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\topmatter
\title
Lifting, Degree and Distributional Jacobian Revisited\endtitle
\author  
{JEAN  BOURGAIN$^{(1)}$,  HA\"IM BREZIS$^{(2),(3)}$ AND PETRU MIRONESCU$^{(4)}$}
\endauthor
\bigskip



  \address
  ${}^{\text {(1)}}$ INSTITUTE FOR ADVANCED STUDY\endgraf
  PRINCETON, NJ  08540\endgraf
  \endaddress
  \email
   bourgain\@math.ias.edu
  \endemail
  \null
  \address 
  ${}^{\text {(2)}}$  LABORATOIRE J. -L. LIONS\endgraf
   UNIVERSIT\'E P. ET M. CURIE, B.C. 187\endgraf
   4 PL. JUSSIEU\endgraf
  75252 PARIS CEDEX 05\endgraf
  \endaddress
  \email
  brezis\@ccr.jussieu.fr
  \endemail
  \address ${}^{\text{(3)}}$ RUTGERS UNIVERSITY\endgraf
  DEPT. OF MATH., HILL CENTER, BUSCH CAMPUS\endgraf
   110 FRELINGHUYSEN RD, PISCATAWAY, NJ 08854\endgraf
  \endaddress
  \email 
   brezis\@math.rutgers.edu\endemail
  \null
  \address ${}^{\text {(4)}}$ D\'EPARTEMENT DE MATH\'EMATIQUES\endgraf
   UNIVERSIT\'E PARIS-SUD\endgraf
   91405 ORSAY\endgraf
    \endaddress
   \email 
   Petru.Mironescu\@math.u-psud.fr\endemail
  \bigskip

\endtopmatter
\document


\subhead 0.  Introduction\endsubhead

  Let $g:I = (0, 1) \to S^1$.  If $g\in VMO$, we may write
  $g=e^{i\varphi}$ for some $\vp\in$VMO ; this $\varphi$ is unique modulo $2\pi$
  (see [BN] and the earlier work [CM]). There is no control of
  $|\varphi|_{BMO}$ in terms of $|g|_{BMO}$, since we always have
  $|g|_{BMO}\leq 2$ and $|\varphi|_{BMO}$ can be arbitrarily large ;
  recall, however, that, when $|g|_{BMO}$ is sufficiently small, there
  is a linear estimate $|\varphi|_{BMO} \leq C|g|_{BMO}$ (see
  [CM], Theorem 4 in [BN], and Remark 1 below).


We are going to establish that a norm slightly stronger than
$|g|_{BMO}$ {\bf does} control $|\varphi|_{BMO}$.  Consider, for $1 < p
< \infty, \;  0 < s < 1,$ the fractional Sobolev space $W^{s,p}(I)$,
equipped with its standard seminorm
$$
|g|_{s,p} = \left( \int\limits_I\int\limits_I \frac{|g(x) -
g(y)|^p}{|x-y|^{1+sp}}  dxdy\right)^{1/p}.
$$
Set
$$
W^{s,p} (I; S^1) = \{ g\in W^{s, p}(I;\Bbb R^2); |g| = 1\ \hbox{\rm a.e.}\}.
$$

Recall (see [BBM1]) that, if $g\in W^{1/p,p} (I;S^1)$, then
$g=e^{i\varphi}$ for some 
$\varphi\in W^{1/p,p} (I;\Bbb R)$ ; this $\varphi$ is unique modulo
$2\pi$ .  Again, there is no estimate of
$|\varphi|_{1/p,p}$ in terms of $|g|_{1/p,p}$.  
The canonical example (see [BBM1]) is the following : let
$$
\varphi_n(x) = \cases 0, & \text{ if }\quad 0<x< 1/2\\
          2n \pi(x-1/2), & \text{ if }\quad  1/2 \leq x \leq 1/2 + 1/n\\
                  2\pi , & \text{ if }\quad  x> 1/2 + 1/n\endcases .
$$
Then 
$|\varphi_n|_{1/p, p} \to \infty$, while $|e^{i\varphi_n}|_{1/p, p}\leq C$.  

In view of the injection
$$
W^{1/p, p} (I) \hookrightarrow VMO (I), \; 1 < p< \infty,
$$
(see, e.g., [BN] and [T]), it is natural to ask whether a control
of $|g|_{1/p,p}$ yields a control of $|\varphi|_{BMO}$.  This is
indeed true :
\proclaim{Theorem 1}  Let $1< p < \infty$.  Let $\varphi \in
W^{1/p,p}(I;\Bbb R)$ and $g = e^{i\varphi}$.  Then 
$$
|\varphi|_{BMO} \leq C_p (|g|^p_{1/p, p} + |g|_{1/p, p}).\tag 0.1
$$
\endproclaim

\remark{Remark 1}  The $p-th$ power growth in (0.1) is optimal when
$|g|_{1/p, p}$ is large.  This is easily seen by choosing $\varphi_n
(x) = n x$.  When $|g|_{1/p,p}$ is small, the linear growth in (0.1) is
a special case of a result of [CM], namely
$$
|\varphi|_{BMO} \leq C|g|_{BMO} \;  \text{ if } \; | g|_{BMO} \leq
 \delta,\tag 0.2
$$
where $\delta$ is a sufficiently small constant.
\endremark
\remark{Remark 2}  When $p=2$, estimate (0.1) can be derived from
Theorem 3 in [BBM4] (announced in [BBM2] ;  see also [BB]) which
asserts that, if $g\in H^{1/2} (I; S^1)$, then we may write
$g=e^{i(\varphi_1 + \varphi_2)}$, with 
$$
|\varphi_1|_{1/2, 2} \leq C|g|_{1/2, 2}\tag 0.3
$$
and
$$
|\varphi_2|_{W^{1,1}} \leq C|g|^2_{1/2, 2}.\tag 0.4
$$
Since
$$
|\varphi_1 + \varphi_2|_{BMO} \leq C(|\varphi_1|_{1/2, 2} +
 |\varphi_2|_{W^{1,1}}),
$$
estimate (0.1) for $p=2$ follows from (0.3), (0.4).

\bigskip
Note that, if Theorem 1 holds for some $p$, it also holds for
every $q\in (1, p)$ ; this follows from (0.1) and (0.2). Hence
Theorem 1 for $1< p \leq 2$ is a consequence of (0.3) - (0.4). The main
novelty concerns the case $p> 2$ ; our argument relies on a completely
different approach.  In fact, we do not know whether (0.3) - (0.4) still
hold when 2 is replaced by $p$ :
\bigskip

\noindent{\bf Open Problem 1.}  Let $\varphi\in C^\infty(\bar I; \Bbb
R)$, $g = e^{i\varphi}$ and $p>2$.  Does there exist a decomposition
$\varphi = \varphi_1 + \varphi_2$, with 
$$
|\varphi_1|_{1/p, p} \leq C|g|_{1/p, p}\tag 0.3$'$
$$
and 
$$
|\varphi_2|_{W^{1,1}} \leq C|g|_{1/p, p}^p\ ?   \tag 0.4$'$ 
$$
Same question when $I$ is replaced by $(0, 1)^N$.

\bigskip
An immediate consequence of Theorem 1 is the following 
\endremark
\proclaim{Corollary 1}  Set $Q=(0,1)^N$.  Let $N<p<\infty$, $\varphi \in
W^{N/p,p}(Q ; \Bbb R)$ and $g= e^{i\varphi}$.  Then 
$$
|\varphi|_{BMO} \leq C_{p,N}(|g|^p_{N/p,p} + |g|_{N/p, p}).\tag 0.5
$$
\endproclaim

We now turn to similar questions for the degree.  If $g\in VMO (S^1;
S^1)$, then $g$ has a well-defined degree (see [BN]).  Clearly, there
is no estimate of the degree in terms of $|g|_{BMO}$ ; however, $\deg g
= 0$ provided $|g|_{BMO}$ is sufficiently small (see [BN]).  An easy
consequence of Theorem 1 asserts that $\deg g$ can be controlled in
terms of $|g|_{1/p, p}$ :

\proclaim{Corollary 2}  Let $1< p< \infty$ and 
$g\in W^{1/p,p}(S^1; S^1)$.  Then 
$$
|\deg g| \leq C_p |g|^p_{1/p, p}.\tag 0.6
$$
\endproclaim

When $p=2$, estimate (0.6) was well-known:  
it may be easily deduced from the degree formula
$$
\deg g = \frac{1}{2i\pi} \int_{S^1} \frac{\dot g}{g} =
\frac{1}{2i\pi} \la \bar g, \dot g\ra_{H^{1/2}, H^{-1/2}},\tag 0.7
$$
which implies that
$$
|\deg g | \leq C|g|^2_{1/2, 2}.
$$

Estimate (0.6) can be obtained from Theorem 1 as follows :  set $h(t) =
g(e^{it})$, $t\in \Bbb R$, and write $h=e^{i\varphi}$.  Note that 
$$
|\deg g| = \frac{1}{ 4\pi^2}\int^{2\pi}_0|\varphi(t+2\pi) -
 \varphi(t)| dt \leq C|\varphi|_{BMO(0, 4\pi)}\tag 0.8
$$
and apply Theorem 1 on $(0, 4\pi)$.

\bigskip
Corollary 2 extends to higher dimensions :
\proclaim{Theorem 2}
Let $p> N$ and $g\in W^{N/p, p} (S^N; S^N)$.  Then
$$
|\deg g |\leq C_{p, N} \;  |g|^p_{N/p, p}.\tag 0.9
$$
\endproclaim

Although the conclusions of Theorems 1 and 2 are different in nature,
the proofs we present below bear some similarities.

\remark{Remark 3}  For $g\in W^{1,N}(S^N; S^N)$, the estimate
$$
|\deg g| \leq C_N \int_{S^N} |\nabla g |^N\tag 0.9$'$
$$
is well-known and follows from  Kronecker's formula
$$
\deg g = \Mint_{S^N} \det (\nabla g)=\Mint_{S^N} \det (\nabla g , g)\ \tag 0.10
$$
(in the first integral, $g$ is regarded as a map from $S^N$ into
itself and ``det'' denotes the determinant
of an $N\times N$ matrix ;  in the second integral, $g$ is considered
as a $\Bbb R^{N+1}$-valued map, and ``det'' denotes the determinant of
an $(N+1)\times (N+1)$ matrix).
\endremark


In fact, we will use (0.10) in the proof of Theorem 2.  It is
presumably possible to rederive (0.9$'$) as a limiting case of (0.9) via a
careful analysis of $C_{p, N}$ as $p\searrow N$, in the spirit of
[BBM3].

Estimate (0.9), which asserts that for every $p>N$,
$$
|\deg g|\leq C_{p,N}\int_{S^N}\int_{S^N} \frac{|g(x) -
 g(y)|^p}{|x-y|^{2N}}dxdy,
$$
suggests the following stronger estimate:

\noindent{\bf Open Problem 2.}  Is it true that, for every $g\in
C^0(S^N; S^N)$,
$$
|\deg g| \leq C_N\iint_{\{(x, y) \in S^N \times S^N; |g(x)-g(y)|>
 1/10\}} |x-y|^{-2N} dxdy \,\quad { } ?
$$
The answer to Open Problem 2 is positive when $N=1$; the proof is
given in [BBM5], where we also present an improvement of Theorem 1 in
the same spirit.

\medskip


We next discuss the distributional Jacobian of maps $g\in
W^{N/p,p}(S^{N+1}; S^N)$. Recall that, if $g$ is a smooth map from
$S^{N+1}$ into $\Bbb R^{N+1}$, its distributional Jacobian is defined
through its action on smooth functions $\zeta\in C^\infty(S^{N+1};\Bbb
R)$ by the formula 
$$
\la\text{\rm Det} (\nabla g), \zeta \ra = -\frac{1}{N+1} \sum^{N+1}_{j=1}
\int_{S^{N+1}} \zeta_{x_j}\det (g_{x_1}, \ldots,
g_{x_{j-1}},g, g_{x_{j+1}},\ldots, g_{x_{N+1}})\ ;\tag 0.11
$$
here, the derivatives are computed pointwise in an orthonormal frame
such that \newline $(x_1, \ldots, x_{N+1}, n)$ is direct, where $n$ is the
outward normal to $S^{N+1}$ (this integrand is frame invariant).

Note that formula (0.11) still makes sense when 
$$
g\in W^{1,N} (S^{N+1};\Bbb R^{N+1})\cap L^\infty
$$
and $\zeta \in W^{1,\infty} (S^{N+1}; \Bbb R)$.  Observe also that
if 
$g\in C^1(S^{N+1}; S^N)$, then its Jacobian determinant vanishes pointwise.
By density, it follows that Det$(\nabla g)=0$ for every 
$g\in W^{1, N+1}(S^{N+1} ; S^N)$.
On the other hand, 
it is standard to construct maps in $W^{1, N}(S^{N+1} ; S^N)$ 
(and even in
$W^{1, q}$, $\forall\ q<N+1$), e.g., with point singularities, such
that Det$(\nabla g)\neq 0$
 (see, e.g., [BCL]).

One of the main goals of this paper is to give a meaning to the
distribution Det$(\nabla g)$ for maps $g : S^{N+1}\to S^N$ which
do not necessarily belong to $W^{1,N}$.  It has been observed in [BBM2] (see also [BBM4]) 
that it is possible to
define Det$(\nabla g)$ for $g\in H^{1/2}(S^2; S^1)$.  The construction
there was painless (using the fact that $H^{1/2}$ is the trace space
of $H^1$).  The same technique allows to define Det$(\nabla g)$ for 
$g\in W^{N/(N+1), N+1}(S^{N+1}; S^N)$. Consequently, Det$(\nabla g)$ makes
sense for $g\in W^{N/p,p}(S^{N+1}; S^N), N\leq p \leq N+1$.  In this
paper, we are able to define Det$(\nabla g)$ for $g\in W^{N/p,
p}(S^{N+1}; S^N)$ in the more delicate case where $N+1<p<\infty$.
The new idea involves an adaptation of the method (and the estimates) 
introduced in the proof of Theorem 2.

\bigskip
Our main result is the following
\proclaim{Theorem 3}  Let $N< p < \infty$.  There exists a (unique)
strongly continuous map
$$
T\, :\, W^{N/p,p}(S^{N+1}; S^N)\to (W^{1, \infty}(S^{N+1}))^*
$$ 
such that, for every $\zeta\in\ W^{1, \infty}(S^{N+1}; \Bbb R)$,
$$
|\la T(g),\zeta\ra |\le C_{p, N}|g|_{N/p,p}^p\| \nabla\zeta\|_{L^\infty},\quad\forall\ g\in W^{N/p,p}\tag 0.12
$$
and
$$
\la T(g),\zeta\ra =\la\text{\rm Det}(\nabla g),\zeta\ra ,\quad\forall\ g \in W^{1,N}\cap W^{N/p,p}.\tag 0.13
$$ 

For each $g\in W^{N/p,p}(S^{N+1};S^N)$, there are sequences $(P_i),
(N_i) \subset S^{N+1}$ such that 
$$
\sum_i |P_i - N_i|\leq C_p|g|^p_{N/p,p}\tag 0.14
$$
and
$$
\la T( g), \zeta \ra=\omega_{N+1} \sum(\zeta(P_i)
-\zeta(N_i)),\quad \forall\ \zeta \in W^{1,\infty}(S^{N+1}; \Bbb R).\tag
 0.15
$$
If $g\in W^{N/p,p}(S^{N+1};S^N)\cap C^0(S^{N+1}\setminus A)$, where $A$ is a finite set, then we may choose $P_i,\, N_i\in A$.

Moreover, we have
$$
\la T( g), \zeta\ra  = \omega_{N+1} \int\limits_\Bbb R\deg(g;\Gamma_\lambda)
d\lambda, \quad \forall\ \zeta\in C^\infty(S^{N+1}; \Bbb R).\tag 0.16
$$
\endproclaim
Here, $\omega_{N+1}$ is the volume of the unit ball in $\Bbb R^{N+1}$ and, for each regular value $\lambda$ of $\zeta$, $\Gamma_\lambda$ is
the level set
$\d
\Gamma_\lambda = \{ x\ ;\ \zeta(x) = \lambda\}
$,
positively oriented with respect to the outward normal  of the open set $\{ x\in S^{N+1}\ ;\ \zeta (x) >\lambda\}$.


Note that, for a.e. $\lambda$, $g_{|\Gamma_\lambda} \in
W^{N/p,p}(\Gamma_\lambda; S^N)\subset$ VMO $(\Gamma_\lambda; S^N),$ so
that $\deg (g;\Gamma_\lambda)$ makes sense (by [BN]).

\remark{ Remark 4} a) Since $W^{1,N}(S^{N+1} ; S^N)\cap W^{N/p,p}$ is dense in $W^{N/p,p}(S^{N+1} ; S^N)$, $N<p<\infty$ (see Appendix), it follows that $T$ is the unique extension of the distributional Jacobian restricted to $W^{1,N}(S^{N+1} ; S^N)\cap W^{N/p,p}$.

\noindent
b) If $N\ge 2$, we have $W^{1,N}\cap L^\infty\subset W^{N/p,p}$, $N<p<\infty$ (see, e.g., [RS]), and thus $W^{1,N}(S^{N+1} ; S^N)\cap W^{N/p,p}=W^{N/p,p} (S^{N+1} ; S^N)$. However, this conclusion fails when $N=1$.

\noindent
c) We will establish in Section 2 that  $T( g)$ is ``intrinsic'' ;
more precisely, if $g\in W^{N/p,p}$, then $g\in W^{N/q,q}$ for every
$q>p$, and the two definitions of $T(g)$ (relative to $p$
and to $q$) coincide.  

\noindent
d) We have reached here the ``largest'' Sobolev
class to which one can extend the distributional Jacobian ; when $sp <N$, there is
no good definition of the distributional Jacobian in the class
$W^{s,p}(S^{N+1}; S^N)$ (see [BBM5]).

\noindent
e) Formula (0.15) has its source in [BCL] for special maps (having a finite
number of singularities) ; the general case (0.15) is an extension of Theorem 1 in [BBM4].
\endremark


\bigskip
%As we are going to see in Section 2, the proof of Theorem 3 relies heavily on the estimates used %in proving Theorem 2.


%When $g$ belongs to the ``natural'' class $W^{1,N}(S^{N+1};S^N)$,
%Det$(\nabla g)$ is a distribution with a remarkable property, namely 
%$$
%\text{ Det}(\nabla g) = \omega_{N+1}\sum^\infty_{i=1}(\delta_{P_i} -
%\delta_{N_i})\ ;\tag 0.12
%$$ 
%here, $\omega_{N+1}$ is the volume of the unit ball in $\Bbb R^{N+1}$
%and $(P_i), (N_i) \subset S^{N+1}$ are such that 
%$$
%\sum|P_i - N_i| \leq C_N\int|\nabla g|^N.\tag 0.13
%$$
%
%Formula (0.12) has its source in [BCL] for special maps (in the class
%$\Cal R$ defined below) ; the general case is an adaptation of the
%proof of Theorem 1 in [BBM4].
%
%One of the main goals of this paper is to give a meaning to the
%distribution Det$(\nabla g)$ for $g$ belonging to a larger class than
%$W^{1,N}$.  It has been observed in [BBM4] that it is possible to
%define Det$(\nabla g)$ for $g\in H^{1/2}(S^2; S^1)$.  The construction
%there was painless (using the fact that $H^{1/2}$ is the trace space
%of $H^1$).  The same idea allows to define Det$(\nabla g)$ for $g\in
%W^{N/(N+1), N+1}(S^{N+1} ; S^N)$. Consequently, Det$(\nabla g)$ makes
%sense for $g\in W^{N/p,p}(S^{N+1} ; S^N)$, $N\leq p \leq N+1$.  In this
%paper, we are able to define Det$(\nabla g)$ for $g\in W^{N/p,
%p}(S^{N+1}; S^N)$ in the more delicate case where $N+1<p<\infty$.
%The proof relies on variants of Theorem 2.
%
%
%As a first step towards defining Det$(\nabla g)$ for $g\in W^{N/p,p}
%(S^{N+1} ; S^N)$, let $g$ belong to the class 
%
%$$\aligned
%\Cal R = \{ & g\in W^{1,q}(S^{N+1} ; S^N), \text{ for every } 1\leq q < N+1\ ;
%\ g\in C^\infty(S^{N+1}\setminus A), \\
%& \text{where }A\text{ is a finite set }\}.\endaligned $$
%
%A straightforward adaptation of Lemma 2 in [BBM4] yields, for every $g\in \Cal R$, the following
%formula :
%$$
%\la \text{\rm Det} (\nabla g), \zeta\ra = \omega_{N+1}\sum_{a\in A} d_a
%\zeta(a), \quad\forall\ \zeta \in W^{1,\infty} (S^{N+1}; \Bbb R)\ ;\tag 0.14
%$$
%here
%$d_a$ is the degree of $g$ computed on small spheres (in $S^{N+1}$)
%around $a$ with the appropriate orientation.
%
%A basic new estimate is, for every $g\in \Cal R, $
%$$
%|\la \text{\rm Det}(\nabla g), \zeta \ra|\leq C_{p,N} |g|^p_{N/p,p} \|\nabla
% \zeta\|_{L^\infty}, \ N<p<\infty, \tag 0.15
%$$
%which in turn is equivalent (via [BCL]) to 
%$$
%L(g) \leq C|g|^p_{N/p, p}.\tag 0.16
%$$
%Here, 
%$$
%L(g) = \underset{\sigma \in S_k}\to{\text{ Min }} |P_i -
%N_{\sigma(i)}|,
%$$
%and the points $P_1, \ldots, P_k, N_1,\ldots, N_k$ are the points
%$a\in A$, repeated with multiplicity $|d_a|$.
%
%As we are going to see in the Appendix, the class $\Cal R$ is dense in
%$W^{N/p,p}(S^{N+1};S^N),$ $N<p< \infty$.  For $N=1$, this property was
%established in [BBM4] (following earlier works of F. Bethuel and
%X. Zheng [BZ], F. Bethuel [B], M. Escobedo [E] and T. Rivi\`ere [R]).
%The argument in [BBM4] can be easily adapted to arbitrary $N$.
%
%We finally prove
%
%\proclaim{Theorem 3}  Let $N< p < \infty$.  The map $\Cal R \ni
%g\longrightarrow\text{\rm Det}(\nabla g)\in (W^{1, \infty}(S^{N+1}))^*$
%extends by density to $W^{N/p,p}(S^{N+1}; S^N)$.
%
%For each $g\in W^{N/p,p}(S^{N+1};S^N)$, there are sequences $(P_i),
%(N_i) \subset S^{N+1}$ such that 
%$$
%\sum_i |P_i - N_i|\leq C_p|g|^p_{N/p,p}
%$$
%and
%$$
%\la\text{\rm Det}(\nabla g), \zeta \ra=\omega_{N+1} \sum(\zeta(P_i)
%-\zeta(N_i)),\quad \forall\ \zeta \in W^{1,\infty}(S^{N+1}; \Bbb R).\tag
% 0.17
%$$
%Moreover, we have
%$$
%\la\text{\rm Det}(\nabla g), \zeta\ra  = \omega_{N+1} \int\deg(g;\Gamma_\lambda)
%d\lambda, \quad \forall\ \zeta\in C^\infty(S^{N+1}; \Bbb R).\tag 0.18
%$$
%\endproclaim
%
%Here, for each regular value $\lambda$ of $\zeta, \Gamma_\lambda$ is
%the level set
%$$
%\Gamma_\lambda = \{ x; \zeta(x) = \lambda\}
%$$
%with the appropriate orientation.
%
%Note that, for a.e. $\lambda$, $g_{|\Gamma_\lambda} \in
%W^{N/p,p}(\Gamma_\lambda; S^N)\subset$ VMO $(\Gamma_\lambda; S^N),$ so
%that $\deg (g;\Gamma_\lambda)$ makes sense (by [BN]).
%
%As we are going to see in Section 2, Det$(\nabla g)$ is ``intrinsic'' ;
%more precisely, if $g\in W^{N/p,p}$, then $g\in W^{N/q,q}$ for every
%$q>p$, and the two definitions of Det$(\nabla g)$ (relative to $p$
%and to $q$) coincide.  We have reached here the ``largest'' Sobolev
%class in which one can define Det$(\nabla g)$ ; when $sp <N$, there is
%no good definition of the distributional Jacobian in the class
%$W^{s,p}(S^{N+1}; S^N)$ (see [BBM5]).


\bigskip

%Finally, we apply Theorem 2 to the study of minimal connections ; this

%is closely related to [BBM2].  Let $p> N$ and $g\in W^{N/p, p}

%(S^{N+1}; S^N)$.  As in [BCL] and [BBM2], we define a distribution

%$T(g)$ which describes the location and degree of the topological

%singularities of $g$.  Its action on a smooth function $\zeta\in

%C^\infty (S^{N+1};\Bbb R)$ is 

%$$

%\la T(g), \zeta\ra = \int \deg (g, \Gamma_\lambda) d\lambda\tag 0.11

%$$

%where $\Gamma_\lambda=\{x; \zeta (x) = \lambda\}$.  It is not clear

%from this definition neither that $\la T(g), \zeta\ra$ makes sense,

%nor that it is linear in $\zeta$.

%

%\proclaim{Theorem 3}  There are points $P_i, N_i \in S^{N+1}$ such

%that 

%$$

%\sum^\infty_{i=1} |P_i - N_i|\leq C(|g|^p_{N/p,p} + 1)\tag 0.12

%$$

%and 

%$$

%\la T(g), \zeta \ra = |S^N|\sum^\infty_{i=1} (\zeta(P_i) -

%\zeta(N_i)), \forall\ \zeta \in C^\infty (S^{N+1}; \Bbb R).\tag 0.13

%$$

%\endproclaim

%

%As a consequence of Theorem 3, we may define  the length of the minimal connection of $g$,

%$$

%L(g)= \frac{1}{|S^N|} \text{\rm Max}\{\la T(g), \zeta\ra; |\nabla

%\zeta|\leq 1\text{ on } S^{N+1}\}\leq C(|g|^p_{N/p,p} + 1).\tag 0.14

%$$

%

%

%The proof of Theorem 3 relies heavily on Theorem 2 combined with a

%density result concerning maps with a finite number of singularities, established in Appendix A.

\subhead 1. Proofs of Theorems 1 and 2
\endsubhead

Let $g\in VMO(S^N; S^N)$ and let $u$ be its harmonic extension to $B^{N+1}$ (with values into $B^{N+1}$).  Let $v(x, \varep) = u((1-\varep) x)$, $x\in S^N$, $0< \varep
\leq 1$.  We have
$$|v(x, \varep)|\to 1\text{ uniformly in $x$ as } \varep \to 0,\tag 1.1$$
%$$\exists \gamma > 0 \text{ such that } |g|_{BMO(B_\varep(x))} \leq \gamma
%\Rightarrow |v(x, \varep)|\geq 1/2,\tag 1.2$$
$$|\nabla v(x, \varep)|\leq C/\varep, \;  \forall\ x \in S^N, \text{
where } C \text{ is an absolute constant}\ ;      \tag 1.2$$
(for the proof of (1.1), see [BN]).

Set, for every $x\in S^N$,

$$
d (x) = \cases 1/2,  \text{ if }\quad |v(x, \varep)
                         |>1/2, \text{ for every } \varep \in (0, 1/2]\\
                         \text{Min}\{\varep \in (0,1/2]\ ;\
|v(x, \varep)|\leq 1/2 \}, \text{ otherwise }\endcases .
$$

\noindent
In other words, $d(x)$=min$(\ell(x),1/2)$, where $\ell(x)$ is the length of the largest radial interval coming from $x\in S^N$, on which $|u|\geq 1/2.$

Clearly,
$$
G = \{ y \in B^{N+1}\ ;\ |u(y)|\leq 1/2\} \subset \bigcup_{x\in S^N} [0, (1-d(x))x].
\tag 1.3
$$
We start with the following ingredient which is of interest in itself.
\proclaim{Theorem 4} For $g\in C^1 (S^N; S^N)$, we have
$$|\deg g | \leq C\ I(g), $$
where
$$ I(g)=\int_{S^N}\frac{1}{(d(x))^N}.$$
\endproclaim
\noindent
The proof of Theorem 4 relies on
\proclaim{Lemma 1}  We have 
$$
\int_G |\nabla u |^{N+1} \leq C \; I(g).\tag 1.4
$$
\endproclaim
\demo{Proof of the lemma}  By (1.2) and (1.3), we have 
$$
\aligned
\int_G |\nabla u |^{N+1} dy &\leq C\int_{S^N}\bigg( \int^{1-d(x)}_{0}
\frac{r^N}{(1-r)^{N+1}} \; dr \bigg) d{x}    \\
&\leq C\int_{S^N} \bigg(
\int^{1-d(x)}_0\frac{1}{(1- r)^{N+1}} \; dr\bigg) d{x}  = C'I(g).
\endaligned
$$
\enddemo
\demo{Proof of Theorem 4} Set, for $y\in B^{N+1}$, 
$$
\tilde u (y) = \cases u(y)/|u(y)|, & \text{ if } \;  |u(y)|> 1/2\\
                      2u (y),      & \text{ if } \;  |u(y)|\leq
                      1/2.\endcases
$$
Note that $\tilde u = g$ on $S^N$ and thus, by Kronecker's formula
(0.10), we have 
$$
\deg g = \Mint_{S^N} \det (\nabla g) = \Mint_{B^{N+1}}\det (\nabla
\tilde u).
$$
[To prove the last equality, consider the vector field
$$
D=(D_1, \ldots, D_{N+1})
$$
where 
$$
D_j = \det (\tilde u_{x_1}, \ldots, \tilde u_{x_{j-1}}, \tilde
u,\tilde u_{x_{j+1}}, \ldots, \tilde u_{x_{N+1}}).
$$
Clearly, we have 
$$ \text{ div } D= (N+1) \det (\nabla \tilde u)
$$
and thus 
$$
\Mint_{B^{N+1}} \det (\nabla \tilde u) =
\frac{(N+1)^{-1}}{|B_{N+1}|}\int_{S^N} D\cdot \nu,
$$
where $\nu$ is the outward normal to $S^N$.  On the other hand, it is
easy to see that $D\cdot \nu= \det(\nabla g)$, where the $N\times N$
Jacobian determinant $\det(\nabla g)$ is computed with respect to any
orthonormal frame in the tangent space to $S^N$ at $x$ and in the tangent space to $S^N$ at $g(x)$.]

Since $|\tilde u(y)|=1$ on $ B^{N+1}\setminus G$ we have $\det(\nabla
\tilde u) = 0$ on $B^{N+1}\setminus G$ and thus 
$$
\deg  g = \frac{1}{|B^{N+1}|}\int_G \det (\nabla \tilde u) =
\frac{2^{N+1}}{|B^{N+1}|} \int_G \det (\nabla u ).
$$
Hence
$$
|\deg g | \leq C\int_G |\nabla u |^{N+1} \leq C' I(g) \text{ by Lemma
 1}.
$$

\noindent
[There is an alternative proof of the first inequality above using differential forms. As is well-known
$$
\deg g = \deg (u, B^{N+1},0).
$$
\noindent
The latter can be given as the integral of the pull back, under the map $u$, of any smooth $(N+1)$-form $\mu$, with compact support in the open ball $B^{N+1}$, and whose integral is 1. Take $\mu= h(z)dz$, where $h$ is any smooth function with support in  $\{z\in\Bbb R^{N+1}; |z|<1/2\}$, and whose integral is 1. Then we find
$$  
\deg g = \deg (u, B^{N+1},0)=\int_{B^{N+1}}h(u(y)) \det(\nabla u(y))dy,
$$
which yields the desired estimate.]

\bigskip
In the proof of Theorem 2  we will also use the following
\enddemo
\proclaim{Lemma 2}  Let $p> N$, $g \in W^{N/p, p}(S^N; S^N)$.  Then 
$$
\int_{S^N}\frac{1}{(d(x))^N} \leq C(|g|^p_{N/p, p} + 1).\tag 1.5
$$
\endproclaim

\demo{Proof}  It suffices to consider only the $x$'s such that $d(x)<
1/2$.  For any such $x$, we have 
$$
1/2 \leq |u ((1-d(x))x) - g(x)| \leq d(x)^{N/p}
|v|_{C^{0,N/p}(\{x\} \times(0,1/2))}\leq
Cd(x)^{N/p}|v|_{(N+1)/p, p (\{x\}\times (0, 1/2))},
$$
by the embedding $W^{s,p} (0,1) \subset C^{0,\alpha}(0,1)$ where $sp >
1$ and $\alpha = s-1/p$.
Thus
$$
\frac{1}{d(x)^N} \leq C|v|^p_{(N+1)/p, p (\{x\} \times (0,
1/2))}.\tag 1.6
$$
Let, for $f$ defined on $B^{N+1}$ and $x\in S^{N}$, $f^x(r)=f(rx)$, $1/2\le r\le 1$. Recall the Besov type inequality
(see, e.g., [A], p. 208-214)
$$
\int_{S^{N}}|f^x|^p_{s,p(1/2 , 1)} dx
\leq C|f|^p_{s,p(B^{N+1})}, \quad\forall\ f \in
W^{s,p}(B^{N+1}).\tag 1.7
$$
Inequality (1.5) follows by combining (1.6) and (1.7) with the
standard estimate

\noindent  $|v|_{(N+1)/p,p(S^N\times (0 , 1/2))}\le C|u|_{(N+1)/p, p} \leq C|g|_{N/p,p}$.
\enddemo
\demo{Proof of Theorem 2}  We want to establish that, for every $g\in
W^{N/p,p}(S^N; S^N)$, 
$$
|\deg g| \leq C|g|_{N/p,p}^p.\tag 1.8
$$
By density of $C^1(S^N;S^N)$ in $W^{N/p,p}(S^N;S^N)$ and continuity of the
degree under $VMO$ convergence, it suffices to prove (1.8) for $g\in
C^1(S^N; S^N)$.
When $|g|_{N/p,p}$ is sufficiently small, we have $\deg g =0$, once
more by
continuity of the degree under VMO convergence, and thus (1.8) holds.  
Otherwise, (1.8) follows from
Theorem 4 and Lemma 2.
\enddemo

\demo{Proof of Theorem 1} We will prove that
$$
|\varphi|_{BMO(I)} \leq C(|g|^p_{1/p,p} + |g|_{1/p, p}).\tag 1.9
$$
As above, we may assume that $g$ is smooth.
When $|g|_{1/p, p}$ is sufficiently small, (1.9) follows from the
estimate
$$
|\varphi|_{BMO(I)}\leq C|g|_{BMO(I)}, \text{ if } |g|_{BMO(I)} \leq
 \delta
$$
($\delta$ small constant) of Coifman and Meyer [CM].  In view of this
and scale invariance, it suffices to
establish the following weaker form of (1.9) 
$$
\Mint_I\Mint_I |\varphi(x) - \varphi(y)|\leq C(|g|^p_{1/p, p} +
1).\tag 1.10
$$
Extending $g$ by symmetry, we may always assume $g$ and $\varphi$
periodic, and thus defined on a circle(with $g$ of degree zero).  We will prove that
$$
\int_{S^1}\int_{S^1} |\varphi(x_1) - \varphi(x_2)|dx_1 dx_2\leq C(|g|^p_{1/p, p} +
1),\tag 1.11
$$
where $\varphi \in W^{1/p, p}(S^1; \Bbb R)$ and $g = e^{i\varphi}$.
As in the proof of Lemma 1, and by Lemma 2, we have 

$$
\int_{\{y=rx ; r\leq 1 - d(x)\}} |\nabla u |^{2}dy\leq C(|g|^p_{1/p, p} + 1).\tag 1.12
$$
By the coarea formula, (1.3), and (1.12), we have 
$$\aligned
\int^{1/2}_{1/3} \left(\int_{\{y\in B^2; |u(y)|= t\}}|\nabla u |\right)
dt & = \int_{\{y\in B^2; 1/3 < |u(y)|< 1/2\}} |\nabla u |\; |\nabla |u||
\\
\\
& 
\leq
\int_G |\nabla u |^2\leq C(|g|^p_{1/p, p} + 1).\endaligned
$$
Thus we may find  some $t\in (1/3, 1/2)$ such that 
$$
\int_\Gamma |\nabla u |  \leq C(|g|^p_{1/p, p} + 1),\tag 1.13
$$
where 
$\Gamma =\{ y\ ;\ |u(y) | = t\}$.  Let $\gamma_1, \gamma_2,\ldots $, be the
connected components of $\Gamma$.  By (1.13), we have 
$$
\sum_j |\deg(u, \gamma_j)|\leq \frac{1}{2\pi t} \sum_j\int_{\gamma_j}
|\nabla u | \leq C(|g|^p_{1/p,p} + 1).\tag 1.14
$$
On the other hand, if $j\neq k$, then the domains enclosed by $\gamma_j$ and $\gamma_k$ have
disjoint interiors, by the maximum principle.

Let now $x, y \in S^1$ and consider the domains 
$$
U=\{ z\ ;\ |u(z)|>t\}, V\text{ as on the figure below and } \tilde W =
U\cap V.
$$
Let $W$ be the connected component of $\tilde W$ 
whose boundary contains $x$ and $y$. Since $\partial U$
is a finite union of analytic curves, $\partial W$ will generically be
a finite union of segments and curves contained in $\Gamma$ :
\vfill

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$$



Let $\gamma$ be the arc from $x$ to $y$ as above.  Let $h$:
$\displaystyle U\to S^1$, $\displaystyle h(z) = \frac{u(z)}{|u(z)|}$.  
Since $u\in W^{2/p, p}$, we have
$h\in W^{2/p, p}$.  Next we note that, since $g\in W^{1/p, p} \cap
L^\infty$, it suffices to establish (1.10) for ${ p\geq 2}$.  Assuming
$p\geq 2$, we have $|h|_{2/p,p} \leq C|u|_{2/p, p} \leq C|g|_{1/p,
p}$, so that it suffices to prove that 
$$
\int_{S^1}\int_{S^1} |\varphi(x) - \varphi(y) |dxdy\leq C(|v|_{2/p, p (S^N\times (0,1/2))}^p +
|g|^p_{1/p, p} + 1) .\tag 1.15
$$
Let $\psi$ be the lifting of $h$ on $\gamma$ such that
$\psi(x)=\varphi(x).$
Then
$$
\varphi(y) - \varphi(x) = \psi(y) - \psi(x) \pm 2 \pi \sum\deg (u,
\gamma_j),
$$
where the above summation is done over the $j$'s such that $\gamma_j\subset W$. By (1.14), we have 
$$
|\varphi(y) - \varphi(x)|\leq |\psi(y) - \psi(x)|+ C(|g|^p_{1/p, p} +
 1).\tag 1.16
$$

We next note that, if $\tilde \gamma$ is an arc on 
$\gamma\cap \Gamma$ with endpoints $a, b$, then
$$|\psi(b) - \psi(a)|\leq 1/t\int_{\tilde
\gamma}|\nabla u |.\tag 1.17$$
We write
$$
\gamma = I_1 \cup \tilde \gamma_1 \cup I_2 \cup\ldots \cup I_n,
$$
where $I_1, \ldots, I_n$ are line segments, $\tilde \gamma_1, \ldots, \tilde \gamma_{n-1}$ are on $\gamma \cap
\Gamma$, $I_1$ has  endpoints $a_1= x$ and $b_1, \tilde \gamma_1$ has
endpoints $b_1$ and $a_2$, etc. By (1.13), (1.16) and (1.17), we find that
$$
|\psi(y) - \psi(x)|\leq C(|g|^p_{1/p, p} + 1) + \sum^n_1 |\psi(b_j)-
 \psi(a_j)|.\tag 1.18
$$
We estimate the terms 
$|\psi(b_1)-\psi(a_1)|$ and $|\psi(b_n)-\psi(a_n)|$ in (1.18) with the help of 
\enddemo

\proclaim{Lemma 3}
Let $\psi\in C^{0,\alpha}((0, l) ; \Bbb R)$ with $0<\alpha\le 1$ and
set 
$h=e^{i\psi}$. Then
$$
|\psi(l) - \psi(0)|\leq 4(l|h|^{1/\alpha}_{C^{0,\alpha}} + l^{\alpha} |h|_{C^{0,\alpha}}).\tag 1.19
$$
\endproclaim
\demo{Proof of Lemma 3} After scaling, we may always take $l=1$.  Suppose first that
$|h|_{C^{0,\alpha}}\le 1$. Then, clearly,
$$
|\psi (1)-\psi (0)|\le 2|h(1)-h(0)|\le 2 |h|_{C^{0,\alpha}}
$$
and the desired conclusion follows.

\noindent
When $|h|_{C^{0,\alpha}}>1$, let $n$ 
be the integer part of $|h|^{1/\alpha}_{C^{0,\alpha}}+1$. For $j=0,\ldots , n$,
set $a_j=j/n$. Since 
$$
|h(a_{j+1})-h(a_j)|\le |h|_{C^{0,\alpha}}(1/n)^\alpha\le 1,
$$
we deduce as above that
$$
|\psi (a_{j+1})-\psi (a_j)|\le 2 |h(a_{j+1})-h(a_j)|\le 
2|h|_{C^{0,\alpha}}(1/n)^\alpha,\quad j=0,\ldots , n-1.
$$
Summing these inequalities for $j=0,\ldots , n-1$, we find
$$
|\psi (1)-\psi (0)|\le 2|h|_{C^{0,\alpha}}
n^{1-\alpha}\le 4 |h|^{1/\alpha}_{C^{0,\alpha}},
$$
since
$\displaystyle
n\le |h|^{1/\alpha}_{C^{0,\alpha}}+1\le 2 |h|^{1/\alpha}_{C^{0,\alpha}}\ ;
$
this is again the desired conclusion.
\enddemo
%\proclaim{Lemma 3}  Let $1< p < \infty$, $h\in W^{2/p, p}((0,l); S^1)$,

%$h=e^{i\psi}$.  Then 

%$$

%|\psi(l) - \psi(0)|\leq C(l^{1/2}|h|^{p/2}_{2/p,p} + l^{1/p} |h|_{2/p,

 %p}).\tag 1.19

%$$

%\endproclaim

% 

%\demo{Proof of Lemma 3}  If $l^{1/p}|h|_{2/p, p}$ is sufficiently

% small, then 

%$$

%|h(x) - h(y)|\leq Cl^{1/p}|h|_{2/p,p} < 1,\ \forall\ x, y\in (0, l),

%$$

%so that $|\psi(l) - \psi(0)|\leq Cl^{1/p}|h|_{2/p, p},$ and the

%conclusion follows.  When $l^{1/p} |h|_{2/p, p}$ is not small, 

%let $\delta > 0$ be such that $l^{1/p}|h|_{2/p, p} \leq \delta

%\Rightarrow |\psi(l) - \psi(0)|\leq 1.$  Consider a partition of

%$(0,l)$ as follows:

%

%\bigskip

%

%insert Figure 2

%

%\bigskip

%

%Pick $l_1$ such that  $l_1^{1/p} |h|_{2/p,p(I_1)} = \delta$, next $l_2$

%such that $l_2^{1/p}|h|_{2/p, p(I_2)} = \delta,$ the last interval of this

%kind being $I_n$ ; therefore, $l^{1/p}_{n+1} |h|_{2/p, p (I_{n+1})}

%\leq\delta$.

%

%Then 

%$$

%|h|^p_{2/p, p(0, l)} \geq \sum^n_1 |h|_{2/p, p(I_j)}^p =

% \delta^p\sum^n_1\frac{1}{l_j}.

%$$

%Since 

%$$

%\sum^n_1 \frac{1}{l_j} \geq n^2\bigg/\bigg(\sum^n_1 l_j\bigg)\geq 
%\frac{n^2}{l},

%$$

%we find

%$$

%|\psi(l) - \psi(0) |\leq n+1 \leq C(l^{1/2} |h|_{2/p,p}^p + 1),

%$$

%and the lemma follows.

%\enddemo

\demo{Proof of Theorem 1 completed} Using Lemma 3, 
the 1-d embedding $W^{2/p,p}\hookrightarrow C^{0,1/p}$ and the inequality 
$$
|\nabla u(y)|\le C\quad\text{ if }|y|\le 1/2,\tag 1.20
$$
we find that
$$
|\psi (b_1)-\psi(a_1)|+|\psi (b_n)-\psi(a_n)|\le 
C(|v|^p_{2/p , p(\{ x\}\times (0,1/2))}+
|v|^p_{2/p , p(\{ y\}\times (0,1/2))}+1).\tag 1.21
$$
The ingredient for estimating the terms $|\psi(b_j)-\psi(a_j)|$, $j=2,\ldots , n-1$, is the inequality
$$
|\psi(b_j)-\psi(a_j)|=\bigg| \int\limits_{[a_j,b_j]}\overline h\frac{\partial h}{\partial\tau}\bigg|\le
C\int\limits_{[a_j,b_j]}|\nabla u|.
\tag 1.22
$$
Estimate (1.22), used in conjunction with (1.20), yields
$$
\sum^{n-1}_2 |\psi(b_j)-
 \psi(a_j)|\le C\bigg(\int\limits_{\{ rx\ ;\ 1/2\le r\le 1-d(x)\}}|
\nabla u|+\int\limits_{\{ ry\ ;\ 1/2\le r\le 1-d(y)\}}|\nabla u|+1\bigg).
\tag 1.23
$$
 By (1.18), (1.21) and (1.23), we find that
$$
\aligned
|\varphi(x) - \varphi(y)|  \leq 
C\bigg( & \int\limits_{\{ rx\ ;\ 1/2\le r\le 1-d(x)\}}|\nabla u|+\int\limits_{\{ ry\ ;\ 1/2\le r\le 1-d(y)\}}|\nabla u|\\
& +|g|^p_{1/p, p} + |v|^{p}_{2/p,
 p(\{x\}\times (0, 1/2))} + |v|^{p}_{2/p, p(\{y\}\times (0,
 1/2))}+1\bigg).
\endaligned
\tag 1.24
$$
\enddemo
The conclusion follows, with the help of (1.7) and (1.12), 
by integrating (1.24).

\demo{Proof of Corollary 1}  Recall that we want to obtain the estimate
$$
|\varphi|_{BMO} \leq C(|g|^p_{N/p,p} + |g|_{N/p,p}).\tag 1.25
$$
 When $|g|_{N/p,p}$ is small, the conclusion follows from Theorem 4 in
 [BN].
  Otherwise, assume, e.g., $N= 2$.  It suffices (after scaling) to prove that 
$$
J= \iint_{(0,1)^2 \times (0,1)^2} |\varphi(x) - \varphi(y)| \leq C
(|g|_{2/p, p}^p+1).
\tag 1.26
$$
This follows from
$$
|\varphi(x) - \varphi(y)|\leq |\varphi(x_1, x_2)-\varphi(x_1, y_2)
|+|\varphi(x_1, y_2) - \varphi(y_1, y_2)|,\tag 1.27
$$
which, combined with Theorem 3, yields
$$
J\leq C
\left(1+\int|g|^p_
{1/p,p (\{s\} \times [0, 1])}
 ds + \int|g|_{1/p,p([0, 1]\times\{t\})}^p dt
\right)
\leq C(|g|^p_{2/p,p} + 1).\tag 1.28
$$
\enddemo

\subhead 2. Proof of Theorem 3 \endsubhead

We want to prove that the distribution Det$(\nabla g)$, initially
defined in (0.11) for  $g\in W^{1,N}(S^{N+1}; S^N)$,
makes sense for $g\in W^{N/p,p}(S^{N+1}; S^N)$, $N<p< \infty$, and
satisfies (0.12)-(0.16).  The strategy of the proof is the
following :
\medskip

(i) we define $\la T(g), \zeta\ra$ for a general $g\in
W^{N/p,p}(S^{N+1}; S^N)$ via an integral formula ;
\medskip

(ii)   with $T$ defined in (i), we prove that (0.12) holds and that the map
$g\mapsto$ $T(g)$ is strongly continuous from $W^{N/p,p}$ into
$(W^{1,\infty})^*$ ;
\medskip

(iii) we establish (0.13) ;
\medskip

(iv)  we note that (0.14)-(0.16) hold for some special $g$'s ; 
for a
general $g\in W^{N/p,p}$, (0.14)-(0.16) will be obtained by density.
\medskip

\noindent
{\bf Step 1.} Definition of $T(g)$ ; continuity of $T(g)$ and proof of (0.12)

\medskip
The  definition of 
$T(g)$ 
relies on a formula which is in the same spirit as the one presented 
in [BBM4] for maps 
in $H^{1/2}(S^2 ; S^1)$.  Let us start with a
smooth map $g:S^{N+1} \to \Bbb R^{N+1}$ and a Lipschitz function $\zeta:
S^{N+1} \to \Bbb R$.  Let $F$ be any smooth extension of $g$ to 
$B^{N+2}$ (with values into $\Bbb R^{N+1}$)  and let
$\xi$ be any Lipschitz extension of $\zeta$ to
$B^{N+2}$.  Set
$$
X(F, \xi) = \sum^{N+2}_{j=1} \int_{B^{N+2}} H_j \xi_{x_j},\tag 2.1
$$
where $H=(H_1, H_2, \ldots, H_{N+2})$ and 
$$
H_j = (-1)^{N+j} F_{x_1} \wedge \ldots \wedge F_{x_{j-1}}\wedge
F_{x_{j+1}}\wedge\ldots\wedge F_{x_{N+2}}.\tag 2.2
$$
It is easy to see that div $H=0$, that $X$ depends only on $g$ and $\zeta$, and 
(after a number of integration by parts) that 
$$
X(F,\xi)= \la \text{\rm Det}(\nabla g), \zeta\ra.\tag 2.3
$$
In the case $N=1$ and $g\in H^{1/2}(S^2 ; S^1)$, we took in [BBM4] an {\bf arbitrary}
extension $F\in H^1(B^3 ; \Bbb R^2)$ of $g$ ; then the corresponding $H$ given by (2.2) 
belongs to $L^1$. Consequently, formula (2.3) allows to define Det$(\nabla g)\in (W^{1,\infty})^*$
for every $g\in H^{1/2}(S^2 ; S^1)$. We may still use the same technique when $g\in W^{N/(N+1), N+1}(S^{N+1} ; S^N)$.
However, this method does not seem to work when $g\in W^{N/p, p}(S^{N+1} ; S^N)$ and $p>N+1$. In this case,
we are going to choose a {\bf special} extension $F$ of $g$ such that :

\noindent
(i) $F\in C^\infty(B^{N+2} ; \Bbb R^{N+1})$ ;

\noindent
(ii) $F\in W^{(N+1)/p, p}(B^{N+2})$ ;

\noindent
(iii) $H$ (defined by (2.2)) belongs to $L^1$. 

\medskip
For every $g\in W^{N/p , p}(S^{N+1} ; S^N)$, let $u$ be the harmonic extension of $g$ to $B^{N+2}$ (with values into $B^{N+1}$).

\medskip
\noindent
[{\bf Warning :} Here, $g$ need not be VMO, in contrast with the situation we encountered in the proofs of Theorems 1 and 2.
In general, $|u(y)|$ does {\bf not} tend to $1$ as $|y|\to 1$ and the set $\{ y\in \overline{B^{N+2}}\ ;\ |u(y)|\le 1/2\}$ is {\bf not} a compact subset of the open ball $B^{N+2}$. This will become particularly transparent later on at the
points of $S^{N+1}$ where $g$ has topological singularities.]

\medskip
Fix any map $\Phi\in C^\infty (\Bbb R^{N+1} ; \Bbb R^{N+1})$ such that $\d\Phi (X)=X/|X|$ if $|X|\ge 1/2$. The {\bf special} $F$ we will use is defined by 
$$
F(y)=\Phi (u(y)),\quad\quad\forall\ y\in B^{N+2}.\tag 2.4
$$
Note that $F\in C^\infty (B^{N+2} ; B^{N+1})$ and that $F(y)\in S^N$ when $|u(y)|\ge 1/2$. Consider the vector-field $H$ defined by (2.2) for this $F$ and observe that $H=0$ in the open set $\{ y\in B^{N+2}\ ;\ |u(y)| > 1/2\}$.

For every $\xi\in W^{1,\infty}(B^{N+2} ; \Bbb R)$, define
$$
Y(\xi)=X(F,\xi)\tag 2.5
$$
as in (2.1)-(2.2). This requires a justification, since it is not clear that $H\in L^1$. A key ingredient in the proof of Theorem 3 is

\proclaim{Lemma 4} For each $g\in W^{N/p,p} (S^{N+1}; S^N)$, we have $H\in L^1(B^{N+2} ; \Bbb R^{N+2})$, so that the
quantity $Y(\xi)$ is well-defined .  Moreover :

\noindent
a) $Y(\xi_1)=Y(\xi_2)$ when $\xi_1=\xi_2$ on $S^{N+1}$.

\noindent
Set $\la T(g) , \zeta\ra =Y(\xi)$, where $\xi$ is any Lipschitz extension of a given  $\zeta\in W^{1,\infty}(S^{N+1} ; \Bbb R)$. Then :


\noindent
b)
$$
|\la T(g) , \zeta\ra|\leq C|g|_{N/p,p}^p\|\nabla\zeta \|_{L^\infty},\quad\forall\ \zeta\in W^{1,\infty}(S^{N+1} ; \Bbb R)\ ;\tag 2.6
$$
c) the map
$g\mapsto T(g)$ is strongly continuous from $W^{N/p,p}$ into  $(W^{1,\infty} (S^{N+1}))^*$. 
\endproclaim
\demo{Proof of Lemma 4}  We start by proving that $H\in L^1$. Assume first that $N<p\leq N+1$.  Then $u$
(the harmonic extension of $g$) belongs to $W^{(N+1)/p,p}(B^{N+2})\cap
L^\infty$, and thus to $W^{1, N+1}$.  Therefore, with our choice of
$F$, we have $H\in L^1$. Moreover, in this case, the map  $g\mapsto H\in L^1$ is clearly 
continuous, so that
c) follows (provided we establish a)).

\noindent
Assume next that $p>N+1$.
In the open set 
$\{y\in B^{N+2}\, ; |u(y)|> 1/2\}$, $F$ is $S^N$-valued, and thus 
$H=0$ pointwise.
Therefore,
$$
\int\limits_{B^{N+2}}|H| = \int_{\{y ; |u(y)|\leq 1/2\}} |H|.
$$
Clearly, $|\nabla F|\le C|\nabla u|$ and therefore $|H|\le C|\nabla u|^{N+1}$.
By the proof of Lemma 1, we have 
$$
\int_{\{y ; |u(y)|\leq 1/2\}} |H|  \leq 
C 
\int_{\{y ; |u(y)|\leq 1/2\}}
 |\nabla u |^{N+1} 
\leq 
  C\int_{S^{N+1}}\frac{1}{(d(x))^N},
$$
where $d(x)$ is defined as in Section 1. 
\enddemo

By the proof of Lemma 2, we further obtain that 
$$
\int_{S^{N+1}}\frac{1}{(d(x))^N} \leq C (|g|^p_{N/p, p} + 1),
$$
and thus 
$$
\int_{\{y ; |u(y)|\leq 1/2\}} |H|  
\leq C(|g|^p_{N/p,p} + 1).  
$$
Hence $H\in L^1$ and consequently $Y(\xi)$ is well-defined.

\medskip
We now turn to the proof of a). Let $\xi_1,\xi_2\in W^{1,\infty}(B^{N+2} ; \Bbb R)$ be
such that $\xi_1=\xi_2$ on $S^{N+1}$ and set $\eta=\xi_1-\xi_2\in W^{1,\infty}_0(B^{N+2})$.
Consider a sequence $(\eta_j)\subset C^\infty_c(B^{N+2})$ such that $\nabla\eta_j\to\nabla\eta$ a.e.
and $\d \|\nabla\eta_j\|_{L^\infty}\le C$. Since div $H=0$, we clearly have $\d\int\limits_{B^{N+2}}H\cdot\nabla\eta_j=0$,
$\forall\ j$, and thus $\d\int\limits_{B^{N+2}}H\cdot\nabla\eta=0$.

\medskip
We next establish b).
It suffices to estimate $\la T(g),\zeta\ra$ when 
$$
\int_{S^{N+1}} \zeta = 0.\tag 2.7
$$
In view of (2.7), we may find an extension $\xi$ of $\zeta$ to
$B^{N+2}$ such that
$$
\|\nabla \xi\|_{L^\infty} \leq C\|\nabla \zeta\|_{L^\infty} \tag 2.8
$$
and 
$$
\text{Supp } \xi \subset \{ y\in \overline{B^{N+2}}\ ; |y|\geq 1/2\}.\tag 2.9
$$
For such a~ $\xi$, we have
$$
|\la T(g) , \zeta\ra |\le \int\limits_{B^{N+2}}|H||\nabla\xi|\le C\|\nabla\zeta\|_{L^\infty}\int\limits_{\{y ; |y| \geq 1/2 \text{ and } |u(y)\leq 1/2\}} |\nabla u |^{N+1}.\tag 2.10
$$
Going back to the proofs of Lemmas 1 and 2, we see that 
$$
\int\limits_{\{y ; |y| \geq 1/2 \text{ and } |u(y)\leq 1/2\}} |\nabla u |^{N+1} \leq C|g|^p_{N/p,p},\tag 2.11 
$$
so that b) is a consequence of (2.10) and (2.11).

\medskip
Finally, we prove c). As we already observed, it suffices to consider the case $p>N+1$.
Let $g_n$, $g\in W^{N/ p,p}(S^{N+1}; S^N)$ be such that $g_n \to g$
in $W^{N/p,p}$ and let $H_n, H$ be the corresponding vector-fields.
We claim that
$$
\int_{B^{N+2}} |H_n - H| \to 0.\tag 2.12
$$
By the uniqueness of the limit, it suffices to establish (2.12) for a subsequence.
With $u_n$, $u$ the corresponding harmonic extensions, we have 
$u_n \to u$ in $C^\infty(B^{N+2})$ and in 
$W^{(N+1)/ p,p}$. For $x\in S^{N+1}$ and $t\in I=(0,1/2)$, set
$$
v_n(x,t)=u_n((1-t)x)\quad\text{ and }\quad v(x,t)=u((1-t)x).
$$
In view of (1.7), we know that 
$$
v_n\to v\quad\quad\text{ in }L^p(S^{N+1} ; W^{s,p}(I)),
$$
where $s=(N+1)/p$. Passing to a subsequence (still denoted $v_n$) we obtain a function $K\in L^1(S^{N+1})$ such that
$$
|v_n(x, \cdot )|^p_{s,p(I)}\le K(x),\quad\forall\ n, \text{ and a.e. }x\in S^{N+1}.\tag 2.13
$$
As in the proof of Lemma 2 we find, using (2.13),
$$
\frac 1{d_n(x)^N}\le CK(x),\quad\forall\ n, \text{ and a.e. }x\in S^{N+1},\tag 2.14
$$
(where $d_n$, corresponding to $g_n$, is defined as in Section 1). Next we have, (using (1.2) and (1.3)), 
$$|H_n(rx)| \leq\cases  0, &\text{ if }\quad 1-d_n(x) < r< 1\\
C/(1-r)^{N+1}, &\text{ if }\quad 0\leq r<
1\endcases.\tag 2.15
$$
Combining (2.14) and (2.15), we obtain 
$$
|H_n(y)|\leq M(y), \quad\quad \forall\ y\in B^{N+2},\tag 2.16
$$
for some $M\in L^1$.  Since, clearly, $H_n \to H$ in $C^\infty (B^{N+2})$, (2.12)
follows from (2.16).

\medskip
\noindent
{\bf Step 2.} Proof of (0.13)

\medskip
 As we already observed, we may still define $T( g)$  if
$g\in W^{N/(N+1), N+1}(S^{N+1} ; \Bbb R^{N+1})$ (note that here $g$ need not be $S^N$-valued). Indeed, for such a $g$, we have $u\in W^{1, N+1}(B^{N+2} ; \Bbb R^{N+1})$ and thus
$H\in L^1$. Similarly, the definition (0.11) of Det$(\nabla g)$ still makes sense for $g\in W^{1, N}(S^{N+1} ; \Bbb R^{N+1})\cap L^\infty$. An easy adaptation of the proof of Lemma 1 in [BBM4] yields, in $(W^{1,\infty})^*$, the equality
$$
\text{\rm Det}(\nabla g) = T(g) ,\quad\forall\ g\in W^{1,N}(S^{N+1} ; \Bbb R^{N+1})\cap W^{N/(N+1), N+1}\cap L^\infty.\tag 2.17
$$
This completes the proof of (0.13) when $N\ge 2$. Indeed, if $N\ge 2$ we have $W^{1,N}(S^{N+1} ; S^N)\subset W^{N/p,p}$, $\forall\ p>N$, so that (0.13) is a special case of (2.17). 

We now turn to the proof of (0.13) when $N=1$, i.e.,
$$
\text{Det}(\nabla g)=T(g),\quad\quad\forall\ p>1,\ \forall\ g\in W^{1,1}(S^2 ; S^1)\cap W^{1/p , p}.\tag 2.18
$$
It is  useful to introduce the class
$$
\aligned
\Cal R= &\{g \in W^{1,q}(S^{N+1}; S^N) \text{ for every } 1\leq q <
N+1\ ;\\
        &\ g\in C^\infty(S^{N+1}\setminus A) \text{ for some finite set
        } A\}.\endaligned 
$$
Note that every $g\in\Cal R$ belongs to $W^{1,N}$ and also to $W^{N/(N+1), N+1}$. Thus (2.17) holds for every $g\in\Cal R$.


Equality (2.18) follows from 

\noindent
a) Lemma 5 below ;

\noindent
b) (2.17) applied to $g\in\Cal R$ ;

\noindent
c) the continuity of $g\mapsto T(g)$ from $W^{1/p,p}(S^2 ; S^1)$ into $(W^{1,\infty})^*$ ;

\noindent
d) the continuity of $g\mapsto\,$Det$(\nabla g)$ from $W^{1,1}(S^2 ; S^1)$ into $(W^{1,\infty})^*$ (which is obvious from (0.11)).

\proclaim{Lemma 5} Let $p>1$. For every $g\in W^{1,1}(S^2 ; S^1)\cap W^{1/p,p}$, there is a sequence $(g_n)\subset\Cal R$ such that $g_n\to g$ in $W^{1,1}$ and in $W^{1/p,p}$.
\endproclaim
The proof of Lemma 5 is given in the Appendix.

\medskip
\noindent
{\bf Step 3.} Proof of (0.14)-(0.16)

\medskip
The proof of (0.14)-(0.15) is a straightforward adaptation --left to the reader-- of the proof of Theorem 1 in [BBM4]. It relies on
four facts :

\noindent
a) $\Cal R$ is dense in $W^{N/p , p}(S^{N+1} ; S^N)$ (see Appendix) ;

\noindent
b) $g\mapsto T(g)$ is continuous from $W^{N/p , p}(S^{N+1} ; S^N)$ into $(W^{1,\infty})^*$ ;

\noindent
c) the equality
$$
\text{Det}(\nabla g)=T(g)=\omega_{N+1}\sum_{\text{\rm finite}}d_a\delta_a,\quad\forall\ g\in\Cal R,\tag 2.19
$$
where $\omega_{N+1}$ is the volume of the unit ball in $\Bbb R^{N+1}$ and $d_a$ denotes the degree of $g$ restricted to a small sphere around $a$ in $S^{N+1}$ (with appropriate orientation). Equality (2.19) is proved as in [BBM4], Lemma 2 ;

\noindent
d) if $g,h\in\Cal R$ and we write 
$$\text{Det}(\nabla g) - \text{ Det}(\nabla h) =
 \omega_{N+1} \sum_{a\in A} d_a \delta_a,\tag 2.20
$$
then (see [BCL])
$$
\|\text{Det}(\nabla g) - \text{Det}(\nabla
h)\|_{(W^{1,\infty})^*}=\omega_{N+1}L, \tag 2.21
$$
where
$$
L= \underset{\sigma \in S_k}\to{\text{Min}} \sum^k_{i=1} d(P_i,
N_{\sigma(i)})\ ;\tag 2.22
$$
here $P_i, N_i$ are the points $a\in A$ repeated according to their
multiplicity and $d$ is the geodesic distance on $S^{N+1}$.


\medskip
The proof of (0.16) relies on the 
following variant of Theorem 4 in [BMP] :

\proclaim{Lemma 6} Let $g,h\in \Cal R$.  Then, for $\zeta\in
C^\infty(S^{N+1};\Bbb R)$, we have
$$
\int|\deg (g;\Gamma_\lambda) - \deg(h;\Gamma_\lambda)|d\lambda \leq
\frac{1}{\omega_{N+1}} \|\nabla\zeta\|_{L^\infty}\|\text{\rm Det}(\nabla g) - \text{\rm Det}(\nabla
h)\|_{(W^{1,\infty})^*}.
$$
\endproclaim 

\demo{Proof}  Let $g,\, h\in \Cal R$. Assume that 
$$
T(g)=\omega_{N+1}\sum_{i=1}^I(\delta_{P_i}-\delta_{N_i}),\quad\quad T(h)=\omega_{N+1}\sum_{j=1}^J(\delta_{\tilde P_j}-\delta_{\tilde N_j}).
$$
If $\lambda$ is a regular value of $\zeta$ such that $\zeta (P_i)\neq\lambda$, $\zeta (N_i)\neq\lambda$, $\zeta (\tilde P_j)\neq\lambda$, $\zeta (\tilde N_j)\neq\lambda$, for every $i$ and $j$, then 
$$
\text{deg}(g ; \Gamma_\lambda)=\text{card}\{ 1\le i\le I\ ;\ \zeta(P_i)>\lambda\}-\text{card}\{ 1\le i\le I\ ;\ \zeta(N_i)>\lambda\},
$$
so that, clearly,
$$
\text{deg}(g ; \Gamma_\lambda)=\frac 12\sum_{i=1}^I(\text{sgn}(\zeta(P_i)-\lambda)-\text{sgn}(\zeta(N_i)-\lambda)). \tag 2.23
$$
It follows from (2.23) that 
$$
\text{deg}(g ; \Gamma_\lambda)-\text{deg}(h ; \Gamma_\lambda)=\frac 12\sum_{k=1}^{I+J}(\text{sgn}(\zeta(P^*_k)-\lambda)-\text{sgn}(\zeta(N^*_k)-\lambda)), \tag 2.24
$$
where the set $\{ P_i\}\cup \{ \tilde N_j\}$, resp. $\{ N_i\}\cup\{ \tilde P_j\}$, is now labeled as $\{ P^*_k\}$, resp. $\{ N^*_k\}$. Assume, e.g., that the length of the minimal connection in (2.22) is given by $L=\d\sum_{k=1}^{I+J}d (P^*_k, N^*_k)$ and let $\gamma_k$ be a geodesic from $P_k^*$ to $N_k^*$, $\forall\ k$. Since, clearly, 
$$
\frac 12|(\text{sgn}(\zeta(P^*_k)-\lambda)-\text{sgn}(\zeta(N^*_k)-\lambda)|\le \text{card}\{ x\in\gamma_k\ ;\ \zeta (x)=\lambda\},
$$
we find, using the area formula and (2.22), that
$$\aligned
\int|\deg (g;\Gamma_\lambda) - \deg(h;\Gamma_\lambda)|d\lambda &\leq \sum_k \int \text{card }\{ x\in\gamma_k\ ;\,\zeta(x) = \lambda\} d\lambda
= \sum_k\int_{\gamma_k} \bigg|\frac{\partial \zeta}{\partial
\tau}\bigg|\\
& \le L\|\nabla\zeta\|_{L^\infty}=\frac{1}{\omega_{N+1}} \|\nabla\zeta\|_{L^\infty}\|\text{\rm Det}(\nabla g) - \text{\rm Det}(\nabla
h)\|_{(W^{1,\infty})^*}
\endaligned
.\tag 2.25
$$
\enddemo
\demo{Proof of (0.16)}  As in [BBM4], we have
$$
\la\text{Det}(\nabla g),\zeta\ra=\int\limits_{\Bbb R}\text{deg}(g; \Gamma_\lambda )d\lambda ,\quad\forall\ \zeta\in C^\infty(S^{N+1} ; \Bbb R),\, \forall\ g\in{\Cal R}.\tag 2.26 
$$
Let $g\in W^{N/p,p}(S^{N+1} ; S^N)$ and let  $(g_n)\subset \Cal R$ be such
that $g_n\to g$ in $W^{N/p,p}$ and
$$
\sum_n \|\text{\rm Det}(\nabla g_{n+1}) - 
\text{\rm Det}(\nabla g_n)\|_{(W^{1, \infty})^*}<\infty.
$$
By Lemma 6 we have, for a fixed $\zeta\in C^\infty(S^{N+1};\Bbb R),$
$$
\sum_n\int\limits_\Bbb R|\deg(g_{n+1}; \Gamma_\lambda) - \deg(g_n; \Gamma_\lambda) |
d\lambda < \infty.\tag 2.27
$$
On the other hand, passing to a subsequence, we have,
 for a.e. $\lambda$,
${g_n}_{|\Gamma_\lambda} \to g_{|\Gamma_\lambda}$ in $W^{N/p,p}$ and thus in VMO.
Therefore,
$$
\deg(g_n; \Gamma_\lambda) \to \deg(g;\Gamma_\lambda)\text{ for a.e. } \lambda .
\tag 2.28
$$
>From (2.27) and (2.28) we obtain
$$
\deg (g_n ;\Gamma_\lambda)\to \deg (g ;\Gamma_\lambda)\quad\quad\text{in }L^1(\Bbb R).\tag 2.29
$$
Property (0.16) follows by combining (2.26), (2.29) and the continuity
of $T$.
\enddemo

\medskip
We conclude this section by showing, in the spirit of [BCL], [BBM4] and
[BMP], that, given points $(P_i)$ and $(N_i)$ in $S^{N+1}$, the minimal
``energy'' (in the $W^{N/p,p}$
sense) required to produce topological singularities at the $P_i$'s
and $N_i$'s, is of the same order as 
 the length of a minimal connection connecting the $P_i$'s to the
$N_i$'s.

Let $\Cal P= (P_i)$, $\Cal N= (N_i) \subset S^{N+1}$ be such that
$\sum_i|P_i - N_i|<\infty$.  We define the length of a minimal
connection to be 
$$
L(\Cal P, \Cal N) =\text{Inf }\left\{\sum d(\tilde P_j,\tilde N_j)\ ;
\sum(\delta_{P_i} - \delta_{N_i}) = \sum (\delta_{\tilde P_j} -
 \delta_{\tilde N_j})\right\}.
$$
As observed in [BBM4], if 
$$
T = \omega_{N+1} \sum (\delta_{P_i} -
\delta_{N_i}),
$$
then
$$
\|T\|_{(W^{1, \infty})^*} = \omega_{N+1}
L(\Cal P, \Cal N).\tag 2.30
$$
\proclaim{Theorem 5}   Given $\Cal P$ and $\Cal N$, we have, for
$N<p<\infty$,
$$
L(\Cal P, \Cal N) \sim \text{\rm Inf}\left\{|g|^p_{N/p,p}\ ;\ g\in
W^{N/p,p}(S^{N+1}; S^N),\ T(g) = \omega_{N+1}
\sum(\delta_{P_i} - \delta_{N_i})\right\}.\tag 2.31
$$
\endproclaim
\noindent
(The equivalence in (2.31) is up to constants depending on $p$ and $N$.)
\demo{Proof}  In view of (0.12) and (2.30), it suffices to find, for $\Cal P$, $\Cal N$ as above, a map 
$g\in W^{N/p,p}(S^{N+1}; S^N)$ such that $\d T(g) = \omega_{N+1}\sum(\delta_{P_i} - \delta_{N_i})$ and $|g|^p_{N/p,p}\le C\, L(\Cal P, \Cal N)$.
We rely on Theorem 5.6 in [ABO], which
asserts that, given ${\Cal P}=(P_i)$, ${\Cal N}=(N_i)\subset S^{N+1}$ such that $\sum |P_i
- N_i|<\infty$, there is some $g\in W^{1,N}(S^{N+1}; S^N)$ such that 
$$
\text{Det}(\nabla g) = \omega_{N+1} \sum(\delta_{P_i} -
\delta_{N_i})\tag 2.32
$$
and
$$\|\nabla g\|^N_{L^N} \leq CL(\Cal P, \Cal N).\tag 2.33
$$
If $N\geq 2$, we have the inclusion $W^{1,N}(S^{N+1};S^N)\hookrightarrow
W^{N/p,p}(S^{N+1}; S^N)$, $N< p < \infty$, and Theorem 5 follows from the inequality
$$
|g|^p_{N/p,p}\leq C\|\nabla g\|^N_{L^N} \leq C L(\Cal P, \Cal N).\tag 2.34
$$
The above inclusion is false when $N=1$.  However, in this case we
rely on the proof of Lemma 16 in [BBM4].  More specifically, given $1<p<\infty$, and given
points $(P_i)$, $(N_i)\subset S^2$ such that $\sum |P_i - N_i|<\infty$,
we constructed in [BBM4] a map $g\in W^{1/p,p}(S^2; S^1)\cap W^{1,1}$, such that Det$(\nabla g) = \pi \sum(\delta_{P_i}
- \delta_{N_i})$ and (2.34) holds.  Estimate (2.34) is established in
[BBM4] only for $p=2$, but the argument there can be easily adapted to every $p$, 
 $1< p < \infty$. For this purpose, one needs to generalize Lemma 17 in [BBM4], with the help
of the obvious inequality
$$
\big| |a+b|^p-|a|^p-|b|^p\big|\le C_p(|a|^{p-1}|b|+|a||b|^{p-1}),\quad\forall\ a,\, b\in \Bbb C, \forall\ p>1.
$$
The proof of Theorem 5 is complete.
\enddemo
\medskip

\noindent{\bf Appendix.  Density of the class $\Cal R$.}

The appendix is devoted to  density results for classes of $S^N$-valued maps. Recall that, if $0<s<1$ , $1<p<\infty$, and $sp\ge N+1$, then $C^\infty (S^{N+1} ; S^N)$ is dense in $W^{s,p}(S^{N+1}; S^N)$ (see, e.g., [B] or [BN], Lemma A.12). We now turn to the remaining case: $sp<N+1$.

\proclaim{Lemma A}  Assume $0<s<1$ , $1<p<\infty$, and $sp<N+1$. Then the class 
$$
\aligned
\Cal R= &\{g \in W^{1,q}(S^{N+1}; S^N) \text{ for every } 1\leq q <
N+1\ ;\\
        &\ g\in C^\infty(S^{N+1}\setminus A) \text{ for some finite set
        } A\}\endaligned
$$
is dense in $W^{s,p}(S^{N+1}; S^N)$.
\endproclaim

For $N=1$, $s=1/2$, and $p=2$, the above result is due to T. Rivi\`ere [R] (following earlier works of F. Bethuel and
X. Zheng [BZ], F. Bethuel [B], and M. Escobedo [E]). A different proof is presented in [BBM4], Lemma 23.  We
explain below how to adapt the proof of [BBM4] to the general case.

\medskip
\def\e{\varepsilon}
Let $g\in W^{s,p}(S^{N+1}; S^N)$ and let $g_\e$ be an $\e$-smoothing of $g$.
Then $g_\e$ satisfies
$$
\|g_\e - g\|_{L^p} \leq C\e^s, \tag A.1
$$
$$
|g_\e|_{s,p} \leq C\tag A.2
$$
and
$$
\|\nabla g_\e \|_{L^p} \leq C\varep^{s-1}.\tag A.3
$$
Given a point  $a\in \Bbb R^{N+1}$ with $|a|\leq 1/10$, let $\pi_a$: $\Bbb
R^{N+1}\setminus \{a\}\to S^N$ be the radial projection onto $S^N$
with vertex $a$.  Using (A.1)-(A.3), we find, with exactly the same
proof as in [BBM4], Lemma 23, that there is a family $(a_\e)$ such that
$|a_\e|\leq 1/10$ and $h_\e = \pi_{a_\e}(g_\e)\to g$ in $W^{s,p}$.
Moreover, as explained in [BBM4], we may choose $a_\e$ to be a regular
value of $g_\e$, and for such a choice we have $h_\e \in \Cal R,
\forall\ n$.
 \proclaim{Corollary A}
For $N<p<\infty$, the class $W^{1,N}(S^{N+1} ; S^N)\cap W^{N/p, p}$ is dense in 
\hfill

\noindent
$W^{N/p,p}(S^{N+1}; S^N)$. 
\endproclaim

\medskip
\demo{Proof of Lemma 5} Let $g_\e$ as above. Then $g_\e$ satisfies (A.1)-(A.3) 
(with $s=1/p$) and, in addition,
$$
\|\nabla g_\e \|_{L^1} \leq C.\tag A.4
$$
On the other hand, we have
$$
\int_{\{ a ; |a|\le 1/10\}}\|\nabla (\pi_a\circ g_\e )\|_{L^1(S^2)}da\le C\|\nabla g_\e \|_{L^1(S^2)} \tag A.5
$$
(this is inequality (5.34) in [BBM4]). By combining (A.1)-(A.5), we find, exactly as in [BBM4],  that there is a family $(a_\e)$ such that
$|a_\e|\leq 1/10$ and $h_\e = \pi_{a_\e}(g_\e)\to g$ in $W^{1/p,p}$ and $\|\nabla h_\e\|_{L^1}\le C$. In order to prove that, in addition, $h_\e\to g$ in $W^{1,1}$, one may adapt the argument in [BBM4]. Convergence in $W^{1/p, p}$ is obtained there with the help of the property (5.43). To establish convergence in $W^{1,1}$, it suffices to note that the analog of (5.43) also holds in $W^{1,1}$ ; this is obtained easily by dominated convergence.  


\enddemo

\medskip

\noindent
{\bf Acknowledgments.}
We thank Louis Nirenberg for making useful remarks on the presentation of the proof of Theorem 2. The first author (J. B.) is partially supported
by NSF Grant 9801013.  The second author (H. B.) is partially sponsored by
an EC Grant through the RTN Program ``Fronts-Singularities'',
HPRN-CT-2002-00274.  He is also a member of the Institut Universitaire
de France.  Part of this work was done during a visit of the third
author (P. M.) at Rutgers University ; he thanks the Mathematics
Department for its support and hospitality.

%new TMR (see Giacomelli Aug. 8)


\input amstex
\NoBlackBoxes
\magnification=\magstep 1
\documentstyle{amsppt}
\loadbold
\loadmsbm

\Refs\nofrills{\bf References}
\widestnumber\key{BBBM-459}


\ref
\key A
\by R. Adams 
\book  Sobolev Spaces
\publ  Acad. Press, 1975
\endref
\medskip

\ref
\key ABO
\by G. Alberti, S. Baldo and G. Orlandi
\paper Functions with prescribed singularities
\jour J. European Math. Soc.
\vol 5
\yr 2003
\page 275--311
\endref
\medskip


\ref
\key B
\by  F. Bethuel 
\paper Approximations in trace spaces defined between manifolds
\jour Nonlinear Anal., Theory, Methods and Appl.
\vol 24
\yr 1995
\pages 121--130
\endref
\medskip

\ref
\key BZ
\by  F. Bethuel and X. Zheng
\paper Density of smooth functions between two manifolds in Sobolev spaces
\jour J. Funct. Anal.
\vol 80
\yr 1988
\pages 60--75
\endref
\medskip

\ref
\key BB
\by J. Bourgain and H. Brezis
\paper On the equation \text{\rm div }$Y=f$ and application to control
of phases
\jour J. Amer. Math. Soc.
\vol 16
\yr 2003
\pages 393--426
\endref
\medskip

\ref
\key BBM1
\by J. Bourgain, H. Brezis and P. Mironescu 
\paper Lifting in Sobolev spaces
\jour J. d'Anal. Math.
\vol 80
\yr 2000
\pages 37--86
\endref
\medskip

\ref
\key BBM2
\by J. Bourgain, H. Brezis and P. Mironescu
\paper On the structure of the Sobolev space $H^{1/2}$ with values
into the circle
\jour C. R. Acad. Sci. Paris, S\'erie I,
\vol 331
\yr 2000
\pages 119--124
\endref
\medskip

\ref
\key BBM3
\by J. Bourgain, H. Brezis and P. Mironescu
\paper \nofrills Another look at Sobolev spaces 
\inbook in Optimal Control and Partial
Differential Equations 
\eds J. L. Menaldi, E. Rofman et A. Sulem 
\publ IOS Press
\vol
\yr  2001
\pages 439--455
\endref
\medskip

\ref
\key BBM4
\by J. Bourgain, H. Brezis and P. Mironescu
\paper $H^{1/2}$ maps with values into the circle : minimal
connections, lifting, and the Ginzburg-Landau equation
\jour Publications math\'ematiques de l' IHES 
\toappear
\endref
\medskip

\ref
\key BBM5
\by J. Bourgain, H. Brezis and P. Mironescu
\jour in preparation
\endref
\medskip

\ref
\key BCL
\by H. Brezis, J.-M. Coron and E. Lieb
\paper Harmonic maps with defects
\jour Comm. Math. Phys.
\vol 107
\yr 1986
\pages 649--705
\endref
\medskip

\ref
\key BMP
\by H. Brezis, P. Mironescu and A. Ponce
\paper\nofrills $W^{1,1}$-maps with values into $S^1$
\inbook in Geometric Analysis of PDE and Several Complex Variables 
(S. Chanillo, P. Cordaro, N. Hanges, J. Hounie and A. Meziani, eds.), 
 Contemporary Mathematics series, AMS
\toappear
\endref
\medskip

\ref
\key BN
\by H. Brezis and L. Nirenberg
\paper Degree Theory and BMO, Part I : Compact manifolds without
boundaries
\jour Selecta Math.
\vol 1
\yr 1995
\pages 197-263
\finalinfo {\it Part II :  Compact manifolds with boundaries}, Selecta
Math. {\bf 2} (1996), 1--60
\endref
\medskip


\ref
\key CM
\by R. Coifman and Y. Meyer
\paper\nofrills Une g\'en\'eralisation du th\'eor\`eme de Calder\`on sur l'int\'egrale de
Cauchy 
\inbook  in Fourier Analysis, Proc. Sem. at El Escorial,
Asoc. Mat. Espa\~ nola, Madrid, 1980, 88--116
\endref
\medskip

\ref
\key E
\by M. Escobedo
\paper Some remarks on the density of
regular mappings in Sobolev classes of $S^M$-valued functions
\jour Rev. Mat. Univ. Complut. Madrid
\vol 1
\yr 1988
\pages 127--144
\endref
\medskip

\ref 
\key R
\by T. Rivi\`ere 
\paper Dense subsets of $H^{1/2}(S^2; S^1)$
\jour Ann. of Global Anal. and Geom.
\vol 18
\yr 2000
\pages 517--528
\endref
\medskip

\ref 
\key RS
\by T. Runst and W. Sickel
\book Sobolev spaces of
fractional order, Nemytskij operators, and nonlinear
partial differential equations
\publ  Walter de Gruyter, Berlin and New York
\yr 1996
\endref
\medskip

\ref 
\key T
\by H. Triebel
\book Interpolation theory. Function spaces. Differential Operators
\publ Johann Ambrosius Barth, Heidelberg, Leipzig
\yr 1995
\endref
\medskip



\endRefs
\enddocument
\end



\by Bethuel F., Bourgain J., Brezis H. and Orlandi G.


\paper $W^{1,p}$ estimate for solutions to the Ginzburg-Landau


equation with boundary data in $H^{1/2}$


\jour C. R. Acad. Sci. Paris, S\'erie I, 


\vol 333 


\yr 2001


\pages 1069--1076


\endref


\medskip  





\ref


\no 6


\by Bethuel F., Brezis H. and Coron J.-M. 


\paper\nofrills Relaxed energies for harmonic maps


\inbook in Variational Problems


\eds Berestycki H., Coron J.-M., Ekeland I.


\publ  Birkh\"auser


\vol


\yr 1990


\pages 37--52


\endref


\medskip





\ref


\no 7


\by Bethuel F., Brezis H. and Orlandi G. 


\paper Small energy solutions to the Ginzburg-Landau equation


\jour $\ \ \ $ C. R. Acad Sc. Paris, S\'erie I,


\vol 331


\yr 2000


\pages 763--770


\endref


\medskip








\ref


\no 8


\by Bethuel F., Brezis H. and Orlandi G.


\paper Asymptotics for the Ginzburg-Landau equation in arbitrary


dimensions


\jour J. Funct. Anal.


\vol 186


\yr 2001


\pages 432--520


\endref


\medskip


\ref


\no 10


\by Bourgain J. and Brezis H.


\paper On the equation \text{\rm div }$Y=f$ and application to control


of phases


\jour J. Amer. Math. Soc.


\toappear


\endref


\medskip



\ref


\no 11
\by Bourgain J., Brezis H. and Mironescu P.





\ref


\no 13


\by Bourgain J., Brezis H. and Mironescu P.


\paper Limiting embedding theorems for $W^{s,p}$ when $s\nearrow 1$ and applications


\jour J. d'Anal. Math.


\vol 87


\yr 2002


\pages 77--101 


\endref


\medskip





\ref


\no 14


\by Bourgain J., Brezis H. and Mironescu P.


\paper


\jour  in preparation


\vol 


\yr 


\pages 


\endref


\medskip





\ref


\no 15


\by Boutet de Monvel A., Georgescu V. and Purice R.


\paper A boundary value problem related to the Ginzburg-Landau model


\jour Commun. Math. Phys.


\vol 142


\yr 1991


\pages 1--23


\endref


\medskip








\ref 


\no 16


\by Brezis H.


\paper\nofrills Liquid crystals and energy estimates for $S^2$-valued maps 


\inbook in Theory and Applications of Liquid Crystals


\eds J. Ericksen and D. Kinderlehrer


\publ Springer, 1987


\vol


\yr


\pages 31--52


\endref


\medskip








\ref


\no 17


\by Brezis H., Coron J.-M. and Lieb E.


\paper Harmonic maps with defects


\jour Commun. Math. Phys.


\vol 107


\yr 1986 


\pages 649--705


\endref


\medskip





\ref


\no 18


\by Brezis H., Li Y. Y., Mironescu P. and Nirenberg L.


\paper Degree and Sobolev spaces


\jour Topol. Meth. in Nonlin. Anal.


\vol 13


\yr 1999


\pages 181--190


\endref


\medskip











\ref


\no 19


\by Brezis H. and Mironescu P.


\paper Gagliardo-Nirenberg, composition and products in fractional


Sobolev spaces


\jour J. Evolution Equations


\vol 1


\yr 2001


\pages 387--404


\endref


\medskip





\ref 


\no 20


\by Brezis H. and Nirenberg L.


\paper Degree Theory and BMO, Part I: Compact manifolds without


boundaries


\jour Selecta Math.


\vol 1


\yr 1995


\pages 197-263


\endref


\medskip





\ref


\no 21


\by Cohen A., Dahmen W., Daubechies I. and DeVore R.


\paper Harmonic analysis of the space BV


\toappear


\endref


\medskip





\ref


\no 22


\by Demengel F.


\paper Une caract\'erisation des fonctions de $W^{1,1}(B^n, S^1)$ qui


peuvent \^ etre approch\'ees par des fonctions r\'eguli\`eres


\jour C. R. Acad. Sci. Paris, S\'erie I,


\vol 310


\yr 1990


\pages 553--557


\endref


\medskip 








\ref 


\no 24


\by Federer H.


\paper Geometric measure theory


\publ Springer, 1969


\endref


\medskip





\ref


\no 25


\by  Giaquinta M., Modica G. and Soucek J.


\book Cartesian Currents in the Calculus of Variations


\publ Springer


\vol II


\yr 1998


\endref


\medskip





\ref


\no 26


\by Hang F. B. and Lin F. H. 


\paper A remark on the Jacobians


\jour Comm. Contemp. Math.


\vol 2


\yr 2000


\pages 35--46


\endref


\medskip








\ref


\no 27


\by Jerrard R. L. and Soner H. M.


\paper Rectifiability of the distributional Jacobian for a class of functions


\jour C. R. Acad. Sci. Paris, S\'erie I,


\vol 329


\yr 1999


\pages 683--688


\endref


\medskip





\ref


\no 28


\by Jerrard R. L. and Soner H. M.


\paper Functions of bounded higher variation


\jour Preprint, 1999 


\vol


\yr


\pages


\endref


\medskip





\ref


\no 29


\by Jerrard R. L. and Soner H. M.


\paper The Jacobian and the Ginzburg-Landau energy


\jour Calc. Var. Partial Differential Equations 


\vol 14


\yr 2002


\pages 151--191


\endref


\medskip





\ref


\no 30


\by Lin F. H. and Rivi\`ere T.


\paper Complex Ginzburg-Landau equations in high dimensions and


codimension two area minimizing currents


\jour J. Eur. Math. Soc.


\vol 1


\yr 1999


\pages 237--311


\moreref


\paper Erratum


\vol 2\yr 2002


\pages 87--91


\endref


\medskip





\ref


\no 31


\by A. Ponce 


\paperinfo (in preparation)


\endref


\medskip







\ref


\no 32


\by Rivi\`ere T.


\paper Line vortices in the $U(1)$-Higgs model


\jour  Control Optim. and


Calc. of Var. 1


\yr 1996


\pages 77--167


\endref


\medskip





\ref


\no 34


\by Sandier E.


\paper Lower bounds for the energy of unit vector fields and applications


\jour J. Funct. Anal.


\vol 152


\yr 1998


\pages 379--403


\endref\medskip





\ref


\no 35


\by Sandier E.


\paper Ginzburg-Landau minimizers from $\Bbb R^{n+1}$ to $\Bbb R^n$


and minimal connections


\jour Indiana Univ. Math. J.


\vol 50


\yr 2001


\pages 1807--1844


\endref


\medskip





\ref


\no 36


\by Schoen R. and Uhlenbeck K. 


\paper Boundary regularity and the Dirichlet problem for harmonic maps


\jour J. Diff. Geom.


\vol 18


\yr 1983


\pages 253-268


\endref


\medskip





\ref 


\no 37


\by Simon L.


\paper Lectures on geometric measure theory


\jour Australian National University, Centre for Mathematical


Analysis, Canberra, 1983


\endref


\medskip





\ref


\no 38


\by Smets D.


\paper On some infinite sums of integer valued Dirac's masses


\jour C. R. Acad. Sc. Paris, S\'erie I, 


\vol 334


\yr 2002


\pages 371--374


\endref


\medskip





\ref


\no 39


\by Solonnikov


\paper Inequalities for functions of the classes ${\vec W}_p (\Bbb


R^n)$


\jour J. Soviet Math.


\vol 3

\yr 1975

\pages 549--564

\endref

\medskip


\endRefs

\enddocument
\end



\address ${}^{\text {(1)}}$ INSTITUTE FOR ADVANCED STUDY\endgraf
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