%Composition in fractional Sobolev spaces
%H.Brezis,P.Mironescu
%November 14,2000
%C.Johnson
% renamed 169-04.tex


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\topmatter
\title{Composition in 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) \,\, DEPARTEMENT 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
\thanks {\bf Acknowledgment:}  The first author (H.B.) is partially
supported by a European Grant ERB FMRX CT98 0201. He is also a member
of the Institut Universitaire de France.  Part of this work was done
when the second author (P.M.) was visiting Rutgers University; he thanks the
Mathematics Department for its invitation and hospitality.  We thank
S. Klainerman for useful discussions.\endthanks
\endtopmatter
\document

\centerline{}
\bigskip
\noindent
{\bf 1.  Introduction}
\medskip
A classical result about composition in Sobolev spaces asserts that if
$u \in W^{k,p}(\Omega) \cap L^\infty(\Omega)$ and $\Phi \in C^k(\Bbb
R)$, then $\Phi \circ u \in W^{k,p}(\Omega)$.  Here $\Omega$ denotes a
smooth bounded domain in $\Bbb R^N$, $k \ge 1$ is an integer and $1
\le p < \infty$.  This result was first proved in [13] with the help
of the Gagliardo-Nirenberg inequality [14].  In particular if $u \in
W^{k,p}(\Omega)$ with $kp > N$ and $\Phi \in C^k(\Bbb R)$ then $\Phi
\circ u \in W^{k,p}$ since $W^{k,p} \subset L^\infty$ by the Sobolev
embedding theorem.  When $kp = N$ the situation is more delicate since
$W^{k,p}$ is not contained in $L^\infty$.  However the following
result still holds (see [2],[3])
\medskip
\noindent
{\bf Theorem 1}.  {\it Assume $u \in W^{k,p}(\Omega)$ where $k \ge 1$
is an integer, $1 \le p < \infty$, and} 
$$
kp = N.\tag 1
$$
\noindent
{\it Let $\Phi \in C^k(\Bbb R)$ with}
$$
D^j \Phi \in L^\infty (\Bbb R) \quad \forall j \le k.  \tag 2
$$ 
\noindent
{\it Then} 
$$
\Phi \circ u \in W^{k,p}(\Omega)
$$
\medskip
The proof is based on the following
\medskip
\noindent
{\bf Lemma 1}.  {\it Assume $u \in W^{k,p}(\Omega) \cap W^{1,kp}(\Omega)$
where $k \ge 1$ is an integer and $1 \le p < \infty$.  Assume $\Phi
\in C^k(\Bbb R)$ satisfies (2).  Then}
$$
\Phi \circ u \in W^{k,p}(\Omega).
$$
\medskip
\noindent
{\it Proof of Theorem 1}.  Since $u \in W^{k,p}$ we have 
$$
Du \in W^{k-1,p} \subset L^q
$$
\noindent
by the Sobolev embedding with
$$
\frac{1}{q} = \frac{1}{p} - \frac{k-1}{N}.
$$
\medskip
\noindent
Applying assumption (1) we find $q = N = kp$ and thus $u \in
W^{1,kp}$.  We deduce from Lemma 1 that $\Phi \circ u \in W^{k,p}$. 
\medskip
\noindent
{\it Proof of Lemma 1}.  Note that if $u \in W^{k,p} \cap L^\infty$
with $k \ge 1$ integer and $1 \le p < \infty$ then $u \in W^{1,kp}$ by
the Gagliardo - Nirenberg inequality [14].  Thus, Lemma 1 is a
generalization of the standard result about composition.  In fact, it
is proved exactly in the same way as in the standard case (when $u \in
W^{k,p} \cap L^\infty)$.  When $k = 2$ the conclusion is trivial.
\medskip
\noindent
Assume, for example that, $k = 3$, then 
$$
W^{3,p} \cap W^{1,3p} \subset W^{2,3p/2}
$$
by the Gagliardo - Nirenberg inequality.  Then 
$$
D^3(\Phi \circ u) = \Phi'(u) D^3 u + 3 \Phi '' (u)D^2u Du +
\Phi'''(u)(Du)^3,
$$
and thus $\Phi \circ u \in W^{3,p}$ since
$$
\align
\int|D^2 u|^p |Du|^p  &\le \big(\int|D^2u|^{3p/2})^{2/3}\big(\int|Du|^{3p}\big)^{1/3}\\
&\le C\|u\|^{p/2}_{W^{3,p}}\|u\|^{3p/2}_{W^{1,3p}}. \endalign
$$
\noindent
A simular argument holds for any $k \ge 4$.
\medskip
Starting in the mid-60's a number of authors considered composition in
various classes of ``Sobolev spaces'' $W^{s,p}$, where $s > 0$ is a
real number and $1 \le p < \infty$.  The most commonly used are the
Bessel potential spaces $L^{s,p}(\Bbb R^N) = \{f = G_s*g; g \in
L^p(\Bbb R^N)\}$ where $\widehat G_s = (1 + |\xi|^2)^{-s/2}$ and the Besov
spaces $B^{s,p}_p(\Bbb R^N)$ (who's definition is recalled below when
$s$ is {\bf not} an integer).  It is well-known (see e.g. [1],[19] and
[20]) that if $k$ is an integer, $L^{k,p}$
coincides with the standard Sobolev space $W^{k,p}$; also if $p = 2$,
the Bessel potential spaces   $L^{s,2}$ and the Besov spaces $B^{s,2}_2$
coincide for every $s$ non-integer and they are usually denoted by
$H^s$.  When $p \ne 2$ the spaces $L^{s,p}$ and $B^{s,p}_p$ are
distinct.
\medskip
The first result about composition in fractional Sobolev spaces seems
to be due to Mizohata [12] for $H^s$,$s > N/2$.  In 1970 Peetre [15]
considered $B^{s,p}_p \cap L^\infty$ using interpolation techniques; a
very simple direct argument for the same class, $B^{s,p}_p \cap
L^\infty$, was given by M. Escobedo [10] (see the proof of Lemma 2
below).
\medskip
Starting in 1980 techniques of dyadic analysis and Littlewood-Paley
decomposition \`a la Bony [5] were introduced.  For example, Y. Meyer
[11] considered composition in $L^{s,p}$ for $sp > N$; see also
[16],[4],[9] for $H^s$ with $s > N/2$ or for $H^s \cap L^\infty$, any
$s > 0$.  We refer to [17],[6],[7],[18] and their bibliographies for
other directions of research concerning composition in Sobolev spaces.
\medskip
In what follow we denote by $W^{s,p}(\Omega)$ the restriction of
$B^{s,p}_p(\Bbb R^N)$ to $\Omega$ when $s$ is not an integer.
\noindent
Our main result is the following
\medskip
{\bf Theorem 2}.  {\it Assume $u \in W^{s,p}(\Omega)$ where $s > 1$ is a
real number, $1 < p < \infty$, and}
$$
sp = N. \tag 3
$$
\noindent
{\it Let $\Phi \in C^k(\Bbb R)$, where $k = [s] + 1$, be such that}
$$
D^j \Phi \in L^\infty(\Bbb R) \quad \forall j \le k. \tag 4
$$
\noindent
{\it Then}
$$
\Phi \circ u \in W^{s,p}(\Omega).
$$
\medskip
The proof of Theorem 2 relies on a variant of Lemma 1 for fractional
Sobolev spaces.
\medskip
\noindent
{\bf Lemma 2}.  {\it Let $u \in W^{s,p}(\Omega)$, where $s > 1$ is a
real number and $1 < p < \infty$.  Assume, in addition, that $u \in
W^{\sigma,q}$ for some $\sigma \in (0,1)$ with}
$$
q = sp / \sigma. \tag 5
$$
\noindent
{\it Let $\Phi \in C^k (\Bbb R)$, where $k = [s] + 1$, be such that (4)
holds}.
\noindent
{\it Then} 
$$
\Phi \circ u \in W^{s,p}
$$
\medskip
\noindent
{\it Proof of Theorem 2}.  By the Sobolev embedding theorem we have
$$
W^{s,p} \subset W^{r,q}
$$
\noindent
with $r < s$ and 
$$
\frac{1}{q} = \frac{1}{p} - \frac{(s-r)}{N}.
$$
\noindent
In view of assumption (3) we find
$$
q = N/r.
$$
\noindent
In particular,
$$
u \in W^{\sigma,q}
$$
\noindent
for {\bf all} $\sigma \in (0,1)$ with
$$
q = \frac{N}{\sigma} = \frac{sp}{\sigma}.
$$
\noindent
Thus we may apply Lemma 2 and conclude that $\Phi \circ u \in W^{s,p}$.
\medskip
\noindent
{\bf Remark 1}.  Theorem 2 is known to be true when the Sobolev spaces 
$W^{s,p}$ are replaced by the Bessel potential spaces $L^{s,p}$ with
$sp = N$; see D. Adams and M. Frazier [3].  Even though the two results
are closely related it does not seem possible to deduce one from the
other.  Their argument relies on a variant of Lemma 2 for Bessel
potential spaces:
\medskip
Let $u\in L^{s,p} \cap L^{1,sp}$ where $s > 1$ is a real number and $1
< p < \infty$.  Let $\Phi$ be as in Lemma 2.  Then $\Phi \circ u \in
L^{s,p}$.
\medskip
\noindent
{\bf Remark 2}.  The assumption in Lemma 2, $u\in W^{s,p} \cap
W^{\sigma,q}$, with $q = sp/\sigma$ for some $\sigma \in (0,1)$, is
{\bf weaker} than the assumption $u \in W^{s,p} \cap L^\infty$ but it is
{\bf stronger} than the assumption $u \in W^{1,sp}$; this is a
consequence of  Gagliardo - Nirenberg type inequalities (see e.g. the
proof of Lemma D.1 in the Appendix D of [8]).
It is therefore natural to raise the following:
\bigskip
\noindent
{\bf Open Problem}.  Is the conclusion of Lemma 2 valid if one assumes
only $u \in W^{s,p} \cap W^{1,sp}$ where $s > 1$ is a (non-integer) real
number?
\medskip
Before giving the proof of Lemma 2 we recall some properties of
$W^{s,p}$ when $s$ is not an integer.
\medskip
When $0 < \sigma < 1$ and $1 < p < \infty$ the standard definition of
$W^{\sigma,p}$ is
$$
W^{\sigma,p}(\Omega) = \{f \in L^p (\Omega);\,\,\, \int\int\frac{|f(x)
- f(y)|^p}{|x-y|^{N +\sigma p}} dxdy < \infty\}.
$$
\noindent
If $s > 1$ is not an integer write $s= [s] + \sigma$ where $[s]$
denotes the integer part of $s$ and $0 < \sigma < 1$.  Then
$$
W^{s,p}(\Omega) = \{ f \in W^{[s],p}(\Omega), D^\alpha f \in
W^{\sigma,p} \text{ for } |\alpha| = [s]\}.
$$
\medskip
There is a very useful characterization of $W^{s,p}$ in terms of
finite differences (see Triebel [20], p.110).  Here it is more
convenient to work with functions defined on all of $\Bbb R^N$ and to
consider their restrictions to $\Omega$.  Set
$$
(\delta_h u)(x) = u(x+h) -u(x), \,\, h \in \Bbb R^N,
$$
\noindent
so that
$$
(\delta^2_h u)(x) = u(x+2h) -2u(x+h) +u(x), \text{ etc...}
$$
\noindent
Given $s > 0$ not integer, fix {\bf any } integer $M >s$.
\noindent
Then
$$
W^{s,p} = \{ f \in L^p ;\,\, \int\int\frac{|\delta^M_h
f(x)|^p}{|h|^{N+sp}} dxdh < \infty\}.
$$
\medskip
\noindent
{\it Proof of Lemma 2}.  It suffices to consider the case where $s$ is
not an integer.  For simplicity we treat just the case where
$1<s<2$.  The same argument extends to general $s>2, s$ noninteger,
using the same type of computations as in Escobedo [10].
\medskip
The key observation is that $\delta^2_h(\Phi \circ u)$ can be
expressed in terms of $\delta^2_h u$ and $\delta_h u$.  This is the
purpose of our next computation.
\medskip
Set
$$
\align
X &=u(x+2h)\\
Y &=u(x+h)\\
Z &=u(x).
\endalign
$$
\medskip
Since $\Phi'' \in L^\infty(\Bbb R)$ we have 
$$
\Phi(X) - \Phi (Y) = \Phi'(Y)(X-Y) + 0(|X-Y|^2) \tag 6
$$
\noindent
and since $\Phi^\prime \in L^\infty(\Bbb R)$ we also have
$$
\Phi(X) -\Phi(Y) = \Phi'(Y) (X-Y) + 0(|X-Y|).  \tag 7
$$
\noindent
Combining (6) and (7) we find
$$
\Phi(X) - \Phi(Y) = \Phi'(Y)(X-Y) + 0(|X-Y|^a)
$$
\noindent
for any $1 \le a \le  2$ ( we will choose a specific value of $a$
later)  Similarly
$$
\Phi (Z) - \Phi (Y) = \Phi'(Y) (Z-Y) + 0(|Z-Y|^a)
$$
\noindent
Since
$$
\delta^2_h(\Phi \circ u)(x) = (\Phi(X) - \Phi(Y)) +(\Phi(Z) -
\Phi(Y)),
$$
\noindent
one finds
$$
|\delta^2_h(\Phi \circ u)(x)| \leq C (|\delta^2_h u(x)| +
|\delta_h u(x+h)|^a + |\delta_h u(x)|^a).  \tag 8
$$
\noindent
This yields
$$
\int\int\frac{|\delta^2_h(\Phi \circ u)(x)|^p}{|h|^{N+sp}} dxdh \leq C
\int\int\frac{|\delta^2_h u(x)|^p}{|h|^{N+sp}} dxdh + C
\int\int\frac{|\delta_h u (x)|^{ap}}{|h|^{N+sp}} dxdh.  \tag 9
$$
\noindent
The first integral on the right-hand side of (9) is finite since 
$u \in W^{s,p}$.  To handle the second integral we argue as follows.
From the assumption $u \in W^{s,p}\cap W^{\sigma,q}$ with $\sigma \in
(0,1)$ and $q$ given by (5) we know that
$$
\int\int\frac{|\delta^2_h u(x)|^p}{|h|^{N+sp}} dxdh < \infty \text{
and } \int\int\frac{|\delta^2_h u (x)|^q}{|h|^{N+sp}} dxdh < \infty.
\tag 10
$$
\noindent
From (10) and H\"older's inequality we derive that
$$
\int\int\frac{|\delta^2_h u (x)|^r}{|h|^{N+sp}} dxdh < \infty  \tag 11
$$
\noindent
for all $r \in [p,q]$, i.e., $u \in W^{\tau,r}$ with $\tau = sp/r$.
We now choose
$$
a = min \{2,s/\sigma\} , \text{ so that }a \in [1,2]
$$
\noindent
and
$r = ap \in [p,q]$.   It follows that
$$
\int\int\frac{|\delta_h u (x)|^{ap}}{|h|^{N+sp}} dxdh < \infty,
$$
\noindent
which is the desired in equality.
\medskip
\noindent  
{\bf Remark 3}.  There could be another natural proof of Theorem 2 by
induction on $[s]$.  One might attempt to prove that 
$$
D(\Phi \circ u) = \Phi^\prime(u) D u \in W^{s-1,p}.
$$
\noindent
Note that $u \in W^{(s-1),N/(s-1)}$ and thus (by induction) we would
have $\Phi^\prime(u) \in W^{(s-1),N/(s-1)}$.  On the other hand $Du \in
W^{s-1,p}$.  In order to conclude we need a lemma about products, but
we are not aware of any such tool.
\medskip
\noindent
{\bf Remark 4}.  When $s$ (or equivalently $p$) is a {\bf rational}
number, and $\Phi \in C^\infty$ with $D^j \Phi \in L^\infty \, \forall
j$, there is a  simple proof of Theorem 2 based on trace theory and Theorem 1.
Assume for simplicity that $\Omega = \Bbb R^N$.  Suppose that $s$ is
not an integer, but that $s_1 = s+1/p$ is an integer.  Then $u$ is the
trace of some function $u_1 \in W^{s_1,p}(\Bbb R^{N+1})$. Then 
$s_1 p = N+1$ and by Theorem 1 we deduce that $\Phi \circ u_1 \in
W^{s_1,p}(\Bbb R^{N+1})$.  Taking traces we find $\Phi \circ u \in
W^{s,p}(\Bbb R^N)$.  If $s_1$ is not an integer we keep extending
$u_1$ to higher dimensions and stop at the first integer $k$ such that
$s_k = s + k/p$ is an integer ( this is possible since $p$ is rational
and $s+k/p = (N+k)/p$ becomes an integer for some integer $k$).  We
have an extension $u_k \in W^{s_k,p}(\Bbb R^{N+k})$ of $u$.  Then $\Phi
\circ u_k \in W^{s_k,p}(\Bbb R^{N+k})$ by Theorem 1.  Taking back
traces yields $u \in W^{s,p}$.

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