
To get around this difficulty we were led to introduce a new class of
functions $f$ in VMO$(\Omega)$ which {\bf does} have a trace $\varphi$
on the boundary, with $\varphi$ in VMO$(\partial\Omega)$.  Our
definition is the following:  Let $\varphi$ be a function in
VMO$(\partial\Omega)$.  Let $\widetilde\varphi$ be the extension of
$\varphi$ in a neighborhood of $\partial\Omega$, constant on normals.
\bigskip
\noindent
{\bf Definition.}  A function $f\in\text{VMO}(\Omega)$ belongs to
VMO$_\varphi(\Omega)$ if
$$
\operatornamewithlimits{\lim}\Sb r\rightarrow 0\\r =
\frac12\text{dist}(x,\partial\Omega)\endSb \;\Mint_{B_r(x)} |f -
\widetilde\varphi| = 0.
$$
\medskip
There are {\bf many natural examples} of functions in VMO$_\varphi$.  Any
function $f$ in $W^{1,n}$ belongs to VMO$_\varphi$ where $\varphi =
\text{trace of $f$}$ (in the sense of Sobolev spaces).  The harmonic
extension in $\Omega$ of a function
$\varphi\in\text{VMO}(\partial\Omega)$ belongs to
VMO$_\varphi(\Omega)$.  Etc. $\ldots$ \;.  For such maps we have
\medskip
\proclaim{Theorem 5}  Assume $u\in\text{VMO}_\varphi(\Omega,\Bbb
R^{n+1})$ and
$$
|\varphi - y|\geq \delta > 0\quad\text{i.e., on $\partial\Omega$}.
$$
Then (6.2) holds and
$$
\deg(u,\Omega,y) = \det\left( \frac{\varphi - y}{|\varphi - y|},
\partial\Omega, S^{n-1}\right).
$$
\endproclaim

The proof is given in Section II.4 of H.~Brezis and L.~Nirenberg~[2].

\bigskip



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\end



          














