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\def\Mint{\diagup\hskip-.50truecm\int}
\def\mint{\diagup\hskip-.38truecm\int}
\def\var{\varepsilon}
\def\om{\Omega}
\def\br{\Bbb R}
\def\bc{\Bbb C}
\def\bz{\Bbb Z}
\def\ga{\Gamma}
\def\la{\Lambda}
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\def\W{W^{s,p}}
\def\fc{\tfrac}
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\def\fourint{\int\!\int\!\int\!\int}
\def\os{\,(\Omega ; S^1)\,}
\topmatter
\title{On some questions of topology for $S^1$-valued fractional
Sobolev spaces}\endtitle
\author{HAIM BREZIS$^{(1),(2)}$ AND PETRU MIRONESCU$^{(3)}$}\endauthor
\medskip
\address (1) \,\, ANALYSE NUM\'ERIQUE\endgraf
\qquad UNIVERSIT\'E P. ET M. CURIE, B.C. 187\endgraf
\qquad 4 PL. JUSSIEU\endgraf
\qquad 75252 PARIS CEDEX 05\endgraf
\endaddress
\address (2) \,\, RUTGERS UNIVERSITY\endgraf
\qquad DEPT. OF MATH., HILL CENTER, BUSCH CAMPUS\endgraf
\qquad 110 FRELINGHUYSEN RD, PISCATAWAY, NJ 08854\endgraf
\endaddress
\email brezis\@ccr.jussieu.fr; brezis\@math.rutgers.edu\endemail
\null
\address (3) \,\, D\'EPARTEMENT DE MATH\'EMATIQUES\endgraf
\qquad UNIVERSIT\'E PARIS-SUD\endgraf
\qquad 91405 ORSAY\endgraf
\endaddress
\email : Petru.Mironescu\@math.u-psud.fr\endemail
\medskip
\medskip
\endtopmatter
\document
\centerline{}
\bigskip
\noindent
{\bf I.  Introduction}
\medskip
The purpose of this paper is to describe the homotopy classes
(i.e., path-connected components) of the space $W^{s,p}\,(\om ; S^1)$.
Here, $0 < s < \infty,\, 1 < p < \infty,\, \om$ is a smooth, bounded,
connected open set in $\br^N$ and 
$$
W^{s,p}\,(\om ; S^1) = \{\,u \in W^{s,p}\,(\om ; S^1) ; \,|u| = 1\,
a.e.\}.
$$
Our main results are
\medskip
\proclaim{Theorem 1}  If $sp <2$, then $W^{s,p}\,(\om ; S^1)$ is
path-connected. \endproclaim
\medskip
\proclaim{Theorem 2}  If $sp \ge 2$, then $W^{s,p}\,(\om ; S^1)$ and
$C^0\,(\bar \om ; S^1)$ have the same homotopy classes in the sense of
[7].  More precisely: 
\medskip
\noindent
a)  each $u \in W^{s,p}\,(\om; S^1)$ is $W^{s,p}$-homotopic to some $v
\in C^\infty\,(\bar {\om}; S^1)$;
\medskip
\noindent
b)  two maps $u,v \in C^\infty\,(\bar \om ; S^1)$ are $C^0$-homotopic if
and only if they are $W^{s,p}$-homotopic. \endproclaim
\medskip
Here a simple consequence of the above results
\medskip
\proclaim{Corollary 1}  If $0 < s < \infty, 1 < p < \infty$ and $\om$
is simply connected, then $\W\,(\om ; S^1)$ is path-connected.
\endproclaim
\medskip
Indeed, when $sp < 2$ this is the content of Theorem 1.  When $sp \ge
2$, we use a) of Theorem 2 to connect $u_1,\, u_2 \in \W\,(\om ; S^1)$ to
$v_1,\, v_2 \in C^\infty\,(\bar \om ; S^1)$; since $\om$ is simply
connected, we may write $v_j = e^{i\varphi_j}$ for $\varphi_j \in
C^\infty\,(\bar \om ; \br)$ and then we connect $v_1$ to $v_2$ via
$e^{i\,[(1 - t) \varphi_1 + t \varphi_2]}$.
\medskip
When $M$ is a compact connected manifold, the study of the topology of
$W^{1,p}\,(\om ; M)$ was initiated in Brezis - Li [7] (see also White [26]
for some related questions).  In particular, these authors proved
Theorems 1 and 2 in the special case $s = 1$.  The analysis of
homotopy classes for an arbitrary manifold $M$ and $s = 1$ was
subsequently tackled by Hang - Lin [15].  The passage to $\W$ introduces
two additional difficulties:
\medskip
\noindent
a)  when $s$ is not an integer, the $\W$ norm is not ``local'';
\medskip
\noindent
b)  when $s \ge 2$ (or more generally $s > 1 + \fc 1 p$), gluing
two maps in $\W$ does not yield a map in $\W$.  
\medskip
In our proofs, we exploit in an essential way the fact that the target
manifold is $S^1$.   (The case of a general target is widely open.)
In particular, we  use the existence of a lifting  of $\W$ unimodular
maps when $s \ge 1$ and $sp \ge 2$ (see Bourgain - Brezis - Mironescu [4]).  
Another important tool is the following
\medskip
\proclaim{Composition Theorem (Brezis - Mironescu [10])}  If $f \in
C^\infty\,(\br ;\br)$ has bounded derivatives and $s \ge 1$, then $\varphi \longmapsto
f \circ \varphi$ is continuous from $\W \cap W^{1,sp}$ into $\W$. \endproclaim
\medskip
\noindent
{\bf Remark 1}.  A very elegant and straightforward proof of this
Composition Theorem has been given by V.Maz'ya and T.Shaposhnikova
[18].
\medskip 
A related question is the description, when $sp \ge 2$, of the homotopy classes of
$\W\,(\om ; S^1)$ in terms of lifting.  Here is a partial result 
\medskip
\proclaim{Theorem 3}  We have
\medskip
\noindent
a) if $s \ge 1,\,N \ge 3$, and $2 \le sp < N$, then 
$$
[u]_{s,p} = \{u e^{i\varphi} ; \varphi \in \W\,(\om ; \br)\,\cap
W^{1,sp}\,(\om ; \br)\};
$$
\noindent
b) if $sp \ge N$, then
$$
[u]_{s,p} = \{u e^{i\varphi} ; \varphi \in \W\,(\om ; \br)\}. 
$$
\endproclaim
\medskip
Theorem 3 is due to Rubinstein - Sternberg [21] in the special case where $s =
1,\, p = 2$ and $\om$ is the solid torus in $\br^3$. 
\medskip
When $0 < s < 1,\,N \ge 3$ and $2 \le sp < N$, there is no such simple
description of $[u]_{s,p}$.  For instance, using the ``non-lifting''
results in Bourgain - Brezis - Mironescu [4], it is easy to see that
$$
[1]_{s,p} \underset \neq \to \supset \{e^{i\varphi} ; \varphi \in
\W\,(\om; \br)\}.
$$ 
Here is an example:  if $N = 3,\, \om = B_1,\, 0 < s < 1,\, 1 < p < \infty,\, 2 \le sp <
3$, then
\medskip
\noindent
a) $u (x) = e^{1/|x|^\alpha} \in [1]_{s,p}$;
\medskip
\noindent
b) there is no $\varphi \in \W\,(B_1; \br)$ such that $u =
e^{i\varphi}$ 
\medskip
\noindent
for $\alpha$ satisfying $\tfrac{3 - sp}{p} \le \alpha <
\tfrac{3 - sp}{sp}$.  
\medskip
However, we conjecture the following result
\medskip
\proclaim{Conjecture 1}  Assume that $0 < s < 1,\,1 < p < \infty,\,N
\ge 3$ and $2 \le sp < N$.  Then  \endproclaim
$$
[u]_{s,p} = u \overline{\{e^{i\varphi} ; \varphi \in \W\,(\om ;
\br)\}}^{\W}.
$$
We will prove below (see Corollary 2) that ``half'' of Conjecture 1 holds, namely
$$
[u]_{s,p} \supset u \overline{\{e^{i\varphi} ; \varphi \in
\W\,(\om ; \br)\}}^{\W}. 
$$
\medskip
In a different but related direction,  we establish some partial
results concerning the density of $C^\infty\,(\bar \om ; S^1)$ into
$\W\,(\om ; S^1)$.
\medskip
\proclaim{Theorem 4}  We have, for $0 < s < \infty,\, 1 < p < \infty$:
\medskip
\noindent
a) if $sp < 1$, then $C^\infty\,(\bar \om ; S^1)$ is dense in
$\W\,(\om ; S^1)$;
\medskip
\noindent
b) if $1 \le sp < 2, N \ge 2$, then $C^\infty\,(\bar \om ; S^1)$ is not
dense in $\W\,(\om ; S^1)$;
\medskip
\noindent
c) if $sp \ge N$, then $C^\infty\,(\bar \om ; S^1)$ is dense in
$\W\,(\om; S^1)$;
\medskip
\noindent
d) if $s \ge 1$ and $sp \ge 2$, then $C^\infty\,(\bar \om ; S^1)$ is
dense in $\W\,(\om ; S^1)$.  \endproclaim
\medskip
There is only one missing case for which we make the following
\medskip
\proclaim{Conjecture 2}  If $0 < s < 1,\, 1 < p < \infty,\, N \ge 3,\, 2
\le sp < N$, then $C^\infty\,(\bar \om ; S^1)$ is dense in $\W\,(\om ;
S^1)$.\endproclaim   
\medskip
\noindent
This problem is open even when $\om$ is a ball in $\br^3$.  We will prove
below the equivalence of Conjectures 1 and 2.  
\medskip
Parts of Theorem 4 were already known.  Part a) is due to Escobedo
[14]; so is part b), but in this case the idea goes back to
Schoen - Uhlenbeck [24] (see also Bourgain - Brezis - Mironescu [5]).
For $s = 1$, part c) is due to Schoen - Uhlenbeck [24]; their argument
can be adapted to the general case (see, e.g., Brezis - Nirenberg [12] or
Brezis - Li [7]).  The only new result is part d).  The proof relies
heavily on the Composition Theorem and Theorems 2 and 3.  We do not
know any direct proof of d).  We also mention that for $s = 1$ and
$\om = B_1$, Theorem 4 was established by Bethuel - Zheng [3].  For a
general compact connected manifold $M$ and for $s = 1$, the question
of density of $C^\infty\,(\bar \om ;M)$ into $W^{1,p}\,(\om ;M)$ was
settled by Bethuel [1] and Hang - Lin [15]. 
\medskip
\noindent
{\bf Remark 2.}  In Theorems 2 and 4, one may replace $\om$ by a
manifold with or without boundary.  The statements are unchanged.
However, the argument in the proof of Theorem 1 does not quite go
through to the case of a manifold without boundary.  Nevertheless, we
make the following  
\medskip
\proclaim{Conjecture 3}  Let $\om$ be a manifold without boundary with
$\dim \om \ge 2$.  Then $\W\,(\om ; M)$ is path-connected for every $0
< s < \infty, 1 < p < \infty$ with $sp < 2$, and for every compact connected
manifold $M$.  \endproclaim
\medskip
Note that the condition $\dim \om \ge 2$ is necessary, since
$\W\,(S^1;S^1)$ is not path-connected when $sp \ge 1$.
\medskip
Finally, we investigate the local path-connectedness of $\W\,(\om ;
S^1)$.  Our main result is 
\proclaim{Theorem 5}  Let $0 < s < \infty, 1 < p < \infty$.  Then
$\W\,(\om ; S^1)$ is locally path-connected.  Consequently, the
homotophy classes coincide with the connected components and they are
open and closed. \endproclaim 
\medskip
The heart of the matter in the proof is the following
\medskip
\noindent
{\bf Claim}.  Let $0 < s < \infty, 1 < p < \infty$.  Then there is some
$\delta > 0$ such that, if $||u - 1||_{\W} < \delta$, then $u$ may be
connected to $1$ in $\W$.  
\medskip
As a consequence of Theorem 5, we have
\medskip
\proclaim{Corollary 2}  Let $0 < s < 1, 1 < p < \infty$.  Then \endproclaim
$$
[u]_{s,p}\supset \, \overline{\{u e^{i\varphi} ; \varphi \in
\W\,(\om ; \br)\}}^{\W}\,\, = u \, \overline{\{ e^{i\varphi} ;
\varphi \in \W\,(\om ; \br)\}}^{\W}.  
$$
\medskip
Equality in Corollary 2 follows from the well-known fact that $\W \cap
L^\infty$ is an algebra.  The inclusion is a consequence of the fact
that, clearly, we have
$$
[u]_{s,p} \,\,\supset \,\,\{u e^{i\varphi};\,\varphi \in \W\,(\om ;
\br)\}
$$
and of the closedness of the homotopy classes.
\medskip
Another consequence of Theorem 5 is
\medskip
\proclaim{Corollary 3}  Conjecture 1 $\Leftrightarrow$ Conjecture
2. \endproclaim
\medskip
{\bf Proof.}  By Corollary 2, we have 
$$
[u]_{s,p} \supset u \overline {\{e^{i\varphi} ; \varphi \in
\W\,(\om ; \br)\}}^{\W}.
$$
We prove that the reverse inclusion follows from Conjecture 1.  By
Proposition 1 a) below,  we may take $u = 1$.  Let $v \in [1]_{s,p}$.
By Theorem 5, there is some $\var > 0$ such that $||v-w||_{\W} < \var
\,\Rightarrow \, w \in [1]_{s,p}$.  Let $(w_n) \subset C^\infty\,(\bar
\om ; S^1)$ be such that $w_n \to v$ in $\W$ and $||w_n - v||_{\W} <
\var$.  By  Theorem 2 b), we obtain that $w_n$ and $1$ are homotopic
in $C^0\,(\bar \om ; S^1)$.  Thus $w_n = e^{i\varphi_n}$ for some {\bf
globally} defined smooth $\varphi_n$.  Hence 
$$
v \in \overline {\{e^{i\varphi} ; \varphi \in \W\,(\om ; \br)\}}^{\W}. 
$$
\medskip
Conversely, assume that Conjecture 2 holds.  Let $u \in \W\,(\om ;
S^1)$.  By Theorem 2 a), there is some $w \in C^\infty\,(\bar \om ;
S^1)$ such that $w \in [u]_{s,p}$.  By Proposition 1 b), we have $u
\bar w \in [1]_{s,p}$.  Thus $u \bar w \in\,\overline
{\{e^{i\varphi} ; \varphi \in \W\,(\om ; \br)\}}^{\W}$, so that clearly $u
\bar w \in\,\overline {\{e^{i\varphi} ; \varphi \in C^\infty(\bar
\om ; \br)\}}^{\W}$.  Finally, $u \in \overline {\{w e^{i\varphi} ; \varphi \in
C^\infty\,(\bar \om ; \br)\}}^{\W}$, i.e. $u$ may be approximated by
smooth maps.  
\medskip
In the same vein, we raise the following 
\medskip
\noindent
{\bf Open Problem 1}.  Let $\om$ be a manifold with or without
boundary.  Is $\W\,(\om ; M)$ locally path-connected for every $s,p$
and every compact manifold $M$?
\medskip
The case $s = 1$ can be settled using the methods of Hang - Lin [15].
We will return to this question in a subsequent work; see Brezis -
Mironescu [11].
\medskip
The reader who is looking for more open problems may also consider the
following
\medskip
\noindent
{\bf Open Problem 2}.  Let $\om \subset \br^2$ be a smooth bounded
domain.  Assume $0 < s < \infty,$ \newline $1 < p < \infty$ and $1 \le sp < 2$
(this is the range where $C^\infty\,(\bar \om ; S^1)$ is not dense in
$\W\,(\om ; S^1)$).  Set 
$$
\Cal R_0 = \{ u \in \W\,(\om ; S^1); u \text{ is smooth except a finite
number of points}\}.  
$$
(Here, the number and location of singular points is left free). Is
$\Cal R_0$ dense in $\W\,(\om ; S^1)$? 
\medskip
\noindent
{\bf Comment.}  $\Cal R_0$ is known to be dense in $\W\,(\om ; S^1)$ in
many cases, e.g.:
\medskip
\noindent
a)  $s = 1$ and $1 \le p < 2$; see Bethuel-Zheng [3]
\medskip
\noindent
b)  $s = 1 - 1/p$ and $ 2 < p < 3$; see Bethuel [2]
\medskip
\noindent
c)  $s = 1/2$ and $p = 2$; see Rivi\`ere [20].


\bigskip
The paper is organized as follows
\bigskip
\noindent
I.   Introduction
\medskip
\noindent
II.  Proof of Theorem 1
\medskip
\noindent
III. Proof of Theorems 2 and 3
\medskip
\noindent
IV.  Proof of Theorem 4
\medskip
\noindent
V.   Proof of Theorem 5
\bigskip
\noindent
Appendix A.  An extension lemma
\medskip
\noindent
Appendix B.  Good restrictions
\medskip
\noindent
Appendix C.  Global lifting
\medskip
\noindent
Appendix D.  Filling a hole - the fractional case
\medskip
\noindent
Appendix E.  Slicing with norm control
\bigskip
\noindent
{\bf II.  Proof of Theorem 1}
\medskip
\noindent
{\bf Case 1:}  $sp < 1$
\medskip
\noindent
When $sp < 1$, we have the following more general result
\proclaim{Theorem 6}  If  $s > 0, \,1 < p < \infty, \,sp < 1$ and $M$ is a
compact manifold, then $\W\,(\om ; M)$ is path-connected.
\endproclaim
\medskip
{\bf Proof.}  Fix some $a \in M$.  For $u \in \W\,(\om ; M)$, let
$$
\tilde u \,=\,\cases u,  &\text{ in } \om \\
a,  &\text{ in } \br^N \backslash \om  \endcases .
$$
\noindent
Since $sp < 1$, we have $\tilde u \in W^{s,p}_{loc}\,(\br^N ; M)$.  Let
$U\,(t,x)\,=\, \tilde u\,(x/(1 - t)) , \,0 \le t < 1, \,x \in \om$ and
$U\,(1,x)\, \equiv a$.  Then clearly $U \in C\,([0,1] ; \W\,(\om ; M))$
and $U$ connects $u$ to the constant $a$ (here we use only $sp < N$).
\medskip
\noindent
{\bf Case 2:}  $1 < sp < 2, N \ge 2$
\medskip
In this case one could adapt the tools developed in Brezis - Li [7],
but we prefer a more direct approach.  
\medskip
Let $\var > 0$ be such that the projection onto $\partial \om$ be well-defined
and smooth in the region $\{ x \in \br^N ;\text{ dist } (x, \partial
\om)\, < 2 \var\}$.  Let $\omega = \{ x \in \br^N \backslash \bar \om
;\text{ dist }\,(x, \partial \om) < \var\}$.  We have  $\partial \omega
\, = \, \partial \om \cup \Lambda$, where $\la \, = \, \{ x \in
\br^N \backslash \om ; \text{ dist }\,(x, \partial \om)\, = \var \}$.
\medskip
Since $1 < sp < 2$, we have $1/p < s < 1 + 1/p$; thus, for $u \in \W$
 we have tr $u \in W^{s - 1/p,p}$.  Let $u \in \W\,(\om ; S^1)$.
Fix some $ a \in S^1$ and define $v \in W^{s - 1/p,p}\,(\partial
\omega ; S^1)$ by
$$
v\, = \, \cases \text{tr }u, \quad &\text{ on } \partial \omega\\
a,  \quad &\text{ on } \Lambda  \endcases .
$$
\medskip
We use the following extension result.  (The first result of this kind
is due to Hardt - Kinderlehrer - Lin [16]; it corresponds to our lemma
when $\sigma = 1 - 1/p,\,p < 2$.)
\medskip
\proclaim{Lemma 1}  Let $0 < \sigma < 1,\, 1 < p < \infty,\, \sigma p < 1$.
Then any $v \in W^{\sigma,p}\,(\partial \omega ; S^1)$ has an
extension $w \in W^{\sigma + 1/p,p}\,(\omega ; S^1)$.
\endproclaim
\medskip
The proof is given in Appendix A; see Lemma A.1.  It relies heavily on
the lifting results in Bourgain - Brezis - Mironescu [4].  
\medskip
Returning to the proof of Case 2, with $w$ given by Lemma 1, set
$$
\tilde u \,=\, \cases u, \,\, \quad &\text{ in } \om\\
w, \,\, \quad &\text{ in } \omega\\
a, \,\, \quad &\text{ in } \br^n \backslash\,(\om \cup \omega)
\endcases .
$$
\medskip
Clearly, $\tilde u \in W^{s,p}_{loc}\,(\br^N ; S^1)$ and $\tilde u$ is
constant outside some compact set.  As in the proof of Theorem 6, we
may use $\tilde u$ to connect $u$ to $a$, since once more we have $sp <
N$.
\medskip
\noindent
{\bf Case 3:}  $sp = 1, \,\, N \ge 2$
\medskip
The idea is the same as in the previous case; however, there is an
additional difficulty, since in the limiting case $s = 1/p$ the trace
theory is delicate - in particular, tr $W^{1/p,p}\, \neq\, L^p$
(unless $p = 1$).  Instead of trace, we work with a notion of ``good
restriction'' developed in Appendix B; when $s = 1/2, \, p = 2$, the
space of functions in $H^{1/2}$ having $0$ as good restriction on the
boundary coincides with the space $H^{1/2}_{00}$ of Lions - Magenes
[17] (see Theorem 11.7, p. 72). 
\medskip
Our aim is to prove that any $u \in W^{1/p,p}\,(\om ; S^1)$ can be
connected to a constant $a \in S^1$.
\medskip
{\bf Step 1:}  we connect $u \in W^{1/p,p}\,(\om ; S^1)$ to some
$u_1 \in W^{1/p,p}\,(\om ; S^1)$ having a good restriction on $\partial \om$
\medskip
Let $\var > 0$ be such that the projection $\Pi$ onto $\partial \om$
be well-defined and smooth in the set $\{x \in \br^N; \text{  dist  }
\,(x, \partial \om)\, < 2 \var)\}$.  For $0 < \delta < \var$, set
$\Sigma_{\delta} = \{x \in \om ; \text{  dist  }\,(x, \partial \om) =
\delta\}$.  By Fubini, for a.e. $0 < \delta < \var$ , we have
$$
u|_{\sum_{\delta}} \in W^{1/p,p}\,(\Sigma_\delta) \text{ and }
\underset {\Sigma_{\delta}}\to\int\,\underset {\om}\to\int \frac{|u(x) -
u(y)|^p}{|x - y|^{N + 1}}\,dy\,ds_x\, < \infty .  \tag 1
$$
By Lemma B.5, this implies that $u$ has a good restriction on
$\Sigma_\delta$, and that Rest $u|_{\sum_{\delta}}\, = \,
u|_{\sum_{\delta}}$ a.e. on $\Sigma_{\delta}$.
\medskip
Let any $0 < \delta < \var$ satisfying (1).  For $0 < \lambda <
\delta$, let $\Psi_\lambda$ be the smooth inverse of
$\Pi|_{\sum_{\lambda}} \, : \, \Sigma_{\lambda} \to \partial \om$.
Let also $\om_\lambda \, = \, \{x \in \om ; \text{ dist }\,(x,
\partial \om) \, > \lambda\}$.  Consider a continuous family of
diffeomorphisms $\Phi_t : \bar \om \to \overline{\om_{t\delta}}, 0
\le t \le 1$, such that $\Phi_0 =$ id and  $\Phi_t |_{\partial \om}\, = \,
\Psi_{t\delta}$.  Then $t \mapsto u \circ \Phi_t$ is a homotopy in
$W^{1/p,p}$.  Moreover, if $u_t \, = \, u \circ \Phi_t$, then $u_0 = u$ and
$u_1|_{\partial \om} = u|_{\sum_\delta} \circ
\bold\Psi_\delta|_{\partial \om}$.  By (1), $u_1$ has a good restriction on $\partial \om$.
\medskip
{\bf Step 2:}  we extend $u_1$ to $\br^N$
\medskip
Let $\omega = \{x \in \br^N \backslash \bar \om ; \text{ dist }\,(x ;
\partial \om)\, < \var\}$.  As in Case 2, we fix some $a \in S^1$ and
set 
$$
v \, = \,\cases u_1, \quad &\text{ on } \partial \om\\
a, \quad &\text{ on } \Lambda  \endcases .
$$  
Clearly, $v \in W^{1/p,p}\,(\partial\omega)$, so that $v \in
W^{\sigma,p}\,(\partial\omega)$  for $0 < \sigma < 1/p$.  We fix any
$0 < \sigma < 1/p$. By Lemma 1, there is some $w \in W^{\sigma +
1/p,p}\,(\omega ; S^1)$ such that $w|_{\partial \omega} = v$.  We define 
$$
\tilde u_1 \, = \,\cases u_1, \quad &\text{ in } \om\\
w, \quad &\text{ in } \omega\\
a, \quad &\text{ in } \br^N \backslash (\om \cup \omega)  \endcases .
$$
We claim that $\tilde u_1 \in W^{1/p,p}_{loc}\,(\br^N ; S^1)$.  Obviously,
$\tilde u \in W^{1/p,p}_{loc}\,(\br^N \backslash \om)$.  It remains to check that $\tilde u_1 \in W^{1/p,p}\,(\om \cup \omega)$.
This is a consequence of 
\medskip
\proclaim{Lemma 2}  Let $0 < s < 1, 1 < p < \infty, sp \ge 1$ and $\rho
>s$.  Let $u_1 \in \W (\om)$ and $ w \in W^{\rho,p} (\omega)$. Assume that
$u_1$ has a good restriction {\rm Rest } $u_1|_{\partial \om}$ on
$\partial \om$ and that $\hskip 10mm$ {\rm tr } $w|_{\partial \om} =$ {\rm Rest }
$u_1|_{\partial \om}$.  Then the map 
$$
\cases u_1,\,\, &\text{ in }\om\\
w,\,\, &\text{ in } \omega \endcases
$$
 belongs to $\W\,(\om \cup \omega)$.  \endproclaim
\medskip
Clearly, in the proof of Lemma 2 it suffices to consider the case of a
flat boundary.  When $\om\,=\, (-1,1)^{N - 1}\, \times\,(0,1)$ and $\omega\,=\,
(-1,1)^{N - 1}\, \times\,(-1,0)$, the proof of Lemma 2 is presented in Appendix
B; see Lemma B.4.
\medskip
Returning to Case 3 and applying Lemma 2 with $s = 1/p,\, \rho =
\sigma + 1/p$, we obtain that $\tilde u_1 \in W^{1/p,p}_{loc}\,(\br^N)$.
As in the two previous cases, this means that $u_1$ is $W^{1/p,p}$-homotopic
 to a constant.
\medskip
\noindent
{\bf Case 4:} $1 \le sp < 2,\, N=1$
\medskip
\noindent
In this case, $\om$ is an interval.  Recall the following result
proved in Bourgain - Brezis - Mironescu [4] (Theorem 1): if $\om$ is
an interval and $sp \ge 1$, then for each 
$u \in \W (\om ; S^1)$ there is some
$\varphi \in \W \,(\om ;\br)$ such that $u = e^{i\varphi}$.  Recall
also that, when $sp \ge N$, then $C^\infty\,(\br;\br)$ functions $f$
with bounded derivatives operate on $\W$; that is, the map $\varphi
\mapsto f \circ \varphi$ is continuous from $\W$ into itself (see,
e.g., Peetre [19] for $sp > N$, Runst - Sickel [23], Corollary 2 and
Remark 5 in Section 5.3.7 or Brezis - Mironescu [9] when $sp = N$;
this is also a consequence of the Composition Theorem).  By combining
these two results, we find that the homotopy $t \mapsto e^{i (1 - t) \varphi}$ connects $u = e^{i\varphi}$ to $1$.
\medskip
The proof of Theorem 1 is complete.




\bigskip
\bigskip
{\bf III. Proof of Theorems 2 and 3}
\medskip
We start with some useful remarks.  For $u \in \W\,(\om ; S^1)$, let
$[u]_{s,p}$ denote its homotopy class in $\W$.
\proclaim{Proposition 1}  Let $0 < s < \infty,\,1 < p < \infty$.  For
$u, v \in \W\,(\om ; S^1)$, we have  \endproclaim
\medskip
\noindent
a) $u [v]_{s,p}\,=\, [uv]_{s,p}$;
\medskip
\noindent
b) $[u]_{s,p}\,=\, [v]_{s,p}\,\Leftrightarrow\,[u \bar
v]_{s,p}\,=\,[1]_{s,p}$;
\medskip
\noindent
c) $[u]_{s,p}\,\,[v]_{s,p}\,=\,[uv]_{s,p}$.
\medskip
The proof relies on two well-known facts:  $\W \cap L^\infty$ is an
algebra; moreover, if $u_n \to u, \,v_n \to v$ in $\W$ and
$||u_n||_{L^\infty}\,\le\,C,\,||v_n||_{L^\infty}\,\le\,C$, then
$u_n\,v_n \to \,uv$ in $\W$.  Here is, for example, the proof of c)
(using a)).  Let first $u_1 \in [u]_{s,p},\,v_1 \in [v]_{s,p}$.  If
$U,V$ are homotopies connecting $u_1$ to $u$ and $v_1$ to $v$, then
$UV$ connects $u_1\,v_1$ to $uv$; thus $[u]_{s,p}\,[v]_{s,p} \subset
[uv]_{s,p}$.  Conversely, if $w \in [uv]_{s,p}$, then $w \in u
[v]_{s,p}$ (by a)), so that $w \bar u \in [v]_{s,p}$.  Therefore, $w =
u(w \bar u) \in [u]_{s,p}\,[v]_{s,p}$. 
\medskip
We next recall the degree theory for $\W$ maps; see Brezis - Li -
Mironescu - Nirenberg [8] for the general case, White [25] when $s =
1$ or Rubinstein - Sternberg [20] for the space $H^1\,(\om ; S^1)$ and
$\om$ the solid torus in $\br^3$.  Let $0 < s < \infty,\, 1 < p < \infty$ be
such that $sp \ge 2$.  Let $ u \in \W\,(S^1 \,\times\, \la ; S^1)$, where $\la$
is some open connected set in $\br^k$.  Clearly, for a.e. $\lambda \in
\la, u\,(\cdot, \lambda) \in \W\,(S^1 ; S^1)$.  For any such $\lambda,
u\,(\cdot, \lambda)$ is continuous, so that it has a winding number
(degree) deg $\bigl( {u\,(\cdot, \lambda)\bigr )}$.  The main result
in [8] asserts that, if $sp \ge 2$, then this degree is constant a.e. and
stable under $\W$ convergence.
\medskip
In the particular case where $s \ge 1$, there is a formula
$$
\text { deg } ({u(\cdot, \lambda))}\,=\, \frac{1}{2\pi}
\underset{S^1}\to \int u\,(x,\lambda)\, \wedge\, \frac{\partial
u}{\partial \tau}\, (x, \lambda)\,ds_x,
$$
where $u \wedge v \,=\, u_1\,v_2 - u_2\,v_1$.  It then follows that,
if $s \ge 1$ and $sp \ge 2$, we have
$$
\text { deg }(u|_{S^1 \times \la})\,=\, \underset\la\to \Mint \, \underset
{S^1}\to \Mint \, u(x,\lambda)\, \wedge \frac{\partial u}{\partial
\tau}\, (x, \lambda)\, ds_x d \lambda.
$$
\medskip
Clearly, the above result extends to domains which are diffeomorphic
to $S^1 \times \la$.  In the sequel, we are interested in the following
particular case: let $\ga$ be a simple closed smooth curve in $\om$
and, for small $\var > 0$, let $\ga_\var$ be the $\var$-tubular
neighborhood of $\ga$.  We fix an orientation on $\ga$.
\medskip
Let $\Phi : S^1 \times B_\var \to \ga_\var$ be a diffeomorphism such that
$\Phi|_{S^1 \times \{0\}} : S^1 \times \{0\} \to \ga$ be an orientation
preserving diffeomorphism; here $B_\var$ is the ball of radius $\var$
in $\br^{N - 1}$.  Then we may define deg $(u|_{\ga_\var})\,=\,$
deg $(u \circ \Phi|_{S^1 \times B_\var})$; this integer is stable under
$\W$ convergence.
\medskip
We now prove b) of Theorem 2, which we restate as 
\medskip
\proclaim{Proposition 2}  Let $0 < s < \infty,\,1 < p < \infty,\, sp
\ge 2$.  Let $ u, v \in C^\infty\,(\bar \om ; S^1)$.  Then
$[u]_{s,p}\,=\,[v]_{s,p}$ if and only if $u$ and $v$ are $C^0$-
homotopic.  \endproclaim
\medskip
{\bf Proof.}  Using Proposition 1, we may assume $v = 1$.  Suppose
first that $u \in C^\infty\,(\bar \om ; S^1)$ and $1$ are
$C^0$-homotopic.  Then $u$ and $1$ are $\W$-homotopic.  Indeed, when
$s = 1$, this is proved in Brezis - Li [7], Proposition A.1; however,
their proof works without modification for any $s$.  We sketch an
alternative proof: since $u$ and $1$ are $C^0$-homotopic, there is
some $\varphi \in C^\infty\,(\bar \om ; \br)$ such that $u =
e^{i\varphi}$.  Then $t \mapsto e^{i\,(1 - t)\,\varphi}$ connects $u $
to $1$ in $\W$.
\medskip
Conversely, assume that the smooth map $u$ is $\W$-homotopic to $1$.
By continuity of the degree, we then have deg $(u|_{\ga_{\var}})\,=\,0$
for each $\Gamma$.  Since $u$ is smooth, we obtain
$$
0 = \text{ deg }(u|_{\ga_\var})\,=\,  \text { deg }(u|_\ga)\, = \frac{1}{2\pi}
\underset\ga\to \int\, u \wedge \frac{\partial u}{\partial \tau} ds.
$$
Thus the closed form $X = u \wedge D u$ has the property that
$\underset\ga\to \int X \cdot \tau ds = 0$ for any simple closed
smooth curve $\ga$.  By the general form of the Poincar\'e lemma, there
is some $\varphi \in C^\infty\,(\bar \om ; \br)$ such that $X =
D\varphi$.  One may easily check that $u = e^{i(\varphi + C)}$ for some
constant $C$.  Then $t \mapsto e^{i(1 - t)\,(\varphi + C)}$ connects
$u$ to $1$ in $C^0\,(\bar \om ; S^1)$.
\medskip
We now turn to the proof of the remaining assertions in Theorems 2 and
3.
\medskip
\noindent
{\bf Case 1:}  $sp \ge N,\,N \ge 2$
\medskip
{\bf Step 1:}  each $u \in \W\,(\om ; S^1)$ can be connected to a
smooth map $v \in C^\infty\,(\bar \om ; S^1)$
\medskip
This is proved in Brezis - Li [7], Proposition A.2, for $s = 1$ and $p
\ge N$; their arguments apply to any $s$ and any $p$ such that $sp \ge
N$.  The main idea originates in the paper Schoen - Uhlenbeck [23];
see also Brezis - Nirenberg [12], [13].
\medskip
{\bf Step 2:}  we have $[u]_{s,p}\,=\,\{u e^{i\varphi}; \varphi \in
\W\,(\om ; \br)\}$
\medskip
Let $\varphi \in \W\,(\om ; \br)$.  Then $t \longmapsto u e^{i(1 -
t)\varphi}$ connects $u e^{i\varphi}$ to $u$ in $\W$.  (Recall that,
if $f \in C^\infty\,(\br ; \br)$ has bounded derivatives and $sp \ge
N$, then the map $\varphi \mapsto f \circ \varphi$ is continuous from
$\W$ into itself.)  This proves ``$\supset$''.  To prove the reverse
inclusion, by Proposition 1, it suffices to show that
$[1]_{s,p}\,\subset\,\{e^{i\varphi}\,;\varphi \in \W\,(\om ;
\br)\,\}$.
\medskip
Let $v \in [1]_{s,p}$.  For each $x \in \om$, let $B_x \subset \om$ be
a ball containing $x$.  We recall the following lifting result from
Bourgain - Brezis - Mironescu [4] (Theorem 2):  if $U$ is simply
connected in $\br^N$ and $sp \ge N$, then for each $w \in \W\,(U ;
S^1)$ there is some $\psi \in \W\,(U ; \br)$ such that $w =
e^{i\psi}$.  Thus, for each $x \in \om$ there is some $\varphi_x \in
\W\,(B_x ; \br )$ such that $v|_{B_x} = e^{i\varphi_x}$.  Note that ,
in $B_x \cap B_y$, we have $\varphi_x - \varphi_y \in \W\,(B_x \cap
B_y; 2 \pi \bz)$.  Therefore, $\varphi_x - \varphi_y \in VMO\,(B_x
\cap B_y; 2\pi \bz)$, since $sp \ge N$.  It then follows that
$\varphi_x - \varphi_y$ is constant a.e. on $B_x \cap B_y$; see Brezis
- Nirenberg [12], Section I.5.
\medskip
By a standard continuation argument, we may thus define a (multi-valued) 
argument $\varphi$ for $v$ in the following way:  fix some
$x_0 \in \om$.  For any $x \in \om$, let $\gamma$ be a simple smooth
path from $x_0$ to $x$.  Then, for $\var > 0$ sufficiently small,
there is a unique function $\varphi^\gamma \in \W\,(\gamma_\var ;
\br)$ such that $v|_{\gamma_\var} \,=\, e^{i\varphi^\gamma}$ and
$\varphi^\gamma|_{B_\var (x_0)} \,=\, \varphi_{x_0}|_{B_\var (x_0)}$;
here, $\gamma_\var$ is the $\var$-tubular neighborhood of $\gamma$.
We then set 
$$
\varphi|_{B_{\var}(x)}\,=\,\varphi^\gamma|_{B_{\var}(x)}.
$$
\medskip
We actually claim that $\varphi$ is single-valued.  This follows  from
\medskip
\proclaim{Lemma 3}  Assume that  $0 < s < \infty,\,1 < p < \infty,\, sp
\ge N,\,N \ge 2$.  If $w \in \W\,(S^1 \times B_1; S^1)$ is such that
{\rm  deg } $\left(w|_{S^1 \times B_1}\right) = 0$, then there is some $\psi
\in \W\,(S^1 \times B_1)$ such that $w = e^{i\psi}$. \endproclaim
\medskip
Here, $B_1$ is the unit ball in $\br^{N-1}$.  The proof of Lemma 3 is
presented in Appendix C; see Lemma C.1.
\medskip
Returning to the claim that $\varphi$ is single-valued, we have that
deg $\left(v|_{\ga_{\var}}\right)\,=\,0$ for each $\ga$, since $v \in
[1]_{s,p}$.  By Lemma 3, a standard argument implies that $\varphi$ is
single-valued.
\medskip
The proof of Theorems 2 and 3 when $sp \ge N$ is complete.
\bigskip
\noindent
{\bf Case 2:}  $s \ge 1,\,1 < p < \infty,\,N \ge 3,\,2 \le sp < N$
\medskip
{\bf Step 1:}  we have $[u]_{s,p} = \{u e^{i\varphi} ; \varphi \in
\W\,(\om ; \br) \cap W^{1,sp}\,(\om ; \br)\}$
\medskip
For ``$\supset$'', we use the Composition Theorem mentioned in the
Introduction, which implies that $t \mapsto u e^{i(1-t)\varphi}$
connects $u e^{i\varphi}$ to $u$ in $\W$.
\medskip
For ``$\subset$'' it suffices to prove that
$[1]_{s,p}\,\subset\,\{e^{i\varphi}; \varphi \in \W\,(\om ; \br)\, \cap
W^{1,sp}\,(\om ; \br)\}$.  We proceed as in Case 1, Step 2.  Let $v
\in [1]_{s,p}$.  The corresponding lifting result we use is the
following (see Bourgain - Brezis - Mironescu [4], Lemma 4):  if $s \ge
1,\,sp \ge 2$ and $U$ is simply connected in $\br^N$, then for each $w
\in \W\,(U ; S^1)$ there is some $\psi \in \W\,(U ; \br)\, \cap
W^{1,sp}\,(U ; \br)$ such that $w\,=\,e^{i\psi}$.  As in Case 1, for
each $x$ there is some $\varphi_x \in \W\,(B_x ; \br)\,\cap
W^{1,sp}\,(B_x ; \br)$ such that $v|_{B_x}\,=\,e^{i\varphi_x}$.
Since $\varphi_x\,-\,\varphi_y \in W^{1,1}\,(B_x \cap B_y; 2\pi
\bz)$, we find that $\varphi_x - \varphi_y$ is constant ae. on $B_x
\cap B_y$ (see [4], Theorem B.1.).  These two ingredients allow the
construction of a multi-valued phase $\varphi \in \W \cap W^{1,sp}$
for $v$.  To prove that $\varphi$ is actually single-valued, we rely
on 
\medskip
\proclaim{Lemma 4}  Assume that $s \ge 1,\,1 < p < \infty,\,N \ge 3,\,2 \le sp <
N$.  If $w \in \W\,(S^1 \,\times\, B_1; S^1)$ is such that
{\rm deg } $(w|_{S^1 \,\times\, B_1})\,=\,0$, then there is some $\psi \in
\W\,(S^1 \,\times B_1; \br)\,\cap W^{1,sp}\,(S^1 \,\times \,B_1; \br)$ such that
$v\,=\,e^{i\psi}$.  \endproclaim
\medskip
The proof of Lemma 4 is given in Appendix C; see Lemma C.2.
\medskip
The proof of Step 1 is complete.
\medskip
{\bf Step 2:} assume $s \ge 1,\,1 < p < \infty,\,sp \ge 2$;  then, for
each $u \in \W\,(\om ; S^1)$, there is some $v \in \W\,(\om ;
S^1)\,\cap C^\infty\,(\om ; S^1)$ such that $v \in [u]_{s,p}$  
\medskip
Consider the form $X\,=\,u \wedge Du$.  Then $X \in
W^{s-1,p}\,(\om)\,\cap L^{sp}\,(\om)$ (see Bourgain - Brezis -
Mironescu [4], Lemmas D.1 and D.2).  Let $\varphi \in \W\,(\om ;
\br)\,\cap W^{1,sp}\,(\om ; \br)$ be any solution of $\Delta
\varphi\,=\,$ div $X$ in $\om$.  By the Composition Theorem, we then
have $e^{- i\varphi} \in \W\,(\om ; S^1)$, and thus $v\,=\,u
e^{- i\varphi} \in \W\,(\om ; S^1)$.  We claim that $v \in
C^\infty\,(\om ; S^1)$.  Indeed, let $B$ be any ball in $\om$.  Since
$s \ge 1$ and $sp \ge 2$, there is some $\psi \in \W\,(B ; \br)\,\cap
W^{1,sp}\,(B; \br)$ such that $u|_B = e^{i\psi}$.  It then follows
that $X|_B \,=\, D\psi$.  Thus $\Delta \varphi\,=\,\Delta \psi$ in
$B$, i.e., $\psi - \varphi$ is harmonic in $B$.  Since in $B$ we have
$v\,=\,u e^{- i\varphi}\,=\,e^{i(\psi - \varphi)}$, we obtain that $v
\in C^\infty (B)$, so that the claim follows.
\medskip
Using Step 1 and the equality $v = u e^{-i\varphi}$, we obtain that $v
\in [u]_{s,p}$.
\medskip
{\bf Step 3:} for each $u \in \W\,(\om ; S^1)$, there is some $w \in
C^\infty\,(\bar \om; S^1)$ such that $w \in [u]_{s,p}$
\medskip
In view of Step 2, it suffices to consider the case where $u \in
\W \os \cap C^\infty \os$.  We use the same homotopy as in Step 1,
Case 3, in the proof of Theorem 1: $t \mapsto u \circ \Phi_t$, where
$\Phi_t$ is a continuous family of diffeomorphisms $\Phi_t : \bar \om
\to \overline{\om_{t \delta}}$ such that $\Phi_0 = id$.  Clearly,
$v\,=\, u \circ \Phi_1 \in C^\infty\,(\bar \om ; S^1)$.
\medskip
The conclusions of Theorems 2 and 3 when $s \ge 1,\,1 < p < \infty,
\, N \ge 3,\,
2 \le sp < N $ follow from Proposition 2 and Steps 1 and 3.
\medskip
We now complete the proof of Theorem 2 with 
\medskip
\noindent
{\bf Case 3:}  $0 < s < 1,\,1 <p < \infty,\, N \ge 3,\, 2 \le sp < N$
\medskip
In this case, all we have to prove is that, for each $u \in \W \os$,
there is some $v \in C^\infty\,(\bar \om ;S^1)$ such that $v \in
[u]_{s,p}$.  The ideas we use in the proof are essentially due to
Brezis - Li [7] (see \S 1.3, ``Filling'' a hole).
\medskip
We may assume that $u$ is defined in a neighborhood $\Cal O$ of $\bar
\om$; this is done by extending $u$ by reflections across the boundary
of $\om$- the extended map is still in $W^{s,p}$
 since $0 < s < 1$.  We next define a good
covering of $\om$: let $\var > 0$ be small enough; for $x \in \br^N$,
we set
$$
\Cal C^x_N\,=\, \bigcup \{x + \var l + (0,\var)^N; l \in \bz^N \text{ and } x +
\var l + (0,\var)^N \subset \Cal O\}.
$$
Define also $\Cal C^x_j, j = 1, ..., N - 1$, by backward induction :
$\Cal C^x_j$ is the union of faces of cubes in $\Cal C^x_{j + 1}$.
\medskip
By Fubini, for a.e. $x \in \br^N$, we have $u|_{\Cal C^x_j} \in \W,\,j
= 1,...,\,N-1$, in the following sense: since $1/p < s < 1$, we have
tr $u|_{\Cal C^x_{N - 1}} \in W^{s - 1/p,p}$ for all $x$.  However,
for a.e. $x$, we have the better property tr $u|_{\Cal C^x_{N-1}} =
u|_{\Cal C^x_{N-1}} \in \W$.  For any such $x$, we have \break\hfill\noindent
tr $\left(u|_{\Cal
C^x_{N-1}}\right)\big|_{\Cal C^x_{N-2}} \in W^{s - 1/p,p}$, but once more for
a.e. such $x$ we have the better property tr
$\left(u|_{\Cal C^x_{N-1}}\right)\big|_{\Cal C^x_{N-2}} = u|_{\Cal
C^x_{N-2}} \in \W$, and so on.  (See Appendix E for a detailed
discussion).
\medskip
We fix any $x$ having the above property and we drop from now on the
superscript $x$.
\medskip
{\bf Step 1:}  we connect $u$ to some smoother map $u_1$ 

Let $k =
[sp]$, so that $2 \le k \le N-1$.  Since $u|_{\Cal C_k} \in \W$ and
$sp \ge k$, there is a neighborhood $\omega$ of $\Cal C_k$ in $\Cal
C_{k +1}$ and an extension $\tilde u \in W^{s + 1/p,p}\,(\omega ;
S^1)$ of  $u|_{\Cal C_k}$.  This extension is first obtained in each
cube $C \subset \Cal C_{k+1}$ starting from $u|_{\partial C}$ (see
Brezis - Nirenberg [12], Appendix 3, for the existence of such an
extension).  We next glue together all these extensions to obtain
$\tilde u;\, \tilde u$ belongs to $W^{s+1/p,p}$ since $1/p < s + 1/p < 1
+ 1/p$.  Moreover, the explicit construction in [12] yields some
$\tilde u \in C^\infty\,(\omega \backslash \Cal C_k)$.  We next extend
$\tilde u$ to $\Cal C_{k+1}$ in the following way: for each $C \subset
\Cal C_{k+1}$, let $\Sigma_C$ be a convex smooth hypersurface in $C
\cap \omega$.  Since $\Sigma_C$ is $k$-dimensional and $k \ge 2, \tilde
u|_{\Sigma_C}$ may be extended smoothly in the interior of $\Sigma_C$ as an
$S^1$-valued map (here, we use the fact that $\pi_k\,(S^1)\,=\,0$).  Let
$\tilde u_C$ be such an extension.  Then the map
$$
v = \cases \tilde u,\quad &\text{ outside the } \Sigma_C \text{'s}\\
\tilde u_C, \quad &\text{ inside } \Sigma_C  \endcases
$$
belongs to $W^{s + 1/p,p}\,(\Cal C_{k+1})$.  To summarize, we have found some
$v \in W^{s+1/p,p}\,(\Cal C_{k+1}; S^1)$ such that $v|_{\Cal C_k} =
u|_{\Cal C_k}$.
\medskip
Pick any $s \, < \, s_1\, <\,$min$\, \{ s + 1/p, 1\}$ and let $p_1$ be such that
$s_1 p_1 = sp + 1$ (note that $1 < p_1 <\infty$).  By Gagliardo -
Nirenberg (see, e.g., Runst [22], Lemma 1, p.329 or Brezis - Mironescu
[10], Corollary 3), we have $W^{s + 1/p,p} \cap L^\infty \subset
W^{s_1,p_1}$.  Thus $v \in W^{s_1,p_1}\,(\Cal C_{k+1})$.
\medskip
We complete the construction of the smoother map $u_1$ in the
following way: if $k = N - 1$, then $v$ is defined in $\Cal C_N$ and
we set $u_1 = v$; if $k < N - 1$, we extend $v$ to $\Cal C_N$ with the
help of 
\medskip
\proclaim{Lemma 5}  Let $0 < s_1 < \infty,\,1 < p_1 < \infty,\,1
<s_1p_1 < N,\, [s_1 p_1] \le j < N$.  Then any $v \in
W^{s_1,p_1}\,(\Cal C_j ;S^1)$ has an extension $u_1 \in
W^{s_1,p_1}\,(\Cal C_N ; S^1)$ such that $u_1|_{\Cal C_l} \in
W^{s_1,p_1}$ for $l = j,...,N -1$.  \endproclaim
\medskip 
When $s_1 = 1$, Lemma 5 is due to Brezis - Li [7], Section 1.3,
``Filling'' a hole; for the general case, see Lemma D.3 in Appendix
D.
\medskip
We summarize what we have done so far: if $k = [sp]$, then there are
some $s_1,p_1$ such that $s < s_1 < 1,\,1 < p_1 < \infty,\,s_1p_1 = sp
+ 1$ and a map $u_1 \in W^{s_1,p_1}\,(\Cal C_N ; S^1)$ such that
$u_1|_{\Cal C_j} \in W^{s_1,p_1},\, j = k,...,N - 1$ and $u_1|_{\Cal
C_k} = u|_{\Cal C_k}$.  By Gagliardo - Nirenberg and the Sobolev embeddings, we have
in particular $u_1|_{\Cal C_j} \in \W,\, j = k,..., N - 1$.  Finally,
$u$ and $u_1$ are $\W$- homotopic by
\medskip
\proclaim{Lemma 6}  Let $0 < s < 1,\,1 < p < \infty,\, 1 < sp < N, [sp]
\le j < N$.  If $u|_{\Cal C_l} \in \W ,\,u_1|_{\Cal C_l} \in \W, l =
j,...,N $, and $u|_{\Cal C_j} = u_1|_{\Cal C_j}$, then $u$ and
$u_1$ are $\W$-homotopic.  \endproclaim
\medskip
The case $s = 1$ is due to Brezis - Li [7]; the proof of Lemma 6 in
the general case is presented in the Appendix D- see Lemma D.4.
\medskip
{\bf Step 2:} 	induction on $[sp]$
\medskip
If $k = [sp] = N - 1$, we have connected in the previous step $u$ to
$u_1 \in W^{s_1,p_1}\,(\Cal C_N ; S^1)$, where $s < s_1 < 1,\,1 < p_1
< \infty$ and $s_1p_1 = sp + 1 \ge N$.  Using Case 1 (i.e., $ sp \ge
N$) from this section, $u_1$ may be connected in $W^{s_1,p_1}$ (and thus
in $\W$, by Gagliardo - Nirenberg and the Sobolev embeddings) to some
$v \in C^\infty\,(\bar \om ; S^1)$.  This case is complete.
\medskip
If $k = [sp] = N - 2$, then $[s_1 p_1] = N - 1$.  By the previous
case, $u_1$ can be connected in $W^{s_1, p_1}$ (and thus in $\W$) to
some $v \in C^\infty\,(\bar \om ; S^1)$.  Clearly, the general case
follows by induction.
\medskip
The proof of Theorems 2 and 3 is complete.
\bigskip
We end this section with two simple consequences of the above proofs; these
results supplement the description of the homotopy classes.
\medskip
\proclaim{Corollary 4}  Let $0 < s < \infty,\,1 < p < \infty,\, sp \ge
2,\,N \ge 2$.
For $u, v \in \W \os$, we have $[u]_{s,p} = [v]_{s,p} \Leftrightarrow$
{\rm deg } $(u|_{\ga_\var}) =$ {\rm deg } $(v|_{\ga_\var})$ for every
$\ga$. \endproclaim
\medskip
\proclaim{Corollary 5}  Let $0 < s_1, s_2 < \infty,\,1 < p_1, p_2 <
\infty, \, s_1p_1 \ge 2,\, s_2 p_2 \ge 2, \, N \ge 2$.  For $u, v \in
W^{s_1,p_1} \os \cap W^{s_2,p_2}\os$, we have $[u]_{s_1,p_1} =
[v]_{s_1,p_1} \Leftrightarrow [u]_{s_2,p_2} = [v]_{s_2,p_2}$. \endproclaim
\medskip
Clearly, Corollary 5 follows from Corollary 4.  As for Corollary 4,
let $u_1,v_1 \in C^\infty\,(\bar \om ; S^1)$ be such that $[u_1]_{s,p}
= [u]_{s,p}$ and $[v_1]_{s,p} = [v]_{s,p}$.  Then, by Theorem 2 b),
$$
[u]_{s,p} = [v]_{s,p} \Leftrightarrow [u_1]_{s,p} = [v_1]_{s,p}
\Leftrightarrow [u_1]_{C^0}=[v_1]_{C^0}
\Leftrightarrow \text{deg }(u_1|_\ga)\,=\,\text{deg }(v_1|_\ga),\quad
\forall \ga .  \tag 2
$$
\medskip
\noindent
Moreover, we have
$$
\text {deg }(u_1|_\ga) =\, \text {deg }(v_1|_\ga) \Leftrightarrow 
\text {deg }(u_1|_{\ga_\var}) 
= \,\text {deg }(v_1|_{\ga_\var}) \Leftrightarrow \text {deg }(u|_{\ga_\var}) = 
\,\text {deg }
(v|_{\ga_\var}),\,\forall \ga,  \tag 3
$$
by standard properties of the degree.
\medskip
We obtain Corollary 4 by combining (2) and (3).

\bigskip
\bigskip
{\bf IV.  Proof of Theorem 4}
\medskip
According to the discussion in the Introduction, we only have to prove
part d).  Let $s \ge 1,\,1 < p < \infty,\,N \ge 3,\,2 \le sp < N$.  Let
$u \in \W \os$.  By Theorem 2 a), there is some $v \in C^\infty\,(\bar
\om ; S^1)$ such that $v \in [u]_{s,p}$.  By Theorem 3 b), there is some
$\varphi \in \W \,(\om ;\br) \cap W^{1,sp}\,(\om ; \br)$ such that $v
= u e^{i\varphi}$.  Let $(\varphi_n) \subset C^\infty\,(\bar \om ; \br)$
be such that  $\varphi_n \to \varphi$ in $\W \cap W^{1,sp}$.  By the
Composition Theorem, the sequence of smooth maps $(v e^{-
i\varphi_n})$ converges to $u$ in $\W \os$.
\medskip
The proof of Theorem 4 is complete.
\bigskip
\bigskip
{\bf V.  Proof of Theorem 5}
\medskip
We start this section with a discussion on the stability of the degree:
recall that if $sp \ge 2$, then deg $(u|_{\ga_{\var}})$ is well-defined
and stable under $\W$ convergence.  However, while the condition $sp
\ge 2$ is optimal for the existence of the degree (see Brezis - Li -
Mironescu - Nirenberg [8], Remark 1), the stability of the degree of
$\W$ maps holds under (the weaker assumption of) $W^{s_1,p_1}$
convergence, where $s_1 p_1 \ge 1$.  This property and Corollary 4
suggest the following generalization of Theorem 5
\medskip
\proclaim{Theorem 7}  Let $0 < s < \infty,\,1 < p < \infty,\, 0 < s_1 <
s,\,1 < p_1 < \infty,\, 1 \le s_1 p_1 \le sp$.  Then for each $u \in
\W \os$ there is some $\delta > 0$ such that  \endproclaim
$$
\{ v \in \W\os ; ||v - u||_{W^{s_1,p_1}} < \delta\}\, \subset
[u]_{s,p}.
$$
\medskip
Note that $\W \os \subset  W^{s_1,p_1}\,\os$, by Gagliardo - Nirenberg and
the Sobolev embeddings, so that Theorem 5 follows from Theorem 7 when
$sp \ge 2$ (when $sp < 2$, there is nothing to prove, by Theorem 1).
\medskip
{\bf Proof of Theorem 7}
\medskip
{\bf Step 1:}  reduction to special values of $s,s_1, p,p_1$
\medskip
We claim that it suffices to prove Theorem 7 when 
$$
0 < s_1 < s < 1 - (N - 1)/p,\,1 < p < \infty,\,1 < p_1 < \infty,
\,s p = 2,\,s_1p_1 = 1, N \ge 2 . \tag 4
$$
Indeed, assume Theorem 7 proved for all the values of $s, s_1, p, p_1$
satisfying (4).  Let $0 < s_0 < \infty,\,1 < p_0 < \infty, N \ge 2$ be
such that $s_0 p_0 \ge 2$ (when $N = 1$ or $s_0 p_0 < 2$, there is
nothing to prove).  Let $u \in W^{s_0,p_0}$ and let $s, s_1, p, p_1$
satisfy (4) and the additional condition $s < s_0$.  By Gagliardo -
Nirenberg and the Sobolev embeddings, there is some $\delta_0 > 0$
such that 
$$
\aligned
M = & \{v \in W^{s_0,p_0} \os ; ||v - u||_{W^{s_0,p_0}} < \delta_0\}
\subset\\
&\{v \in \W \os; ||v - u||_{W^{s_1,p_1}} < \delta\}.  \endaligned \tag 5
$$
By the special case of Theorem 7, we have $v \in M \Rightarrow v \in
[u]_{s,p}$. By Corollary 5, we obtain $M \subset [u]_{s_0,p_0}$, i.e., $
[u]_{s_0,p_0}$ is open.
\medskip
In conclusion, it suffices to prove Theorem 7 under assumption (4).
Moreover, by Proposition 1 we may assume $u = 1$.
\medskip
{\bf Step 2:}  construction of a good covering 
\medskip
We fix a small neighborhood $\Cal O$ of $\bar \om$.  By reflections
across the boundary of $\om$, we may associate to each $u \in \W \os$
an extension $\tilde u \in \W (\Cal O ; S^1)$ satisfying
$$
||\tilde u - \tilde v||_{\W (\Cal O)} \le C_1\,\, ||u - v||_{\W (\om)}
\tag 6
$$
and
$$
||\tilde u - \tilde v||_{W^{s_1,p_1} (\Cal O)} \le C_1 ||u -
v||_{W^{s_1,p_1} (\om)}.\tag 7
$$
\medskip
\noindent
In this section, $C_1,C_2,...$ denote constants independent of $u,v,...$.
\medskip
We fix some small $\var > 0$.  By Lemma E.2 in Appendix E, for each $v
\in \W \os$ there is some $x \in \br^N$ (depending possibly on $v$)
such that the covering $\Cal C^x_N$ has the properties
$$
v|_{\Cal C^x_j} \in \W, \,\,j = 1,..., N-1  \tag 8
$$
and 
$$
||v|_{\Cal C^x_1} - 1||_{W^{s_1,p_1}\,(\Cal C^x_1)} \le C_2 ||v -
1||_{W^{s_1,p_1}\,(\Cal O)} \le C_2 C_1||v - 1||_{W^{s_1,p_1}(\om)}
\tag 9
$$
(the last inequality follows from (7)).
\medskip
While $x$ may depend on $v$, the covering $\Cal C^x_N$ has two
features independent of $v$:
$$
\text{ the number of squares in } \Cal C^x_2 \text{ has a uniform upper bound } K ;  \tag 10
$$
\medskip
$$
\align
&\text{if } C^1, C^2 \text{ are two squares in } \Cal C^x_2,
\text{ there is a path of squares in } \Cal C^x_2\\  &\text{each one having an edge in common with its neighbours, connecting}\tag 11\\ 
& C^1 \text{ to } C^2.      
\endalign
$$
\medskip
{\bf Step 3:} choice of $\delta$
\medskip
We rely on
\medskip
\proclaim{Lemma 7}  Let $C = (0,\var)^2$ and $0 < s_1 < 1,\,1 < p_1 <
\infty,\, s_1 p_1 = 1.$  Then for each $\delta_1 > 0$ there is some
$\delta_2 > 0$ such that every map $v \in W^{s_1,p_1}\,(\partial C ;
S^1)$ satisfying 
$$
||v - 1||_{W^{s_1,p_1}(\partial C)}\, < \delta_2  \tag 12
$$
has a lifting $\varphi \in W^{s_1,p_1}\,(\partial C; \br)$ such that 
$$
||\varphi||_{W^{s_1,p_1} (\partial C)}\, < \delta_1.  \tag 13 
$$
\endproclaim
\medskip
Clearly, in Lemma 7, $C$ may be replaced by the unit disc.  For the
unit disc, the proof of Lemma 7 is given in Appendix C; see Lemma C.3.
\medskip
In particular, if (12) holds, then we have
$$
||\varphi||_{L^1\,(\partial C)} < C_3 \delta_1  \tag 14
$$
for some $C_3$ independent of the $\delta^\prime$s.
\medskip
We now take $\delta_1$ such that 
$$
\delta_1 < \pi \var/C_3.  \tag 15
$$
With $\delta_2$ provided by Lemma 7, we choose
$$
\delta =\, \text{min}\, \{\delta_2 /C_0, \delta_2 /C_1 C_2\} . \tag 16
$$
\medskip
{\bf Step 4:} construction of a global lifting for $v|_{\Cal C^x_1}$
\medskip
Let $v \in \W \os$ satisfy $||v - 1||_{W^{s_1,p_1}} < \delta$.  
Since $\delta \le \delta_2 / C_1 C_2$, (9) implies that the conclusion
of Lemma 7 holds for $v|_{\partial C}$ and every square $C$ in $\Cal
C^x_2$. Thus, for every  $C\in \Cal C_2^x$, $v|_{\partial C}$ has a lifting
$\varphi_C$ satisfying (14) and $\varphi_C \in W^{s_1,p_1}\,(\partial
C)$.
\medskip
We claim that $\varphi_C \in \W \,(\partial C)$.  The statement being
local, it suffices to prove that $\varphi_C \in \W\,(L)$, where $L$ is
the union of three edges in $\partial C$. Since $L$ is Lipschitz
homeomorphic with an interval, by Theorem 1 in [4] there is some $\psi
\in \W\,(L)$ such that $v = e^{i\psi}$ in $L$ (here we use $0 < s <
1$ and $sp = 2 \ge 1$).  In $L$, we have $\psi - \varphi_C \in (\W +
W^{s_1,p_1})\,(L; 2 \pi \bz)$; thus $\psi - \varphi_C$ is constant
a.e. in $L$ (see [4], Remark B.3), so that the claim follows.
\medskip
Since $sp > 1$ and $v|_{\Cal C^x_1} \in \W, \varphi_C \in \W$, we may
redefine $v|_{\Cal C^x_1}$ and $\varphi_C$ on null sets in order to
have continuous functions.  We claim that the function $\varphi(y) =
\varphi_C(y)$, if $y \in C$ is well-defined on $\Cal C^x_1$ (and thus
continuous and $\W$).  By (11), it suffices to prove that, if $C^1,C^2$
are squares in $\Cal C^x_2$ having the edge $\Cal E$ in common, then
$\varphi_{C^1} = \varphi_{C^2}$ on $\Cal E$.  Clearly, on $\Cal E$ we
have $\varphi_{C^2} = \varphi_{C^1} + 2l \pi$ for some $l \in \bz$. Thus
$$
||\varphi_{C^1} + 2l \pi||_{L^1\,(\Cal E)} =
||\varphi_{C^2}||_{L^1\,(\Cal E)} < C_3 \delta_1,  
$$
by (14).  It follows that
$$
2|l|\pi \varepsilon = ||2 l \pi||_{L^1\,(\Cal E)} \le
||\varphi_{C^1}||_{L^1\,(\Cal E)} + C_3 \delta_1 < 2 C_3 \delta_1,  \tag 17 
$$
which implies $l = 0$ by (15) and (16).
\medskip
In conclusion, $v|_{
\Cal C^x_1}$ has a global lifting $\varphi \in \W\,(\Cal C^x_1; \br)$.
\medskip
{\bf Step 5:} construction of a good extension $w$ of $v|_{\Cal
C^x_1}$ 
\medskip
Let $\varphi_2 \in W^{s + 1/p,p}\,(\Cal C^x_2; \br)$ be an extension of
$\varphi$,  $\varphi_3 \in W^{s + 2/p,p} \,(\Cal C^x_3; \br)$ an
extension of $\varphi_2$, and so on; let $\varphi_N \in W^{s +
(N-1)/p,p}\,(\Cal C^x_N; \br)$ be the final extension.  Note that
these extensions exist since $s < 1 + (N - 1)/p$, so that trace
theory applies.  We set $w = e^{i\varphi_N} \in
W^{s+(N-1)/p,p}\,(\Cal C^x_N; S^1)$.  Since $(s + (N-1)/p) \cdot p = N +
1 > N$, we obtain by Theorem 3 that $w \in [1]_{s + (N-1)/p,p}$.
By Corollary 5, we also have $w \in [1]_{s,p}$.
\medskip
We complete the proof of Theorem 7 by proving
\medskip
{\bf Step 6:} $w \in [v]_{s,p}$
\medskip
We rely on the following variant of Lemma 6
\medskip
\proclaim{Lemma 8}  Let $0 < s < 1,\,1 < p < \infty,\,1 < sp < N,\, [sp]
\le j < N$.
Let $v, w \in \W\,(\Cal C_N; S^1)$ be such that $v|_{\Cal C_l}
\in \W, w|_{\Cal C_l} \in \W,\, l= j,..., N - 1$.  Assume that
$v|_{\Cal C_j}$ and $w|_{\Cal C_j}$ are $\W$-homotopic.  Then $v$ and
$w$ are $\W$-homotopic. \endproclaim  
\medskip
The proof of Lemma 8 is given Appendix D; see Lemma D.5.
\medskip
When $N\ge 3$, we are going to apply Lemma 8 with $j=2$. In order to prove that 
$v|_{\Cal C_2}$ and $w|_{\Cal C_2}$ are $\W$-homotopic,
it suffices to find, for each $C\in \Cal C_2$, a homotopy
$U_C$ from $v|_C$ to $w|_C$ preserving the boundary 
condition on $\partial C$; we next glue together these homotopies (this works since $0<s<1$). We construct $U_C$
using the
lifting: since $sp = 2 =$ dim $C$ and $C$ is simply
connected, by Theorem 2 in [4] there is some $\psi \in \W\,(C ; \br)$
such that $v = e^{i\psi}$ in $C$.  By taking traces, we find that
$v|_{\partial C} = e^{i \text{tr } \psi} = e^{i\varphi_C}$; thus 
tr $\psi - \varphi_C$ $\in (W^{s-1/p,p} + \W )(\partial C;2\pi\bz )$.
Therefore, tr $\psi - \varphi_C$ is constant a.e., by Remark B.3 in
[4].  We may assume that tr $\psi = \varphi_C =\,$ tr $\varphi_2$.  Then $
t \longmapsto e^{i((1-t) \psi + t\varphi_2)}$ is the desired homotopy
$U_C$.

When $N=2$, the above argument proves directly (i.e., without the help of Lemma 8)
that $w \in [v]_{s,p}$.
\medskip
The proof of Theorem 7 is complete.
\bigskip
{\bf Appendix A.  An extension lemma}
\medskip
In this appendix, we investigate, in a special case, the question
whether a map in $W^{\sigma,p}\,(\partial \omega ;S^1)$ admits an
extension in $W^{\sigma + 1/p,p}\,(\omega ;S^1)$.
\medskip
\proclaim{Lemma A.1}  Let $0 < \sigma < 1,\,1 < p < \infty,\,\sigma p
< 1,\,N \ge 2$.  Let $\omega$ be a smooth bounded domain in $\br^N$.
Then every $v \in 
W^{\sigma,p}\,(\partial \omega ;S^1)$
has an extension $w\in W^{\sigma + 1/p,p}\,(\omega ;S^1)$.  \endproclaim
\medskip
{\bf Proof.}  We distinguish two cases: $\sigma \le 1 - 1/p$ and
$\sigma > 1 - 1/p$.
\medskip
\noindent
{\bf Case $\sigma \le 1 - 1/p$:}  since $\sigma p < 1,\,v$ may be
lifted in $W^{\sigma,p}$ (see Bourgain - Brezis - Mironescu [4]),
i.e. there is some $\psi \in W^{\sigma,p}\,(\partial \omega ; \br)$
such that $v = e^{i\psi}$.  Let $\varphi \in W^{\sigma + 1/p,p}\,(\omega
; \br)$ be an extension of $\psi$.  Then $w = e^{i\varphi} \in
W^{\sigma + 1/p,p}\,(\omega ; S^1)$ (since $\sigma + 1/p \le 1$ and
$x \mapsto e^{ix}$ is Lipchitz).  Clearly, $w$ has all the required
properties.
\medskip
\noindent
{\bf Case $\sigma > 1 - 1/p$:}  the argument is similar, but somewhat
more involved.  The proof in [4] actually yields a lifting which is
better than $W^{\sigma,p}$; more specifically, this lifting $\psi$
belongs to $W^{t\sigma, p/t}$ for $0 < t \le 1$, see Remark 2, p.41,
in the above reference.  On the other hand, since $\sigma > 1 - 1/p$,
we have $t = p/(\sigma p + 1) < 1$.  For this choice of $t$, we obtain
that $v$ has a lifting $\psi \in W^{\sigma,p} \cap W^{1 - 1/(\sigma p +
1), \sigma p + 1}$.  This $\psi$ has an extension $\varphi \in
W^{\sigma + 1/p,p} \cap W^{1, \sigma p + 1}$.  By the Composition
Theorem stated in the Introduction, the map $w = e^{i\varphi}$ belongs
to $W^{\sigma + 1/p,p}\,(\omega ; S^1)$.  Clearly, we have tr $w = v$.
\medskip
\noindent
{\bf Remark A.1.}  The special case $p < 2$ and $\sigma = 1 - 1/p$
was originally treated by Hardt - Kinderlehrer - Lin [16] via a
totally different method.  Their argument extends to the case $p < 2$
and $\sigma p < 1$, but does not seem to apply when $p \ge 2$. 
\bigskip
{\bf Appendix B.  Good restrictions}
\medskip
In this appendix, we describe a natural substitute for the trace
theory when $s = 1/p$; it is known that the standard trace theory is
not defined in this limiting case.
\medskip
For simplicity, we consider mainly the case of a flat boundary.
However, we state Lemma B.5 (used in the proof of Theorem 1) for a
general domain.  We start by introducing some
\medskip
\noindent
{\bf Notations:}  let $Q =\,(0,1)^{N - 1},\, \om_+ = Q \times \,(0,1),\,\om_-
= Q \times \, (-1,0),\, \om = \om_+\,\cup \om_-\,= Q \times\,(-1,1)$.  If $v$ is a
function defined on $Q$, we set $\tilde v\,(x^\prime,t)\,=\, v(x)$ for
$(x^\prime,t) \in \om$.
\medskip
\proclaim{Lemma B.1}  Let $0 < s < 1,\,1 < p < \infty$.  Then for $u
\in \W\,(\om_+)$ and for any function $v$ defined on $Q$, the
following assertions are equivalent: 

\medskip
\noindent
a) $v \in \W\,(Q)$ and 
$$
I = \underset{\om_+}\to \int \,\,\frac{|u(x) - \tilde
v(x)|^p}{x^{sp}_N}\,dx < \infty ;  \tag B.1
$$
\medskip
\noindent
b) the map $w_1\,=\,\cases u, \quad &\text{ in } \om_+\\ 
\tilde v, \quad &\text{ in } \om_- \endcases$  belongs to $\W\,(\om)$; 
\medskip
\noindent
c) the map $w_2\,=\,\cases u -\tilde v, \quad &\text{ in } \om_+\\
0, \quad &\text{ in } \om_-  \endcases$  belongs to $\W\,(\om)$.
\endproclaim
\medskip
{\bf Proof.}  Recall that, if $U$ is a smooth or cube-like domain,
then an equivalent (semi-) norm on $\W\,(U)$ is given by 
$$
f \longmapsto \left ( \sum^N_{j = 1}\,\,\int_0^\infty\,\underset{\{x
\in U;\, x + t e_j \in U\}}\to \int\,\, \frac{f(x+te_j) - f(x)|^p}{t^{sp
+ 1}}\,dx dt \right )^{1/p}  \tag B.2
$$
\noindent
(see, e.g., Triebel [25]). 
\medskip
Clearly, both b) and c) imply that $v \in \W\,(Q)$.  Conversely, for
$v \in \W\,(Q)$ we have to prove the equivalence of (B.1), b) and c).
We consider the norm given by (B.2).  Taking into account the fact that
$w_1, w_2$ belong to $\W$ in $\om_+$ and $\om_-$, we see that 
$$
w_1 \in \W\,(\om)\,\Leftrightarrow\, J = \int_{\om_+} \int_{-1}^0 \,
\frac{|u(x) - \tilde v(x)|^p}{(x_N - t)^{sp + 1}}\, dt dx < \infty
\tag B.3
$$
\noindent
and
$$
w_2 \in \W\,(\om)\,\Leftrightarrow\, J < \infty  .\tag B.4
$$
\medskip
\noindent
The lemma follows from the obvious inequality
$$
\frac{1 - 2^{-sp}}{sp}\,I\,\le\,J\,\le \frac{1}{sp}\,I.
$$
\medskip
We now assume in addition that $sp \ge 1$ and derive the following 
\medskip
\proclaim{Corollary B.1}  Let $0 < s < 1,\,1 < p < \infty$ be such
that $sp \ge 1$.  
Then, for every $u \in \W\,(\om_+)$ we have
\medskip
\noindent
a)  for each $0 \le t_0 < 1$, there is at most one function $v$
defined on $Q$ such that the maps
$$
w^{t_0}_1 = \cases u, \quad &\text{ in }Q\, \times\,(t_0,1)\\
\tilde v, \quad &\text{ in }Q\,\times\,(-1,t_0)  \endcases
$$
and 
$$
w^{t_0}_2 = \cases u - \tilde v, \quad &\text{ in } Q\,\times\,(t_0,1)\\
0, \quad &\text{ in } Q\,\times\,(-1,t_0)  \endcases
$$
belong to $\W\,(\om)$;
\medskip
\noindent
b) for a.e. $0 \le t_0 < 1$, the function $v = u\,(\cdot ,t_0)$ has the
property that $w^{t_0}_1, w^{t_0}_2 \in \W\,(\om)$.
\endproclaim
\medskip
\noindent
(As usual, the uniqueness of $v$ is understood a.e.)
\medskip
The above corollary suggests the following 
\medskip
\noindent
{\bf Definition:} let $0 < s < 1,\,1 < p < \infty,\,sp \ge 1,\,0 \le
t_0 < 1$.  Let $u \in \W\,(\om_+)$ and let $v$ be a function defined on
$Q$.  Then $v$ is the downward good restriction of $u$  to $ \{x_N =
t_0\}$ if $w^{t_0}_1, w^{t_0}_2 \in \W\,(\om)$; we then write $v =
\text{ Rest } u|^-_{x_N = t_0}$.
Similarly, for $0 < t_0 < 1$ we may define an upward good restriction
Rest $u|^+_{x_N = t_0} = v$ as the unique function
 $v$ defined on $Q$ satisfying  the two equivalent conditions
\medskip
\noindent
a) $W^{t_0}_1 = \cases \tilde v, \quad &\text{ in } Q \,\times\,(t_0,1)\\
u, \quad &\text{ in } Q \,\times\,(0,t_0) \endcases \in \W\,(\om_+)$
\medskip
\noindent
and
\medskip
\noindent
b) $W^{t_0}_2 = \cases 0, \quad &\text{ in } Q \,\times\,(t_0,1)\\
u - \tilde v, \quad &\text{ in } Q \,\times\,(0,t_0)  \endcases \in
\W\,(\om_+).$
\medskip
\noindent
If $v$ is both an upward and a downward good restriction, we call it a
good restriction and we write $v = \text{ Rest } u|_{x_N = t_0}$.
\medskip
\proclaim{Corollary B.2}  Let $0 < s < 1,\,1 < p < \infty,\,sp \ge 1$.
Let $u \in \W\,(\om_+)$.  Then, for a.e. $0 < t_0 < 1$, we have
{\rm Rest } $u|_{x_N = t_0} = u\,(\cdot,t_0)$.  \endproclaim
\medskip
\noindent
{\bf Remark B.1.}  If $sp > 1$, then functions $u \in \W\,(\om_+)$
have traces for {\bf all} $0 \le t_0 \le 1$.  However, these traces
need not be good restrictions.  Here is an example: For $N = 2$, one
may prove that the map $x \mapsto (x - 1/2 e_1)/ |x - 1/2
e_1|$ belongs to $\W\,(\om)$ if $0 < s < 1,$ \newline $1 < p < \infty, sp < 2$.
However, if $sp > 1$, its trace 
 
$$
\text{ tr } u|_{x_2 = 0}\,\,=\,\,\cases 1, \quad &\text{ if } x_1 > 1/2\\
-1, \quad &\text{ if } x_1 < 1/2 \endcases
$$
does not belong to $\W\,(0,1)$, so that it is not a good restriction.
\medskip
\noindent
{\bf Remark B.2.}  In the limiting case $s = 1/p$, functions in $\W$
do not have traces.  However, they do have good restrictions a.e.
\medskip
Here is yet another simple consequence of Lemma B.1
\medskip
\proclaim{Corollary B.3}  Let $0 < s < 1,\,1 < p < \infty,\,sp \ge 1$.
Let $ u_\pm\in \W\,(\om_\pm)$ be such that {\rm Rest }$u_+|^-_{x_N = 0}\,=\,$
{\rm Rest }$u_-|^+_{x_N = 0}$.  

\medskip
\noindent
Then the map  $w = \cases u_+, \quad &\text{ in } \om_+\\
u_-, \quad &\text{ in } \om_-  \endcases$  belongs to $\W$.
\endproclaim
\medskip
The following results explain the connections between good
restrictions and traces.
\medskip
\proclaim{Lemma B.2}  Let $0 < s < 1,\,1 < p < \infty,\, sp > 1$.  Let
$u \in \W\,(\om_+)$.  Assume that there exists $v =$ {\rm Rest }
$u|^-_{x_N = 0}$.   Then $v = $ {\rm tr } $u|_{x_N = 0}$.
\endproclaim
\medskip
{\bf Proof.}  Let $w = \cases u - \tilde v, \quad &\text{ in } \om_+\\
0, \quad &\text{ in } \om_-  \endcases$.  By Lemma B.1, we have $w \in
\W\,(\om)$.  By trace theory and continuity of the trace, we have $0=$ tr 
$w|_{x_N=0}$, so that tr $u|_{x_N=0}=v$.
\medskip
\proclaim{Lemma B.3}  Let $0 < s < 1,\,1 < p < \infty,\, sp \ge 1$.
Let
$u \in W^{s+1/p,p}\,(\om_+)$.
Then, considered as a $\W$ function, $u$ has a good
downward restriction to $\{x_N = 0\}$ which coincides with {\rm tr }$u|_{x_N
= 0}$.
\endproclaim
\medskip
{\bf Proof.}  Let $v =$ tr $u|_{x_N = 0}$.  Then $v \in \W\,(Q)$, by the
trace theory.  By Lemma B.1, it remains to prove that
$$
\int_{\om_+}\quad \frac{|u(x) - \tilde v(x)|^p}{x_N^{sp}}\, dx < \infty .
\tag B.5
$$
\medskip
\noindent
Assume first that $ s + 1/p = 1$.  Then (B.5) follows from the
well-known Hardy inequality
$$
\int_Q\,\int^1_0 \quad \frac{|u(x^\prime ,t) - u(x^\prime ,0)|^p}{t^p} dt
dx \le C \|Du\|^p_{L^p},\,\forall u \in W^{1,p}\,(\om_+).  \tag B.6
$$
\medskip
\noindent
Consider now the case where $s + 1/p \neq 1$.  Let $\sigma = s + 1/p$.
We are going to prove that
$$
\int_{\om_+}\quad \frac{|u(x) - \tilde v(x)|^p}{x^{sp}_N}\, dx \le
C\|u\|^p_{W^{\sigma,p}}  \tag B.7
$$
for some convenient equivalent (semi-) norm on $W^{\sigma,p}$.  It is
useful to consider the norm
$$
f \mapsto \left ( \sum^N_{j = 1} \int^\infty_0\,\underset{\{ x \in U;\,x +
te_j \in U,\,x + 2te_j \in U\}}\to\int  \frac{|f(x+2te_j) - 2f(x +te_j) +
f(x)|^p}{t^{\sigma p+1}} dxdt \right )^{1/p}   \tag B.8
$$
\noindent
(see, e.g., Triebel [24]).
\medskip
\noindent
For any $x^\prime \in Q$ such that $u_{x^\prime} = u(x^\prime,\cdot) \in 
W^{\sigma,p}\,(0,1)$, 
the map
$$
f_{x^\prime} (t) = \cases u(x^\prime ,t), \quad &\text{ if } t > 0\\
v(x^\prime ), \quad &\text{ if } t<0  \endcases
$$
\noindent
belongs to $W^{\sigma,p}\,(-1,1)$, by standard trace theory.  Moreover,
for any such $x^\prime$ we have
$$
\|f_{x^\prime} \|^p_{W^{\sigma,p}\,(-1,1)} \, \le \,
C\|u_{x^\prime}\|^p_{W^{\sigma,p}\,(0,1)},  \tag B.9
$$
\noindent
i.e.
$$
\aligned
&\int^\infty_0\,\,\underset{\{ h\in (-1,1);\, h+t\in (-1,1),\, h + 2t\in (-1,1)\}}\to\int\,
\frac{|f_{x^\prime} (h+2t) - 2f_{x^\prime} (h+t) + f_{x^\prime} (h)|^p}{t^{\sigma p+ 1}}
\,dh dt \le\\ 
&C \int^\infty_0 \, \underset{\{h\in (0,1);\, h + t\in (0,1),\, h + 2t\in
(0,1)\}}\to\int 
\frac{|u_{x^\prime}(h+2t) - 2u_{x^\prime}(h+t) + u_{x^\prime}(h)|^p}{t^{\sigma p+ 1}} dh
dt.   \endaligned  
$$
\noindent
In particular,
$$
I = \int^{1/2}_0\,\,\int^{-t}_{-2t} \, \frac{|f_{x^\prime}(h+2t) - 2f_{x^\prime}(h+t) +
f_{x^\prime} (h)|^p}{t^{\sigma p+ 1}} dh dt \le C\|u_{x^\prime}\|^p_{W\sigma,p}.
\tag B.10
$$
\noindent
Since
$$
I \ge C \int^{1/3}_0 \frac{|u(x^\prime ,t) - v(x^\prime )|^p}{t^{\sigma p}} dt = C
\int^{1/3}_0 \frac{|u(x^\prime ,t) - v(x^\prime )|^p}{t^{sp + 1}} dt,  \tag B.11
$$
\noindent
we find that
$$
\int^{1/3}_0 \frac{|u(x^\prime ,t) - v(x^\prime )|^p}{t^{sp + 1}} dt \le
C\|u_{x^\prime }\|^p_{W^{\sigma,p}}.  \tag B.12
$$
\noindent
On the other hand, we clearly have
$$
\int^1_{1/3} \frac{|u(x^\prime ,t) - v(x^\prime )|^p}{t^{sp + 1}} dt \le
C\|u_{x^\prime }\|^p_{L^p} + C|v(x^\prime )|^p.  \tag B.13
$$
\noindent
By combining (B.12), (B.13) and integrating with respect to $x^\prime$, we
obtain (B.7).  The proof of Lemma B.3 is complete.
\medskip
A simple consequence of Lemma B.3 is the following 
\proclaim{Lemma B.4}  Let $0 < s < 1,\,1 < p < \infty,\, sp \ge 1$ and
$\rho > s$.  Let $u_1 \in \W\,(\om_+)$ and  $u_2 \in W^{\rho,p}\,(\om_-)$.  
Assume that
$u_1$ has a good downward restriction $v =$ {\rm Rest } $u_1|^-_{x_N =
0}$ and that $v =$ {\rm tr } $u_2|_{x_N = 0}$.  Then the map 
$$
w = \cases u_1,\quad \text{ in } \om_+\\
u_2, \quad \text{ in } \om_- \endcases
$$
\noindent
belongs to $\W(\om)$.
\endproclaim
\medskip
{\bf Proof.}  Let $u_3 \in W^{s+ 1/p,p}\,(\om_-)$ be an extension of
$v$.
Then $w = w_1 + w_2$, where
$$
w_1 = \cases u_1,\quad \text{ in } \om_+\\
u_3, \quad \text{ in } \om_- \endcases
$$
and
$$
w_2 = \cases 0, \quad &\text{ in } \om_+\\
u_2 - u_3, \quad &\text{ in } \om_-  \endcases .
$$
\medskip
By Lemma B.3 and the assumption $v =$ Rest $u_1|^-_{x_N = 0}$,  we have Rest
$u_1|_{x_N = 0}^-\, =\, \text{ Rest } u_3|_{x_N = 0}^+$.  By Corollary B.3,
we find that $w_1 \in \W\,(\om)$.  It remains to prove that $w_2 \in
\W\,(\om)$.  Let $\sigma =$ min $\{\rho, s + 1/p,\,1\}$.  Then  $w_2 \in
 W^{\sigma,p}\,(\om)$, by standard trace theory.  Thus $w_2 \in
\W\,(\om)$.
\medskip
We conclude this section by stating the following precised form of
Corollary B.1, b) in the case of a general boundary.  We use the same
notations as in the proof of Theorem 1, Case 4.
\medskip
\proclaim{Lemma B.5}  Let $u \in W^{1/p,p}\,(\om)$.  Then 
\medskip
\noindent
a) for a.e. $ 0 < \delta < \var$ we have 
$$
u|_{\Sigma_\delta} \in W^{1/p,p}\,(\Sigma_\delta) \text{ and }
\int_{\Sigma_\delta} \int _\Omega \frac{|u(x) - u(y)|^p}{|x - y|^{N +
1}} dy ds_x < \infty; \tag B.14
$$
\medskip
\noindent
b) for any such $\delta$, $u$ has a good restriction to $\Sigma_\delta$
which coincides (a.e. on $\Sigma_\delta$) with $u|_{\Sigma_\delta}$.
\endproclaim
\bigskip
{\bf Appendix C.  Global lifting}
\medskip
In this appendix, we investigate the existence of a global lifting in
some domains with non-trival topology.
\proclaim{Lemma C.1}  Let $0 < s < \infty,\,1 < p < \infty,\,sp \ge N,
N \ge 2$.  Let $u \in \W\,(S^1 \times B_1; S^1)$ be such that {\rm deg}
$(u|_{S^1 \times B_1}) = 0$.  Then there is some $\varphi \in \W\,(S^1
\times B_1; S^1)$ such that $u = e^{i\varphi}$. \endproclaim
\medskip
\noindent
Here, $B_1$ is the unit ball in $\br^{N-1}$.
\medskip
{\bf Proof.}  Let $v : \br \times B_1 \to S^1,\, v(t,x) = u(e^{it},x)$.
Then $v \in W^{s,p}_{loc} \, (\br \times B_1 ; S^1)$, where ``loc''
refers only to the variable $t$.  By Theorem 2 in
Bourgain - Brezis - Mironescu [4], there is some $\psi \in
W^{s,p}_{loc}\,(\br \times B_1; \br)$ such that $v = e^{i\psi}$.  We
claim that $\psi$ is $2\pi$-periodic in the variable $t$.  Indeed, for
a.e. $x \in B_1$, we have $u \in W^{s,p}\,(S^1 \times \{x\}; S^1)$ and
deg $(u|_{S^1 \times \{x\}})= 0$.  In particular, for any such $x$ the
map $u|_{S^1 \times \{x\}}$ has a continuous lifting $\eta_x$.  On
the other hand, for a.e. $x \in B_1$ we have $\psi_x = \psi(\cdot ,x)
\in W^{s,p}_{loc}\,(\br \times \{x\}; \br)$.  Thus, with $\lambda_x
(t) = \eta_x (e^{it})$, we find that for a.e. $x \in B_1$ the function
$\psi_x - \lambda_x$ is continuous and $2\pi \bz$ -valued; therefore it
is a constant.  Since $\lambda_x$ is $2\pi$-periodic, so is $\psi_x$
for a.e. $x \in B_1$.  We obtain that $\psi$ is $2\pi$-periodic in the
variable $t$.  Thus the map $\varphi: S^1 \times B_1 \to \br,\, \varphi
(e^{it},x) = \psi(t,x)$ is well-defined and belongs to $W^{s,p}\,(S^1
\times B_1;\br)$.  Moreover, we clearly have $u = e^{i\varphi}$.
\medskip
In the same vein, we have
\medskip
\proclaim{Lemma C.2}  Let $s \ge 1,\,1 < p < \infty,\,N \ge 3,\,2 \le
sp < N$.  Let $u \in W^{s,p}\,(S^1 \times B_1;S^1)$ be such that {\rm deg }
 $(u|_{S^1 \times B_1}) = 0$.  Then there is some $\varphi \in
W^{s,p}\,(S^1 \times B_1;\br) \cap W^{1,sp}\,(S^1 \times B_1;\br)$
such that $u = e^{i\varphi}$. \endproclaim
\medskip
The proof is similar to that of Lemma C.1; one has to use Lemma 4 in
[4] instead of Theorem 2 in [4].
\medskip
\proclaim{Lemma C.3}  Let $1 < p < \infty$ and $\delta_1 > 0$.  Then
there is some $\delta_2 > 0$ such that every $v \in
W^{1/p,p}\,(S^1;S^1)$ satisfying $\|v-1\|_{W^{1/p,p}(S^1)} < \delta_2$
has a global lifting $\varphi \in W^{1/p,p}\,(S^1;\br )$ such that 
$\|\varphi\|_{W^{1/p,p}(S^1)}<\delta_1$.
\endproclaim
\medskip
{\bf Proof.}  Recall that if $I$ is an interval, then every $w \in
W^{1/p,p}\,(I;S^1)$ has a lifting $\psi \in W^{1/p,p}\,(I;\br)$ (see
Bourgain - Brezis - Mironescu [4], Theorem 1).  Moreover, this lifting may
be chosen to be (locally) continuous with respect to $w$, i.e. for
every $w_0 \in W^{1/p,p}(I;S^1)$ there is some $\delta_0 > 0$ such that in
the set 
$$
\{w; \|w - w_0\|_{W^{1/p,p}(I;S^1)} < \delta_0\}
$$
there is a lifting $w \mapsto \psi$ continuous for the $W^{1/p,p}$
norm.  (This assertion can be established using the same argument as
in Step 7 of the proof of Theorem 4 in Brezis - Nirenberg [12]; it can
also be derived from the explicit construction of $\psi$ in the proof
of Theorem 1 in [4]; see also Boutet de
Monvel-Berthier - Georgescu - Purice [6] when $p = 2$).
\medskip
Let $I = [-2\pi,2\pi]$.  To each $v \in W^{1/p,p}\,(S^1;S^1)$ we
associate the map $w \in W^{1/p,p}\,(I;S^1)$,  $w(t) = v(e^{it})$.  By
the above considerations, for every $\delta_3 > 0$ there is some
$\delta_4 > 0$ such that, if $\|v-1\|_{W^{1/p,p}(S^1)} < \delta_4$,
then $w$ has a lifting $\psi$ such that $\|\psi\|_{W^{1/p,p}(I)} <
\delta_3$.  We claim that $\psi$ is $2\pi$-periodic if $\delta_3$ is 
small enough. Indeed, the function $\xi (t)=\psi (t-2\pi )-\psi (t)$ 
belongs to $W^{1/p,p} ([0,2\pi ];2\pi\bz )$, so that $\xi$ is constant 
a.e.  (see [4], Theorem
B.1). Since $\|\xi\|_{L^1} \le \|\psi\|_{L^1} < C\delta_3$, we have 
$\xi =0$ (i.e. $\psi$ is $2\pi$-periodic) if $C\delta_3<2\pi$.
\medskip
Thus, for $\delta_3$ small enough, the map $\varphi (e^{it}) = \psi
(t)$ is well-defined, belongs to $W^{1/p,p}$ and satisfies
$\|\varphi\|_{W^{1/p,p}(S^1) }< \delta_1$ and $u = e^{i\varphi}$.



\bigskip
{\bf Appendix D.  Filling a hole - the fractional case}
\medskip
We adapt to fractional Sobolev spaces the technique of Brezis - Li [7],
Section 1.3.
\medskip
The first two results are preparations for the proofs of Lemmas 5,6
and 8 (see Lemmas D.3, D.4 and D.5 below).
\medskip
\proclaim{Lemma D.1}  Let $0 < s < 1,\,1 < p < \infty,\,1 < sp < N$.
Let $C = (-1,1)^N$ and $u \in W^{s,p}\,(\partial C)$.  Then $\tilde u
\in W^{s,p}\,(C)$; here,  $\tilde u(x) = u(x/|x|)$ and $|\,\,\,|$ is the
$L^\infty$ norm in $\br^N$.  Moreover, the map $u \mapsto \tilde u$ is
continuous from $W^{s,p}\,(\partial C)$ into $W^{s,p}\,(C)$.  \endproclaim
\medskip
{\bf Proof.}  Clearly, we have $\|\tilde u\|_{L^p (C)} \le C_0
\|u\|_{L^p (\partial C)}$.  Thus it suffices to prove, for the
Gagliardo semi-norms in $W^{s,p}$, the inequality
\medskip
$$
\|\tilde u\|_{W^{s,p}(C)}^p \le C_1 (\|u\|^p_{W^{s,p}(\partial C)} +
\|u\|_{L^p(\partial C)}^p).  \tag D.1
$$
\noindent
We have
$$
\underset{C}\to\int \underset{C}\to\int \frac{|\tilde u(x) - \tilde u(y)|^p}{|x-y|^{N+sp}}dxdy =
\int^1_0 \int^1_0 \underset{\partial C}\to\int\,\,
 \underset{\partial C}\to\int \frac{|u(x) -
u(y)|^p}{|\tau x - \sigma y|^{N+sp}} \tau^{N-1}\sigma^{N-1} ds_x
ds_y d\tau d\sigma. \tag D.2
$$
\noindent
We claim that
$$
I = \int^1_0 \int^1_0 \frac{\tau^{N-1} \sigma^{N-1}}{|\tau x - \sigma
y|^{N+sp}} d\tau d\sigma \le C_2/|x-y|^{N+sp}. \tag D.3
$$
\noindent
Indeed,
$$
\aligned
I = &\int^1_0 \int^{1/\tau}_0 \,\,\frac{\tau^{N-1} (\lambda
\tau)^{N-1}}{|\tau x - \lambda \tau y|^{N+sp}} d\lambda d\tau \quad = \quad \\
&\int^1_0 \int^{1/\tau}_0  \tau^{N - sp - 1}
\frac{\lambda^{N-1}}{|x-\lambda y|^{N+sp}} d\lambda d\tau \le I_1 + I_2,
\endaligned     \tag D.4   
$$
\noindent
where $I_1 = \int^1_0 \int^2_0$ and $I_2 =  \int^1_0 \int^\infty_2$.

\medskip
On the one hand, we have
$$
\aligned
I_1 &= \int^1_0 \int^2_0  \tau^{N - sp-1}
\frac{\lambda^{N-1}}{|x-\lambda y|^{N+sp}} d\lambda d\tau\\ 
&\le C_3 \int^1_0 \int^2_0 \tau^{N - sp- 1} \, \frac{\lambda^{N-1}}{|x-
y|^{N+sp}} d\lambda d\tau \le C_4/|x-y|^{N + sp}.   \endaligned     \tag D.5  
$$
\noindent
On the other hand, we have
$$
\aligned
I_2 &= \int^1_0 \int^\infty_2  \tau^{N - sp-1}
\frac{\lambda^{N-1}}{|x-\lambda y|^{N+sp}} d\lambda d\tau\\ 
&\le C_5 \int^1_0 \int^\infty_2 \tau^{N - sp- 1} \, \frac{\lambda^{N-1}}
{\lambda^{N+sp}} d\lambda d\tau 
 = C_5 \int^1_0 \int^\infty_2 \tau^{N - sp -1} \lambda^{-sp
-1}\,d\lambda d\tau \le C_6. \endaligned\tag D.6
$$
\noindent
We obtain (D.3) by combining (D.4), (D.5) and (D.6).  Finally, (D.1)
follows from (D.2) and (D.3).
\medskip
The proof of Lemma D.1 is complete.
\medskip
\proclaim{Lemma D.2}  Let $0 < s < 1,\,1 < p < \infty,\,1 < sp < N$.
Let $v, w \in W^{s,p}\,(C;S^1)$ be such that $v|_{\partial C} =
w|_{\partial C} \in W^{s,p}\,(\partial C)$.  Then, there is a homotopy
$U \in C^0 ([0,1];W^{s,p}\,(C;S^1))$ such that $U(0, \cdot) = v, \,
U(1,\cdot) = w$ and $U(t, \cdot)|_{\partial C} = v|_{\partial C},
\forall t \in [0,1]$.\endproclaim
\medskip
{\bf Proof.}  Let $u = v|_{\partial C}$.  It clearly suffices to prove
the lemma in the special case $w = \tilde u$.  In this case, let, for
$0 \le t < 1$,
$$
U(t,x) = \cases v(x/(1-t)),\quad &\text{ if } |x| \le 1 - t\\
\tilde u(x),\quad &\text{ if } 1 - t < |x| \le 1  \endcases ;
$$
\noindent
set $U(1, \cdot) = \tilde u$.  Clearly, $U \in C^0([0,1);
W^{s,p}\,(C; S^1))$.  It remains to prove that $U(t,\cdot) \to \tilde
u$ as $t \to 1$.  Let
$$
f(x) = \cases v(x),\quad &\text{ if } |x| \le 1\\
\tilde u (x),\quad &\text{ if } |x| > 1 \endcases 
$$
\noindent
and $g = f - \tilde u$.  Then $f,\, \tilde u \in
W^{s,p}_{loc}\,(\br^N)$, so that $g \in W^{s,p}_{loc}\,(\br^N)$.
Since $g = 0$ outside $C$, we actually have $g \in W^{s,p}\,(\br^N)$.
Thus 
$$
\aligned
&\|U(t, \cdot) - \tilde u\|^p_{W^{s,p}(C)} = \|g(\cdot
/(1-t))\|^p_{W^{s,p}(C)} \le\\
&\|g(\cdot/(1-t))\|^p_{W^{s,p}(\br^N)} = (1 - t)^{N - sp}
\|g\|^p_{W^{s,p}(\br^N)} \to 0
\endaligned
$$
\noindent
as $t \to 1$. The proof of Lemma D.2 is complete.
\medskip
We introduce a useful notation: let $u \in W^{s_1,p_1}\,(\Cal C_k)$,
where $0 < s_1 < 1,\,1 < p_1 < \infty,\,1 < s_1p_1 < N$.  We extend,
for each $C \in \Cal C_{k+1}, u|_{\partial C}$ to $C$ as in Lemma D.1.
Let $\tilde u$ be the map obtained by gluing these extensions.  We
next extend $\tilde u$ to $\Cal C_{k+2}$ in the same manner, and so
on, until we obtain a map defined in $\Cal C_N$; call it $H_k (u)$.
\medskip
\proclaim{Lemma D.3}  Let $0 < s_1 < 1,\,1 < p_1 < \infty,\,1 <
s_1p_1 < N,\,[s_1p_1] \le j < N$.  Then every $v \in
W^{s_1,p_1}\,(\Cal C_j;S^1)$ has an extension $u_1 \in W^{s_1,p_1}\,(\Cal
 C_N;S^1)$ such that $u_1|_{\Cal C_l} \in W^{s_1,p_1}$ for $l =
j,...,N-1$.  \endproclaim
\medskip
{\bf Proof.}  We take $u_1 = H_j(v)$.  We may use repeatedly Lemma
D.1, since for $l = j+1,...,N$ we have $1 < s_1p_1 < l$.
\medskip
\proclaim{Lemma D.4}  Let $0 < s < 1,\,1 < p < \infty,\,1 < sp < N,\,
[sp] \le j < N$.  If $u|_{\Cal C_l} \in W^{s,p}, u_1|_{\Cal C_l} \in
W^{s,p}, l=j,..., N-1$, and $u|_{\Cal C_j} = u_1|_{\Cal C_j}$, then
$u$ and $u_1$ are $W^{s,p}$-homotopic.  \endproclaim
\medskip
{\bf Proof.}  We argue by backward induction on $j$.  If $j = N-1$,
then for each $C \in \Cal C_N$ Lemma D.2 provides a $W^{s,p}$-homotopy
of $u|_C$ and $u_1|_C$ preserving the boundary condition.  By gluing
together these homotopies we find that $u$ and $u_1$ are
$W^{s,p}$-homotopic (here we use $1/p < s < 1$).  Suppose now that the
conclusion of the lemma holds for $j+1$; we prove it for $j$, assuming
that $j \ge [sp]$.  By assumption, $u$ and $H_{j+1}(u|_{\Cal C_{j+1}})$
are $W^{s,p}$-homotopic, and so are $u_1$ and $H_{j+1}(u_1|_{\Cal C_{j+1}})$. 
It suffices therefore to prove that $v =  H_{j+1}(u|_{\Cal C_{j+1}})$
and $v_1 =  H_{j+1}(u_1|_{\Cal C_{j+1}})$ are $W^{s,p}$-homotopic.  For
each $C \in \Cal C_{j+1}$, we have $v|_{\partial C} = v_1|_{\partial C}
= u|_{\partial C} = u_1|_{\partial C}$.  By Lemma D.2, $v|_C$ and
$v_1|_C$ are connected by a homotopy preserving the trace on $\partial
C$.  Gluing together these homotopies, we find that $v|_{\Cal
C_{j+1}}$ and $v_1|_{\Cal C_{j+1}}$ are $W^{s,p}$-homotopic.  If $U$
connects  $v|_{\Cal C_{j+1}}$ to $v_1|_{\Cal C_{j+1}}$,  then Lemma
D.1 used repeatedly implies that $t \mapsto H_{j+1} (U(t))$ connects in
$W^{s,p}\,(\Cal C_N;S^1)$ the map $H_{j+1}\,(v|_{\Cal C_{j+1}})$ to
$H_{j+1}\,(v_1|_{\Cal C_{j+1}})$, i.e., $v$ to $v_1$.
\medskip
The proof of Lemma D.4 is complete.
\proclaim{Lemma D.5}  Let $0 < s < 1,\,1 < p < \infty,\,1 < sp < N,\,
[sp] \le j < N$.  Let $v, w \in W^{s,p}\,(\Cal C_N;S^1)$ be such that
$v|_{\Cal C_l} \in W^{s,p}, w|_{\Cal C_l} \in W^{s,p},l = j,...,N-1$.
Assume that $v|_{\Cal C_j}$ and  $w|_{\Cal C_j}$ are
$W^{s,p}$-homotopic.  Then $v$ and $w$ are $W^{s,p}$-homotopic.
\endproclaim
\medskip
{\bf Proof.}  By Lemma D.4, $v$ and $H_j(v|_{\Cal C_j})$ (respectively
$w$ and $H^j(w|_{\Cal C_j})$) are $W^{s,p}$-homotopic.  If $U$ connects
$v|_{\Cal C_j}$ to $w|_{\Cal C_j}$ in $W^{s,p}$, then as in the
proof of Lemma D.4, we obtain that $t \mapsto H_j(U(t))$ connects 
$H_j(v|_{\Cal C_j})$ to $H_j(w|_{\Cal C_j})$ in $W^{s,p}$.  Thus $v$
and $w$ are $W^{s,p}$-homotopic.
\bigskip
\bigskip
{\bf Appendix E.  Slicing with norm control}
\medskip
In this section, we prove the existence of good coverings for
$W^{s,p}$ maps.  The arguments are rather standard.
\medskip
Without loss of generality, we may consider maps defined in $\br^N$.
Throughout this section, we assume $\var = 1$, i.e. we consider a
covering with cubes of size 1.  We start by introducing some useful
notations: for $x \in C^N = (0,1)^N$ and for $j = 1,...,N - 1$, let
$$C_j = \bigcup\,\bigg\{\sum^j_{k=1}\,t_k\, e_{i_k}\, + \,\sum^{N-j}_{l=1}\,
\lambda_l e_{j_l} ; \, t_k \in \br , \lambda_l \in \bz , \{e_{i_k}\} \,
\cup \{e_{j_l}\} = \{e_1,...e_N\} \bigg\} 
$$
\noindent
and $C_j (x) = x + C_j$.
(With the notations introduced in Section 3, we have $C_j (x) = \Cal
C_j^x$ when $\om = \br^N$).
\medskip
\noindent
For a fixed set $\Lambda \subset \{1,..,N\}$ such that $|\Lambda| =
j$, let also
$$
C_j^\Lambda = \bigg\{ \sum_{i \in \Lambda} t_i e_i + \sum_{j \notin \Lambda}
\lambda_j  e_j ;\, t_i \in \br, \lambda_j \in \bz\bigg\},
$$
\noindent
so that 
$$
C_j = \cup \{ C^\Lambda_j ; \Lambda \subset \{1,...,N\}, |\Lambda|=j\},
$$
\noindent
and with obvious notations
$$
C_j(x) = \cup \{ C^\Lambda_j(x) ; \Lambda \subset \{1,...,N\},
|\Lambda|=j\}.
$$
\medskip
Instead of considering a fixed (semi-) norm on $W^{s,p}, 0 < s < 1,\,1
< p < \infty$, it is convenient to consider a family of equivalent
norms
$$
|f|^p_j\, = \, \sum \Sb \Lambda \subset \{1,...,N\}\\ |\Lambda| = j
\endSb \int_{\br^N} \, \int_{\br^j}\, \frac{|f(x + \sum_{i \in
\Lambda} t_i e_i) - f(x)|^p}{|t|^{j + sp}} dt dx
$$
\noindent
(see, e.g., Triebel [24]).  An obvious computation yields, for the
usual Gagliardo

\noindent
(semi-) norm on $C^\Lambda_j (x)$,
\medskip
\proclaim{Lemma E.1}  Let $0 < s < 1,\,1 < p < \infty$ and $u \in
W^{s,p}$.  Then 
$$
\sum \Sb \Lambda \subset \{1,...,N\}\\ |\Lambda| = j  \endSb \,\,
\int_{C^N} \, \|u\|^p_{W^{s,p}\, (C^\Lambda_j(x))} dx \le |u|_j^p
$$
for some $C$ independent of $u$.
\endproclaim
\medskip
We next define the norm $\|u\|_{W^{s,p}\,(C_j(x))}$ by the formula
$$
\|u\|^p_{W^{s,p}\,(C_j(x))} \, = \, \sum_{C \in C_{j+1}(x)}\,
\|u\|^p_{W^{s,p}\,(\partial C)}.
$$
\medskip
\proclaim{Lemma E.2}  Let $0 < s < 1,\,1 < p < \infty$.  Then, for $u
\in W^{s,p}$, we have  

\medskip
\noindent
a) for a.e. $x \in C^N, u|_{C_j(x)} \in W^{s,p}_{loc}, j = 1,...,N -
1$;
\medskip
\noindent
b) there is a fat set (i.e., with positive measure) $A \subset C^N$
such that
$$
\|u\|^p_{W^{s,p}\,(C_j(x))} \, \, \le C \, |u|^p_j ,\quad \forall x \in A.
\tag E.2
$$
\endproclaim
\medskip
\noindent
{\bf Remark E.1.}  Here, $u|_{C_j(x)}$ are restrictions, not traces.
However, when $sp > 1$ we may replace restrictions by traces, by a
standard argument.  We obtain  
\medskip
\proclaim{Corollary E.1}  Let $0 < s < 1,\,1 < p < \infty,\, sp > 1$.
Let $u \in W^{s,p}$.  Then, for a.e. $x \in C^N$, {\rm tr }
$u|_{C_{N-1}(x)}\in W^{s,p}$. Moreover,  for a.e. $x \in C^N$, {\rm tr }
$u|_{C_{N-1}(x)}$
has a trace on $C_{N-2}(x)$ which belongs to
$W^{s,p}$, and so on. \endproclaim
\medskip
{\bf Proof of Lemma E.2.}  In order to avoid long computations, we
treat only the case $j=1, N=2$.  The general case does not bring any
additional difficulty.  Let $C \in C_1(x)$; denote its lower (resp. upper,
left, right) edge by
$C^l$ (resp. $C^u,\,C^L,\,C^R$).  By (E.1), we have
$u|_{C^l } \in W^{s,p}$ for a.e. $x \in C^2$ and, for $x$ in a fat
set, $\sum_{C \in C_1(x)} \|u\|^p_{W^{s,p}(C^l)} \le \text{
const. } |u|^p_1$.  Similar statements hold for the other edges.
\medskip
It remains to control the cross - integrals in the Gagliardo norm,
e.g. to prove  
$$
I = \int_{C^2} \sum_{C \in C_1(x)} \int_{C^l} \int_{C^L}
\frac{|u (y) - u (z)|^p}{|y-z|^{2+sp}} dy dz \le \text{ const. }
\|u\|^p_{W^{s,p}}  \tag E.3
$$
\noindent
(here, we take the usual Gagliardo norm in $W^{s,p}\,(\br^2)$).  We
have
$$ 
\aligned
I &= \int_{C^2} \,\, \sum_{m \in \bz^2} \int^1_0 \int^1_0
\frac{|u(x+m_1 e_1+m_2e_2+\tau e_1) - u (x +
m_1e_1+m_2e_2+\sigma e_2)|^p}{|\tau e_1 - \sigma e_2|^{2+sp}} d\sigma d\tau dx\\
&= \int_{\br^2} \,\int^1_0 \int^1_0 \frac{|u(y+\tau e_1) - u(y+\sigma
e_2)|^p}{|\tau e_1 - \sigma e_2|^{2+sp}} d\sigma d\tau dy\\
&= \int_{\br^2} \,\int^1_0 \int^1_0 \frac{|u(z) - u(z - \tau e_1 +
\sigma e_2)|^p}{|\tau e_1 - \sigma e_2|^{2 + sp}} d\sigma d\tau dz\\
&\le \int_{\br^2} \int_{\br^2} \frac{|u(z+h) -u(z)|^p}{|h|^{2+sp}} dh
dz = \|u\|^p_{W^{s,p}}.  \endaligned
$$
\medskip
The proof of Lemma E.2 is complete.
\bigskip
{\bf Acknowledgements}.  The first author (H.B.) warmly thanks Yanyan
Li for useful discussions.  He is partially supported by a European
Grant ERB FMRX CT980201, and is also a member of the Institut
Universitaire de France.  This work was initiated when the second
author (P.M.) was visiting Rutgers University; he thanks the
Mathematics Department for its invitation and hospitality.  It was
completed while both authors were visiting the Isaac Newton Institute
in Cambridge, which they also wish to thank.
\bigskip
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