%10/04 10/14/04 10/16/04, 10/18 with Brezis corrections
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\topmatter
\title{New questions related to the topological degree}
\endtitle
\author{Ha\"im Brezis$^{(1),(2)}$}\endauthor
\medskip
\medskip
\address
${}^{\text {(1)}}$  LABORATOIRE J.-L. LIONS\endgraf
UNIVERSIT\'E P. ET M. CURIE, B.C. 187\endgraf
4 PL. JUSSIEU\endgraf
75252 PARIS CEDEX 05, FRANCE\endgraf
\null
and\endgraf
\null
INSTITUT UNIVERSITAIRE DE FRANCE\endgraf
\endaddress
\null
\address
${}^{\text {(2)}}$  RUTGERS UNIVERSITY\endgraf
DEPT. OF MATH., HILL CENTER, BUSCH CAMPUS\endgraf
 110 FRELINGHUYSEN RD, PISCATAWAY, NJ 08854, USA\endgraf
\endaddress

\email
brezis\@ccr.jussieu.fr,
brezis\@math.rutgers.edu
\endemail
\null
%\abstract
%\endabstract
\endtopmatter
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\centerline{To I. M. Gelfand with admiration}
\bigskip
\subhead 1. Topological degree and VMO\endsubhead
\medskip

Degree theory for continuous maps has a long history and has been
extensively studied, both from the point of view of Analysis and
Topology.  If $f\in C^0(S^n, S^n), \deg f$ is a well-defined element of
$\Bbb Z$, which is stable under continuous deformation.  Starting in
the early 80's, the need to define a degree for some classes of
discontinuous maps, emerged from the study of some nonlinear PDE's
(related to problems in liquid crystals and superconductors).  These
examples involved Sobolev maps in the limiting case of the Sobolev
imbedding, see Sections 2 and 3 below (topological questions for Sobolev maps strictly below the limiting exponent have been investigated in [14] and [13]).  In these cases the Sobolev
imbedding asserts only that such maps belong to the space VMO (see
below) and {\it need not be continuous.}

In connection with degree for $H^{1/2} (S^1, S^1)$, L. Boutet de
Monvel and O. Gabber suggested a concept of degree for maps
in VMO$(S^1, S^1)$(see [2] and Section 3 below).  In our joint work with L. Nirenberg [15] we
followed-up on their suggestion and established on firm grounds a
degree theory for maps in VMO$(S^n, S^n)$.  Here is a brief summary
of our contribution.

First recall the definition of BMO (bounded mean oscillation), a
concept originally introduced by F. John and L. Nirenberg in 1961.  Let
$\Omega$ be a smooth bounded open domain in $\Bbb R^n$, or a smooth,
compact, $n$-dimensional Riemannian manifold (with or without
boundary).  An integrable function $f: \Omega \to \Bbb R$ belongs to
BMO if
$$
|f|_{\text{BMO}} = \underset{B\subset \Omega}\to{\text{ Sup }}
 \Mint_B\Mint_B |f(x) - f(y)|dx\ dy < \infty,
$$
where the Sup is taken over all (geodesic) balls in $\Omega$.  It is
easy to see that an equivalent semi-norm is given by
$$
\underset{B\subset\Omega}\to{\text{ Sup}}\quad
\Mint_B \Bigg| f(x) - \Mint_B
f(y)dy
\Bigg|
dx.
$$
A very important subspace of BMO, introduced by L. Sarason, consists of VMO (vanishing mean
oscillation) functions, in the sense that
$$
\lim_{|B|\to 0} \Mint_B\Mint_B |f(x) - f(y)|dx dy = 0.
$$
It is easy to see that
$$
\text{VMO }(\Omega, \Bbb R) = \overline{C^0(\overline{\Omega}, \Bbb
R)}^{\text{ BMO}}.
$$
The space VMO is equipped with the BMO semi-norm $|f|_{\text{ BMO}}$.
Clearly $L^\infty\subset \text{ BMO}$.  It is well-know that BMO is
strictly bigger than $L^\infty$ (a standard example is $f(x) = |\log
|x|\big|$); however, as a consequence of the classical
John-Nirenberg inequality,
$$
\text{BMO } \subset \underset{ p< \infty}\to{\cap} L^p.
$$
Thus, BMO is ``squeezed'' between $L^\infty$ and $\underset{ p<
\infty}\to{\cap} L^p$, and for many purposes serves as an interesting
``substitute'' for $L^\infty$.

Concerning VMO, it is easy to see that $L^\infty \not\subset$ VMO, but
of course $C^0 \subset $ VMO.  A useful example, showing that the
inclusion is strict, is the function
$$
f(x) = |\log |x| |^\alpha,
$$
which belongs to VMO for every $\alpha < 1$.  In some sense, VMO serves
as a ``substitute'' for $C^0$.  The Sobolev space $W^{1, n}$ provides
an important class of VMO functions.  Recall that for every $1\leq p <
\infty$,
$$
W^{1,p} (\Omega, \Bbb R) = \{ f\in L^p(\Omega); \nabla f\in L^p
(\Omega)\}.
$$
Poincar\'e's inequality asserts that
$$
\int_B \Bigg|f-\Mint_B f\Bigg| \leq C|B|^{1/n} \int_B|\nabla f|,
$$
from which we deduce, using H\"older, that
$$
\Mint_B \Bigg|f-\Mint_B f
\Bigg| \leq C\left[ \int_B |\nabla f|^n\right]^{1/n}
$$
and thus $W^{1,n} \subset $ VMO.

Similarly, the fractional Sobolev space $W^{s,p}(\Omega)$ is
contained in VMO for all $0< s< 1$ and all $1< p < \infty$ with $sp =
n$ (the limiting case of the Sobolev imbedding).  Indeed, in the
Gagliardo characterization, we have
$$
W^{s,p} (\Omega) = \{ f\in L^p(\Omega); \int_\Omega\int_\Omega
\frac{|f(x) - f(y) |^p}{|x-y|^{n+sp}} dx\ dy < \infty \}.\tag 1.1
$$
Clearly
$$
\int_B\int_B |f(x) - f(y)| dx \ dy = \int_B \int_B \frac{|f(x) -
f(y)|}{|x-y|^{(n/p) + s}} |x-y|^{(n/p) + s} dx \ dy
$$
$$
\leq C |B|^{(1/p) + (s/n)} \int_B \int_B \frac{|f(x) -
f(y)|}{|x-y|^{(n/p) + s}} dx\ dy.
$$
Using H\"older we deduce that
$$
\int_B \int_B |f(x) -f(y)|dx\ dy \leq C|B|^{(1/p) + (s/n) + 2 - (2/p)}
\left[\int_B \int_B \frac{|f(x) - f(y)|^p}{|x-y|^{n+sp}}
dx\ dy\right]^{1/p}.
$$
and thus, when $sp = n$,
$$
\Mint_B\Mint_B |f(x) - f(y)|dx\ dy \leq
C\left[\int_B\int_B\frac{|f(x) - f(y)|^p}{|x-y|^{n+sp}} dx\ dy
\right]^{1/p},
$$
which implies that $W^{s,p} \subset$ VMO.
\medskip

One of the basic results in [15] is the following

\proclaim{Theorem 1}{\rm (\text{H. Brezis - L. Nirenberg [15]})}.  Every map
$f\in$ VMO$(S^n, S^n)$ has a well-defined degree.  Moreover:

(a)  this degree coincides with the standard degree when $f$ is
continuous,
\medskip

(b)  the map $f\mapsto \deg f$ is continuous on VMO$(S^n, S^n)$
under BMO-convergence.
\endproclaim
\medskip
It is quite easy to define the VMO-degree.  For any given measurable
map $f: S^n \to S^n$ and $0< \varep < 1$, set
$$
{\bar f}_\varep (x) = \Mint_{B_\varep(x)} f(y) dy.
$$


Next, an elementary lemma which is extremely useful

\proclaim{Lemma 1}  If $f \in$ VMO$(S^n, S^n)$, then
$$
|{\bar f}_\varep (x) |\to 1 \text{ as } \varep \to 0, \text{ uniformly
 in } x \in S^n.
$$
\endproclaim

\demo{Proof}  Set
$$
\rho_\varep (x) = \Mint_{B_\varep(x)} \Mint_{B_\varep(x)} |f (y) - f(z)
 |dy\ dz,
$$
so that $\rho_\varep(x) \to 0$ as $\varep \to 0, \text{
uniformly in } x\in S^n$, since $f \in$ VMO.  Then, observe that
$$
1 - \rho_\varep (x) \leq |\bar f_\varep (x) |\leq 1.
$$
\enddemo

If $f\in$ VMO$(S^n, S^n)$ we may now set
$$
f_\varep(x) = \frac{\bar f_\varep (x)}{|\bar f_\varep (x)|} , x\in
S^n, 0< \varep < \varep_0 (f).
$$
Using $\varep$ as a homotopy parameter we see that $\deg f_\varep$ is
well-defined and {\it independent of }$\varep$ for $\varep > 0$ sufficiently
small.  This integer is, by definition, $\text{VMO-}\deg f$.  The proof
of (a) in Theorem 1 is straightforward.  For the proof of (b) we
refer to [15].
\medskip


The space VMO$(S^n, S^n)$ is larger than $C^0(S^n, S^n)$.  However its
structure, from the point of view of connected (or equivalently
path-connected) components, is similar to $C^0(S^n, S^n)$.  More
precisely, there is a VMO version of the celebrated Hopf result:

\proclaim{Theorem 2} The homotopy classes (i.e. the
path-connected components) of \newline VMO$(S^n, S^n)$ are characterized by
their VMO-degree.
\endproclaim

\remark{Remark 1}  By contrast, it is {\it not} possible to define a
degree for maps in $L^\infty(S^n, S^n)$.  In fact, the space
$L^\infty(S^n, S^n)$ is path-connected (see [15], Section I. 5).
\endremark


\subhead 2.  Degree for $H^1(S^2, S^2)$ and beyond\endsubhead

In my earlier paper with J. M. Coron [11](see also [8],[9]) we were led to a concept of
degree for maps in $H^1(S^2, S^2)$.  Our original motivation came from solving a nonlinear
elliptic system, proposed in [16],which amounts to finding critical points of the
Dirichlet integral
$$
E(u) = \int_\Omega |\nabla u |^2
$$
subject to the constraint
$$
u\in H^1_\varphi(\Omega, S^2)=\{ u \in H^1(\Omega; S^2); u =\varphi
\text{ on } \partial \Omega\},
$$
where $\Omega$ denotes the unit disc in $\Bbb R^2$ and $\varphi:
\partial \Omega \to S^2$ is given (smooth).  In the process of finding
critical points it is natural to the study the connected components of
$H^1_\varphi(\Omega, S^2)$, a question which is closely related to
the study of the components of $H^1(S^2, S^2)$.  The way we defined a
degree for $H^1(S^2, S^2)$ was with the help of an {\it integral
formula}.  Recall that if $f\in C^1(S^n, S^n)$, Kronecker's formula
asserts that
$$
\deg f = \Mint_{S^n} \det (\nabla f)\tag 2.1
$$
where $\det(\nabla f)$ denotes the $n\times n$ Jacobian determinant of
$f$.  When $n=2$, the right-hand side of (2.1) still makes sense when
$f$ is not
$C^1$, but merely in $H^1(S^2, S^2)$ because $\det(\nabla f) \in
L^1$.  We were able to prove (via a density argument) that the RHS in
(2.1) belongs to $\Bbb Z$ and we took it as a definition of the $H^1$-degree of $f$.
Similarly, one may use (2.1) to define a degree for every map $f\in
W^{1,n}$.  In view of the discussion in Section 1, we know that
$W^{1,n}\subset$ VMO and thus any $f\in W^{1,n}(S^n, S^n)$ admits a VMO-degree
in the sense of Section 1.  Fortunately, the two definitions coincide.
In fact, we have

\proclaim{Lemma 2}  For every $f\in W^{1, n}(S^n, S^n)$,
$$
W^{1, n} \text{-}\deg f = \text{VMO-}\deg f.
$$
Moreover the components of $W^{1, n}(S^n, S^n)$ are characterized by
their degree.
\endproclaim

Using this concept of degree we managed to prove in [11] that if
$\varphi$ is not a constant, then $E$ achieves its minimum on two
distinct components of $H^1_\varphi(\Omega, S^2)$.  A very interesting
question remains open:

\noindent{\bf Open problem 1. }  Does $E$ admit a critical point in
each component of $H^1_\varphi(\Omega, S^2)$ when $\varphi$ is not
a constant?

Even the special case
$$
\varphi(x, y) = (Rx, Ry, \sqrt{1-R^2}),\,\ 0 < R < 1,\,\ x^2 + y^2 = 1,
$$
is open.

\medskip

It is also interesting to study the homotopy structure of $W^{1,
p}(S^n, S^n)$ for values of $p \neq n$.  This was done in my joint paper
with Y. Li [13]:

\proclaim{Theorem 3} When $p> n$, the standard $(C^0)$ degree of
maps in $W^{1, p}$ is well-defined and the components of $W^{1, p}$
are characterized by their degree.  When $1\leq p < n, W^{1, p}$ is
path-connected.
\endproclaim

Following the earlier paper[14] we started to investigate with Y.Li[13] the homotopy
structure of $W^{1, p} (M, N)$ when $M$ and $N$ are general Riemannian
manifolds ($M$ possibly with boundary, while $\partial N = \phi$).
When $p \geq \dim M$ the homotopy structure of $W^{1, p} (M, N)$ is
identical to the one of $C^0(M, N)$.  When $\dim M > 1$ and $1\leq p <
2$, we proved in [13] that $W^{1, p}(M, N)$ is {\it always} path-connected.  When $p$ decreases
from $\dim M$ to 2, the set $W^{1,p}(M, N)$ becomes larger and larger while
various surprising phenomena may occur:
\medskip

(a) some homotopy classes persist below the Sobolev threshold $p=\dim
M$ , where maps need not belong to VMO.
\medskip

(b) as $p$ decreases, the set $W^{1,p}(M, N)$ increases and in this
process some of the homotopy classes ``coalesce'' as $p$ crosses
distinguished {\it integer}
values - and usually there is a cascade of such levels where the
homotopy structure undergoes ``dramatic'' jumps.
\medskip

(c) as $p$ decreases new homotopy classes may ``suddenly" appear, at some
(integral) levels; every map in these new classes must have ``robust''
singularities: they cannot be erased via homotopy.
\medskip

We refer the interested reader to [13] and to the subsequent
remarkable paper by F.B. Hang - F.H. Lin [17].


\subhead 3.  Degree for $H^{1/2}(S^1, S^1)$.  Can one hear the degree of
continuous maps?\endsubhead

Another important example which motivated my work with L. Nirenberg [15]
was the concept of degree for maps in $H^{1/2} (S^1, S^1)$ due to
L. Boutet de Monvel - O. Gabber (it is presented in the Appendix of [2]).
The motivation in [2] came from a Ginzburg-Landau model arising in
superconductivity.  This $H^{1/2}$-degree also plays an important role
in our study of the Ginzburg-Landau Vortices with F. Bethuel - F.
H\'elein (see [1]).  For example it is at the heart of the proof of

\proclaim{Lemma 3}  Let $\Omega$ be the unit disc in $\Bbb R^2$ and
let $\varphi$ be a smooth map from $\partial \Omega = S^1$ into $S^1$.
then
$$
[H^1_\varphi(\Omega, S^1) \neq \phi] \Leftrightarrow [\deg \varphi =
0].
$$
\endproclaim

The way Boutet de Monvel and Gabber originally defined a degree for $H^{1/2} (S^1, S^1)$ went as
follows.  First, observe that if $f\in C^1(S^1, \Bbb C\setminus
\{0\})$, then the Cauchy formula asserts that
$$
\deg f = \frac{1}{2 i \pi} \int_{S^1} \frac{\dot f}{f}.\tag 3.1
$$
In particular, if $f\in C^1(S^1, S^1)$ we may write (3.1) as
$$
\deg f = \frac{1}{2 i \pi} \int_{S^1} \bar f \dot f = \frac{1}{2 \pi}
\int_{S^1} \det (f, \dot f)\tag 3.2
$$
(which is the simplest form of Kronecker's formula (2.1)).  Then
Boutet de Monvel-Gabber observed that the right-hand side in (3.2) still makes
when $f$ is not $C^1$, but merely in $H^{1/2}$.  To do so, they
interpret the RHS in (3.2) as a scalar product in the duality
$H^{1/2} - H^{-1/2}(\bar f \in H^{1/2}, \dot f \in H^{-1/2})$.  Using
a density argument they prove that the RHS in (3.2) belongs to
$\Bbb Z$ and they take it as definition for the $H^{1/2}$-degree of $f$.  On the other hand
recall (see Section 1) that $H^{1/2} (S^1) \subset $VMO$(S^1)$.
Therefore any $f\in H^{1/2} (S^1, S^1)$ admits a VMO-degree in the
sense of Section 1 and in fact we have


\proclaim{Lemma 4}  For every $f\in H^{1/2}(S^1, S^1)$
$$
H^{1/2}\text{-}\deg f =\text{VMO - }\deg f.
$$
\endproclaim

Lemma 2 and Lemma 4 show the unifying character of the VMO-degree, putting
various concepts of degree (for continuous maps, for $W^{1,n}
(S^n, S^n)$ maps, for \newline $H^{1/2} (S^1, S^1)$ maps, etc) under a common roof.

In 1996, I. M. Gelfand invited me to present at his seminar the
VMO-degree theory we had just developed with Louis Nirenberg.  He
asked me to elaborate on the special case of the $H^{1/2} (S^1, S^1)$ -
degree.  I wrote down Gagliardo's characterization of $H^{1/2}$, which, in this
special case, takes the form
$$
H^{1/2} (S^1) = \{ f\in L^2(S^1); \int_{S^1}\int_{S^1} \frac{|f(x) -
f(y)|^2}{|x-y|^2} dx \ dy < \infty \}.
$$
Since I. M. Gelfand was not fully satisfied with Gagliardo's
formulation, I also wrote down the characterization of $H^{1/2}$ in
terms of the Fourier coefficients $(a_n)$ of $f$:
$$
H^{1/2} (S^1) = \{ f\in L^2(S^1); \sum^{+\infty}_{n=-\infty} |n|
|a_n|^2 < \infty\}
$$
(see also Lemma 5 below). At that point I. M. Gelfand asked whether there is a connection
between the degree and the Fourier coefficients.  At first I was
surprized by his question, but I realized shortly afterwards that if
one inserts the Fourier expansion
$$
f(\theta)= \sum^{+\infty}_{n=-\infty} a_n e^{in\theta}
$$
into (3.2) one finds
$$
\deg f = \sum^{+\infty}_{n=-\infty} n | a_n |^2.\tag 3.3
$$
Formula (3.3) is easily justified when $f\in C^1 (S^1, S^1)$.  The
density of $C^1 (S^1, S^1)$ into $H^{1/2}(S^1, S^1)$ and the stability
of degree under VMO-convergence (and thus under $H^{1/2}$-convergence) yield

\proclaim{Theorem 4}  For every $f\in H^{1/2}(S^1, S^1)$
$$
\text{VMO-} \deg f = \sum^{+\infty}_{n=-\infty} n | a_n|^2.\tag 3.4
$$
\endproclaim

Formula (3.4) raises some intriguing questions. But, first, a consequence
of Theorem 4:

\proclaim{Corollary 1}  Let $(a_n)$ be a sequence of complex numbers
satisfying
$$
\sum^{+\infty}_{n=-\infty} |n|\ |a_n|^2 < \infty,\tag 3.5
$$
$$
\sum^{+\infty}_{n=-\infty} |a_n|^2 = 1 \tag 3.6
$$
and
$$
\sum^{+\infty}_{n=-\infty} a_n \bar a_{n+k} = 0 \quad \forall k \neq
0.\tag 3.7
$$
Then
$$
\sum^{+\infty}_{n=-\infty} n|a_n|^2\in \Bbb Z.\tag 3.8
$$
\endproclaim

\demo{Proof}  Set
$$
f(\theta) = \sum^{+\infty}_{n=-\infty} a_n e^{in\theta},
$$
so that $f\in H^{1/2}(S^1, \Bbb C)$.  Moreover we have
$$
\int_{S^1}(|f(\theta)|^2 - 1)e^{ik\theta} d\theta = 0\quad\forall
k.\tag 3.9
$$
Indeed, for $k=0$, (3.9) follows from (3.6) and for $k\neq 0$, (3.9)
follows from (3.7).  Thus we obtain
$$
|f(\theta)| = 1 \ a.e.\tag 3.10
$$
Applying Theorem 4 we find (3.8).
\enddemo

\noindent{\bf Pedagogical question.}  Is there an elementary proof of
Corollary 1 which does not rely on Theorem 4?
\medskip
Suppose now $f\in C^0(S^1, S^1)$ and $f\notin H^{1/2}$.  Then the
series
$$
\sum^{+\infty}_{n =- \infty} |n|  |a_n|^2
$$
is divergent.  The LHS in (3.4) is well-defined, but the RHS is not.
It is natural to ask whether $\deg f$ may still be computed as a
``principal value'' of the series $\sum^{+\infty}_{n=-\infty}
n|a_n|^2$ (which is not absolutely convergent).  In [10] we raised the
question whether standard summation processes can be used to compute
the degree of a general $f\in
C^0(S^1, S^1)$.  Let for example
$$
\sigma_N = \sum^{+N}_{n=-N} n |a_n|^2
$$
or
$$
P_r = \sum^{+\infty}_{n=-\infty} n|a_n|^2 r^{|n|},\,\ 0 < r < 1.
$$
Is it true that, for any $f\in C^0(S^1, S^1)$,
$$
\deg f = \underset{N \to +\infty}\to{\lim} \sigma_N \text{ or } \deg
f = \underset{r\downarrow 1}\to{\lim}\,\ P_r?
$$
J. Korevaar [19] has shown that the answer is negative.  He has
constructed interesting examples of maps $f\in C^0(S^1, S^1)$, of
degree zero, such that $\sigma_N$ (resp. $P_r$) need not have a limit as $N \to \infty$ (resp. $r \to 1$) or may converge to any
given real number $\lambda \neq 0$, including $\pm \infty$.  In view
of this fact we now propose a more ``modest'' question: do the
absolute values of the Fourier coefficients determine the degree?
More precisely
\medskip

\noindent{\bf Open problem 2.}  (Can one hear the degree of continuous
maps?).  Let $f, g\in C^0(S^1, S^1)$ and let $(a_n), (b_n)$ denote the
Fourier coefficients of $f$ and $g$ respectively.  Assume
$$
|a_n| = |b_n|\quad \forall n \in \Bbb Z.\tag 3.11
$$
Can one conclude that
$$
\deg f = \deg g\,\ ?
$$
Same question if one assumes only that $f,g\in VMO (S^1, S^1)$.
\medskip
Of course, the answer to Open problem 2 is positive if $f, g\in
H^{1/2} (S^1, S^1)$.  This is a consequence of
Theorem 4.  The answer is still positive in a class of functions
strictly larger than $H^{1/2}$. The proof is based on

\proclaim{Theorem 5}  For every $f\in W^{1/3, 3} (S^1, S^1)$ we have
$$
VMO\text{-}\deg f = \underset{\varep \downarrow 0}\to{\lim}
\frac{1}{\varep^2} \sum_{\Sb n \in \Bbb Z\\ n\neq
0\endSb}|a_n|^2\frac{\sin^2 n\varep}{n}.\tag 3.12
$$
\endproclaim
\proclaim{Corollary 2}  Assume $f, g \in W^{1/3,3} (S^1, S^1)$ satisfy
(3.11).  Then
$$
VMO\text{-}\deg f = VMO\text{-}\deg g.
$$
\endproclaim
\proclaim{Corollary 3} {\rm (\text{J. P. Kahane [18]})}.  Assume
$f,g\in C^{0,\alpha} (S^1, S^1)$, with $\alpha > 1/3$, satisfy (3.11).
Then
$$
\deg f = \deg g.
$$
\endproclaim

Note that $C^{0,\alpha}\subset W^{1/3, 3}\quad\forall \alpha > 1/3$.
(This is an obvious consequence of Gagliardo's characterization
(1.1)).  Thus Corollary 2 implies Corollary 3.  Our proof of Theorem 5
is a straightforward adaptation of the ingenious argument of
J. P. Kahane [18] for $C^{0,\alpha}, \alpha > 1/3$.


\remark{Remark 2}  The conclusion of Theorem 5 holds if $f\in
W^{1/p, p}(S^1, S^1)$ with $1< p \leq 3$ (since $W^{1/p, p}\cap
L^\infty\subset W^{1/3,3}\,\ \forall p\leq 3$). [Note that when $1 < p
\leq 2$ the conclusion of Theorem 5 is an immediate consequence of Theorem 4 since
$\sum |n| |a_n|^2 < \infty$.  However in the range $2< p \leq 3$ the
conclusion is far from obvious since the series $|n| |a_n|^2$ may be
divergent].  It is interesting to point out that formula (3.12) {\it fails}
if one assume only $f\in W^{1/p, p} (S^1, S^1)$ with $p>3$.  In fact,
J. P. Kahane [18] has constructed an example of a function $f\in C^{0,
1/3} (S^1, S^1)$ such that $\deg f = 0$ while
$$
\underset{\varep \downarrow 0}\to{\lim } \frac{1}{\varep^2} \sum_{\Sb n\in
\Bbb Z\\ n\neq 0\endSb}|a_n|^2\frac{\sin^2 n\varep}{n} = \lambda,
$$
where $\lambda$ could be any real number $\lambda \neq 0$.  The heart
of the matter is the existence of a $2\pi$-periodic function $\varphi
\in C^{0, 1/3}(\Bbb R, \Bbb R)$ such that
$$
\int^{2\pi}_0 (\varphi(\theta + h) - \varphi(\theta))^3 d\theta =
\sin h\quad \forall h.
$$
This still leaves open the question whether Corollary 2 holds when
$W^{1/3,3}$ is replaced by $W^{1/p,p}, p > 3$.
\endremark
\medskip

Taking $p\to 1$ in Remark 2 suggests that Theorem 5 holds for
$f\in W^{1,1}$.  This is indeed true and there is even a stronger
statement:

\proclaim{Theorem 6}  For every $f\in C^0 (S^1, S^1) \cap BV (S^1,
S^1)$ we have
$$
\deg f = \underset{\varep\downarrow 0}\to{\lim} \frac{1}{\varep}
\sum^{+\infty}_{n=-\infty} |a_n|^2 \sin n\varep.
$$
\endproclaim


Consequently, we also have

\proclaim{Corollary 4}  Assume $f,g \in C^0(S^1,S^1)\cap BV (S^1,
S^1)$ satisfy (3.11).  Then
$$
\deg f = \deg g.
$$
\endproclaim
\remark{Remark 3}  It was already observed by J. Korevaar in [19] that
for every $f\in C^0\cap BV$ one has
$$
\deg f = \underset{ N\to \infty}\to{\lim} \sum^{+N}_{n =-N}
n|a_n|^2,
$$
which also implies Corollary 4.
\endremark

\demo{Proof of Theorem 5}  We follow the argument of
J. P. Kahane [18], except that we work in the fractional Sobolev space
$W^{1/3, 3}$ instead of the smaller H\"older space $C^{0, \alpha},
\alpha > 1/3$.  Set
$$
d= VMO-\deg f.
$$
By Theorem 3 (and Remark 10) in [15] we may write
$$
 f(\theta) = e^{i(\varphi(\theta) + d\theta)}
$$
for some $\varphi \in VMO (S^1, \Bbb R)$.  Applying Theorem 1 from [13] 
and the uniqueness of the lifting in VMO we know that $\varphi \in
W^{1/3, 3}$.

Write
$$
\int^{2\pi}_0 f(\theta + h) \bar f (\theta) d\theta = 2 \pi
\sum^{+\infty}_{n=-\infty} |a_n|^2 e^{inh} = \int^{2\pi}_0 e^{idh}
e^{i(\varphi(\theta+ h) - \varphi(\theta))} d\theta,\tag 3.13
$$
$$
e^{idh} = 1 + idh + O(|h|^2),\tag 3.14
$$
and
\medskip
$$
e^{i(\varphi(\theta + h) - \varphi(\theta))} = 1 + i(\varphi(\theta+
h) - \varphi(\theta))-\frac{1}{2} (\varphi(\theta + h) -
\varphi(\theta))^2 + O(|\varphi(\theta + h) - \varphi(\theta)|^3).\tag
3.15
$$

Thus
$$\aligned
&{\text I}{\text m}[e^{idh} e^{i(\varphi(\theta + h) - \varphi(\theta))}]={\text I}{\text m}[(1+idh)e^{i(\varphi(\theta + h) - \varphi(\theta))}] +O(|h|^2) \\
&=
(\varphi(\theta+ h) - \varphi(\theta)) + dh + O|h|^2) + O(|h|
|\varphi(\theta + h) - \varphi(\theta)|^2) + O(|\varphi(\theta + h) -
\varphi(\theta)|^3).\endaligned\tag 3.16
$$
Integrating (3.16) with respect to $\theta$ yields
$$
\Bigg|\sum^{+\infty}_{n =- \infty} |a_n|^2 \sin nh - dh\Bigg| \leq
C|h|^2 + C\int^{2\pi}_0|\varphi(\theta + h) - \varphi(\theta)|^3
d\theta.\tag 3.17
$$
Next, integrating (3.17) with respect to $h$ on $(0, 2\varep)$ gives
$$
\Bigg|\sum_{\Sb n\in \Bbb Z\\ n\neq 0\endSb} |a_n|^2
\left(\frac{1-\cos 2 n \varep}{n}\right) - 2d \varep^2\Bigg| \leq C
\varep^3 + C\int^{2 \varep}_0 d h \int^{2 \pi}_0 |\varphi(\theta + h)
-\varphi(\theta)|^3 d\theta
$$
and therefore
$$
\aligned &
\Bigg|\frac{1}{\varep^2} \sum_{\Sb n\in \Bbb Z\\ n\neq 0\endSb}
|a_n|^2 \ \frac{\sin^2 n\varep}{n} - d\Bigg|\\
&
\leq C\varep + \frac{C}{\varep^2} \int^{2\varep}_0 \int^{2\pi}_0
|\varphi (\theta + h) - \varphi(\theta) |^3 dh\ d \theta\\
&\leq C\varep + C\int^{2\varep}_0\int^{2\pi}_0\frac{|\varphi (\theta + h)
- \varphi(\theta) |^3}{|h|^2} dh\ d \theta,\endaligned
\tag 3.18
$$
which implies (3.12) since $\varphi\in W^{1/3, 3}$.
\enddemo

\demo{Proof of Theorem 6}  Since $f\in C^0\cap BV$, the corresponding
$\varphi$ satisfies $\varphi\in C^0 \cap BV$.  We return to (3.17)
with $h=\varep,$
$$
\Big| \frac{1}{\varep} \sum^{+\infty}_{h = - \infty} |a_n|^2 \sin
n\varep - d \Big|\leq C\varep + \frac{C}{\varep} \int^{2 \pi}_0
|\varphi(\theta+ \varep) - \varphi(\theta)|^3 d \theta.\tag 3.19
$$
Next we have
$$
\int^{2\pi}_0 |\varphi(\theta + \varep) - \varphi(\theta)
|d\theta\leq \varep \| \varphi\|_{BV}.\tag 3.20
$$
Inserting (3.20) in (3.19) gives
$$
\Big|\frac{1}{\varep} \sum^{+\infty}_{n=-\infty} |a_n|^2 \sin n\varep
- d\Big|\leq C\varep + C\underset{\theta}\to{\text{ Sup }}
  \|\varphi(\theta + \varep) - \varphi(\theta)\|_{L^\infty}^2\tag 3.21
$$
and the conclusion follows since $\varphi \in C^0$.
\enddemo
\subhead 4. New estimates for the degree \endsubhead
\medskip
Going back to (3.3) we see that, for every $f\in C^1(S^1, S^1)$
$$
|\deg f |\leq \sum |n|\ |a_n|^2.\tag 4.1
$$
Combining (4.1) with Gagliardo's characterization (1.1) of $H^{1/2}$
we find
$$
|\deg f| \leq C \int_{S^1}\int_{S^1} \frac{|f(x) - f(y)|^2}{|x-y|^2}
 dx\ dy.\tag 4.2
$$
In fact, the sharp estimate
$$
|\deg f | \leq \frac{1}{4\pi^2} \int_{S^1}\int_{S^1}
\frac{|f(x) - f(y)|^2}{|x-y|^2}
 dx\ dy\tag 4.3
$$
is an immediate consequence of (4.1) and
\proclaim{Lemma 5}  For every $f\in H^{1/2}$ one has
$$
\int_{S^1}\int_{S^1} \frac{| f(x) - f(x)|^2}{|x-y|^2} dx\ dy = 4 \pi^2
\sum^{+\infty}_{n =- \infty} |n |\ |a_n|^2\tag 4.4
$$
\endproclaim
\demo{Proof}  Write
$$
\aligned
\int_{S^1} \int_{S^1} \frac{|f(x) - f(y)|^2}{|x-y|^2} dx\ dy &= \int_0^{2 \pi}
\int^{2 \pi}_0 \frac{|\sum a_n e^{ in\theta} - \sum a_n
e^{in\psi}|^2}{|e^{i\theta} - e^{i\psi} |^2} d\theta \ d\psi \\
&= \int^{2\pi}_0 \frac{d\gamma}{|e^{i \gamma} - 1|^2} \int^{2\pi}_0
\big|\sum a_n (1-e^{in \gamma})e^{in \theta}\Big|^2 d\theta\\
&= 2 \pi \sum |a_n|^2\int^{2\pi}_0 \frac{|e^{in
\gamma} - 1|^2}{|e^{i\gamma} - 1|^2} d\gamma.\endaligned
$$
But, for $|n|\geq 1,$
$$
\frac{|e^{in \gamma} - 1|^2}{|e^{i\gamma} - 1|^2} = (e^{i(n-1)\gamma} +
\ldots + 1) (e^{-i(n-1)\gamma} + \ldots + 1)
$$
and thus
$$
\int^{2 \pi}_0 \frac{|e^{in \gamma} - 1|^2}{|e^{i\gamma} - 1|^2}
\  d\gamma = 2 \pi |n|.
$$
Inserting this into the previous equality yields (4.4).
\enddemo

\remark{Remark 4}  Inequality (4.3) can be viewed as an estimate for
the ``least amount of $H^{1/2}$-energy'' necessary to produce a map
$f:S^1-S^1$ with assigned degree.  More precisely we have
$$
\underset{\Sb f: S^1 \to S^1\\ \deg f = n\endSb}\to{\text{ Inf }}
\int_{S^1}\int_{S^1} \frac{|f(x)-f(y)|^2}{|x-y|^2} dx\ dy = 4\pi^2
|n|\tag 4.5
$$
and the Inf in
(4.5) is achieved when $f(\theta) = e^{in\theta}$.  The existence of a
minimizer for similar problems where the standard $H^{1/2}$ norm is
replaced by equivalent norms (e.g. the trace of an $H^1$ norm on the
disc with variable coefficients) is a very delicate question because
of ``lack of compactness''; we refer to [20]
\endremark

\remark{Remark 5}  Estimate (4.2) serves as a building block in the
study of the least $H^{1/2}$-energy of maps $u: S^2\to S^1$ with
prescribed singularities.  Such a question has been investigated in
[5].  More precisely, recall that

$$
\|u \|^2_{H^{1/2}(S^2)} = \int_{S^2}\int_{S^2} \frac{|u(x)-
u(y)|^2}{|x-y|^3} dx\ dy.
$$
Given points $\Sigma = \{ p_1, p_2, \ldots, p_k\} \cup\{ n_1, n_2,
\ldots, n_k\}$ consider the class of maps
$$
A= \{ u \in C^1(S^2\setminus \Sigma, S^1); \deg (u, p_i)= + 1 \text{ and
}\deg (u, n_i) = - 1, \quad \forall i\}.
$$
\endremark

\proclaim{Theorem 7} {\rm (\text{Bourgain-Brezis-Mironescu [5]})}.  There exist
absolute constants \newline $C_1, C_2 > 0$ such that
$$
C_1 L(\Sigma) \leq \underset{u\in A}\to {\text{\rm{Inf}}} \|u\|^2_{H^{1/2}(S^2)}\leq C_2 L
(\Sigma)\tag 4.6
$$
where $L(\Sigma)$ is the length of a minimal connection connecting the
points $(p_i)$ to the points $(n_i)$.
\endproclaim

Theorem 7 is the $H^{1/2}$-version of an earlier result [12]
concerning $H^1$ maps from $S^3$ into $S^2$ with singularities which
had been motivated by questions arising in liquid crystals with
point defects, while the analysis in [5] has its source in the
Ginzburg-Landau model for superconductors.  It is the LHS inequality in (4.6)
which is related to (4.2).  The RHS inequality in (4.6) comes from a ``brute
force" construction called the ``dipole construction''.

\remark{Remark 6}  An immediate consequence of (4.3) is the estimate
$$
|\deg f| \leq \frac{1}{\pi^2} \int_{S^1}\int_{S^1} \frac{|f(x) -
 f(y)|^p}{|x-y|^2}\quad \forall f \in C^1(S^1, S^1), \forall p \in
 (1,2).\tag 4.7
$$
Estimate (4.7) deteriorates as $p\downarrow 1$ since the RHS in (4.6)
tends to $+\infty$ unless $f$ is constant (see [4]).  It would be
desirable to improve the constant $(1/\pi^2)$ and establish that
$$
|\deg f|\leq C_p\int_{S^1}\int_{S^1} \frac{|f(x) - f(y)|^p}{|x-y|^2}
 dx \ dy\quad \forall f \in C^1(S^1, S^1), \forall p \in (1,2).\tag
 4.8
$$
with a constant $C_p \sim (p-1)$ as $p\downarrow 1$.  In the limit as
$p\downarrow 1$, one should be able to recover (in the spirit of [4])
the obvious inequality
$$
|\deg f| \leq \frac{1}{2\pi} \int |\dot f|.\tag 4.9
$$

Inequality (4.8) is also valid for $p>2$, but it cannot be deduced
from (4.3) and its proof requires much work.
\endremark

\proclaim{Theorem 8}  {\rm(\text{Bourgain-Brezis-Mironescu [6]})}.  For every
$p>1$, there is a constant $C_p$ such that for any (smooth) $f: S^1
\to S^1$
$$
|\deg f | \leq C_p \int_{S^1}\int_{S^1} \frac{|f(x) -
 f(y)|^p}{|x-y|^2} = C_p\|f\|^p_{W^{1/p, p}}.\tag 4.10
$$
\endproclaim

The proof of (4.10) we present in [6] makes use of the harmonic
extension of $f$ inside the disc (and the machinery of $W^{s,p}$-trace
theory).  It would be desirable to find a more direct proof of (4.10).

There is an estimate stronger than (4.10):

\proclaim{Theorem 9} {\rm(\text{Bourgain-Brezis-Mironescu [7]})}.  For any $\delta
> 0$ sufficiently small there is a constant $C_\delta$ such that,
  $\forall f \in C^0 (S^1, S^1)$,
$$
|\deg f | \leq C_\delta \underset{ [|f(x) -
 f(y)|>\delta]}\to{\int_{S^1}\int_{S^1}} \frac{1}{|x-y|^2} dx\ dy.\tag
 4.11
$$
\endproclaim

The only proof we know for (4.11) is very involved and it is natural
to raise
\medskip

\noindent{\bf Open problem 3.}  Find a simpler proof for (4.11). Also, is there a more precise estimate of the form
$$
|\deg f| \leq C\delta
\underset{[|f(x) - f(y)|>\delta]}\to{\int_{S^1}\int_{S^1}}
\frac{1}{|x-y|^2} dx\ dy\tag 4.12
$$
with $C$ independent of $\delta$?

In the spirit of [4] one might then be able to recover (4.9) as
$\delta \to 0$.
\medskip

\noindent{\bf Higher dimensional analogs.}


Theorem 8 can be extended to higher dimensions:

\proclaim{Theorem 8$'$} {\rm(\text{Bourgain-Brezis-Mironescu [6]})}.
Let $n\geq 1$.  For every $p> n$ there is a constant $C(p,n)$
such that for any (smooth) $f: S^n \to S^n$,
$$
|\deg f | \leq C(p, n) \int_{S^n}\int_{S^n}\frac{|f(x) -
 f(y)|^p}{|x-y|^{2n}} dx \ dy = C(p,n) \|f\|^p_{W^{n/p, p}}.\tag 4.13
$$
\endproclaim

We have not been able to generalize Theorem 9 to higher dimensions.  A
natural analog would be
\medskip

\noindent{\bf Open problem 4.}  Are there constants $\delta \in (0,1)$ and
$C$ such that, $\forall f \in C^0 (S^1, S^n)$,
$$
|\deg f | \leq C\underset{ [|f(x) - f(y)|>
 \delta]}\to{\int_{S^n}\int_{S^n}} \frac{1}{|x-y|^{2n}} dx \ dy?\tag
 4.14
$$

In a different direction, it might be interesting to estimate other
topological invariants in terms of fractional Sobolev norms.  One of
the simplest examples could be 
\medskip

\noindent{\bf Open problem 5}.  Does one have
$$
|\text{ Hopf-degree } f |\leq C_p\int_{S^3}\int_{S^3} \frac{|f(x) -
 f(y)|^p}{|x-y|^6} \quad \forall p > 3, \forall f \in C^1(S^3, S^2)?
$$
\medskip


\noindent{\bf Acknowledgements.}  I am very grateful to J. P. Kahane
 for a personal communication [18] which has led me to Theorem 5 (and
 Corollary 2).  Special thanks to H.Furstenberg and to P. Mironescu for useful
 conversations.  This work is partially supported by an EC Grant
 through the RTN Program ``Front-Singularities'' HPRN - CT - 2002 - 00274.



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