%Composition in fractional Sobolev spaces %H.Brezis,P.Mironescu %November 14,2000 %C.Johnson % renamed 169-04.tex \magnification=\magstep 1 \input amstex \NoBlackBoxes \documentstyle{amsppt} \pageheight{8.5truein} \pagewidth{6.5truein} \loadbold \def\varep{\varepsilon} \def\Mint{\diagup\hskip-.50truecm\int} \def\mint{\diagup\hskip-.39truecm\int} \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 $12, s$ 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 $\Phi \circ u \in W^{s,p}(\Bbb R^N)$. 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