\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{Another look at Sobolev spaces}\endtitle \author{JEAN BOURGAIN$^{(1)}$, HAIM BREZIS$^{(2),(3)}$ AND PETRU MIRONESCU$^{(4)}$}\endauthor \medskip \medskip \address (1) INSTITUTE FOR ADVANCED STUDY\endgraf PRINCETON, NJ 08540\endgraf \endaddress \email bourgain\@ias.edu\endemail \null \address (2) ANALYSE NUM\'ERIQUE\endgraf UNIVERSIT\'E P. ET M. CURIE, B.C. 187\endgraf 4 PL. JUSSIEU\endgraf 75252 PARIS CEDEX 05\endgraf \endaddress \address (3) RUTGERS UNIVERSITY\endgraf DEPT. OF MATH., HILL CENTER, BUSCH CAMPUS\endgraf 110 FRELINGHUYSEN RD, PISCATAWAY, NJ 08854\endgraf \endaddress \email brezis\@ccr.jussieu.fr; brezis\@math.rutgers.edu\endemail \null \address (4) D\'EPARTEMENT DE MATH\'EMATIQUES\endgraf UNIVERSIT\'E PARIS-SUD\endgraf 91405 ORSAY\endgraf \endaddress \email Petru.Mironescu\@math.u-psud.fr\endemail \medskip \medskip \thanks {\bf Acknowledgment:} The second 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 third author (P.M.) was visiting Rutgers University; he thanks the Mathematics Department for its invitation and hospitality.\endthanks \endtopmatter \document \centerline{Dedicated to Alain Bensoussan with esteem and affection} \bigskip \noindent {\bf 1. Introduction} \medskip Our initial concern is to study the limiting behavior of the norms of fractional Sobolev spaces $W^{s,p}, 0 < s < 1$, $1 < p < \infty$, as $s \to 1$. Recall that a commonly used (semi-) norm on $W^{s,p}$, introduced by Gagliardo, is given by $$ \|f \|_{W^{s,p}}^p = \int\limits_{\Omega} \int\limits_{\Omega} \frac{|f(x) - f(y)|^p}{|x-y|^{N+sp}} dxdy $$ \noindent where $\Omega$ is a smooth bounded domain in $\Bbb R^N$ (see e.g. Adams [1]). A well-known ``defect'' of this scale of norms is that $\|f\|_{W^{s,p}}$ does not converge, as $s\nearrow 1$, to $\|f\|_{W^{1,p}}$, given by the (semi-) norm $$ \|f\|_{W^{1,p}}^p = \int\limits_{\Omega} |\nabla f |^p dx, $$ \noindent where $|\ \ |$ denotes the euclidean norm. \medskip In fact, it is clear that if $f$ is any smooth nonconstant function, then $\|f\|_{W^{s,p}} \to \infty$ as $s \nearrow 1$. The factor$(1-s)^{1/p}$ in front of $\|f\|_{W^{s,p}}$ ``rectifies'' the situation (see Corollary 2 and Remark 6). This analysis leads us to a new characterization of the Sobolev space $W^{1,p},1 < p < \infty$. \medskip First, an easy observation \medskip \noindent \proclaim{Theorem 1} Assume $f \in W^{1,p}(\Omega)$, $1\leq p < \infty$ and let $\rho \in L^1 (\Bbb R^N)$, $\rho\geq 0$. Then $$ \int\limits_{\Omega}\int\limits_{\Omega}\frac{|f(x) - f(y)|^p}{|x - y|^p} \rho (x-y) dxdy \leq C \|f\|_{W^{1,p}}^p \|\rho\|_{L^1} $$ \noindent where C depends only on $p$ and $\Omega$.\endproclaim \medskip Next, we take a sequence $(\rho_n)$ of radial mollifiers, i.e. $$ \rho_n(x) = \rho_n(|x |),\quad \rho_n \geq 0, \quad \int \rho_n(x) dx =1 $$ \noindent and $$\lim_{n \to\infty}\;\;\int^\infty_\delta \rho_n(r) r^{N-1} dr = 0 \quad\text{ for every } \delta > 0. $$ \medskip \noindent \proclaim{Theorem 2} Assume $f \in L^p (\Omega), 1 < p < \infty$. Then $$ \lim_{n\to\infty}\quad\int\limits_{\Omega} \int\limits_{\Omega}\frac{|f(x) - f(y)|^p}{|x-y|^p} \rho_n (x-y) dxdy = K_{p,N} \|f\|_{W^{1,p}}^p, $$ \noindent with the convention that $\|f\|_{W^{1,p}} = \infty$ if $f \notin W^{1,p}$. Here $K_{p,N}$ depends only on $p$ and $N$.\endproclaim \medskip When $p = 1$ we have the following variants \medskip \noindent \proclaim{Theorem 3} Assume $f\in W^{1,1}(\Omega )$. Then $$ \lim_{n \to \infty} \int\limits_{\Omega} \int\limits_{\Omega}\frac{|f(x) - f(y)|}{|x - y|} \rho_n (x-y) dxdy = K_{1,N} \|f\|_{W^{1,1}}, $$ \noindent where $K_{1,N}$ depends only on $N$.\endproclaim \medskip \noindent \proclaim{Theorem 3'} Assume $f\in L^1(\Omega)$. Then $f\in BV (\Omega)$ if and only if $$ \liminf\limits_{n \to \infty} \int\limits_{\Omega} \int\limits_{\Omega}\frac{|f(x) - f(y)|}{|x - y|} \rho_n (x-y) dxdy < \infty, $$ \noindent and then $$ \align C_1 \|f\|_{BV} &\leq \liminf\limits_{n\to\infty} \int\limits_{\Omega} \int\limits_{\Omega}\frac{|f(x) - f(y)|}{|x - y|} \rho_n (x-y) dxdy\\ &\leq \limsup\limits_{n\to\infty} \int\limits_{\Omega} \int\limits_{\Omega}\frac{|f(x) - f(y)|}{|x - y|} \rho_n (x-y) dxdy \leq C_2 \|f\|_{BV}. \tag 1\\\endalign $$ \endproclaim \noindent Here $C_1$ and $C_2$ depend only on $\Omega$, and $$ \|f\|_{BV} = \int\limits_{\Omega} |\nabla f| = \sup \left\{\int\limits_{\Omega} f \text{ div }\varphi \,\, ;\,\,\varphi\in C^\infty_0 (\Omega; \Bbb R^N),\, |\varphi(x)| \leq 1 \text{ on }\Omega\right\}. $$ \remark{Remark 1} In dimension $N = 1$ we can prove that for every $f \in B V(0,1)$ $$ \lim\limits_{n \to\infty} \int_0^1 \int_0^1\frac{|f(x) - f(y)|}{|x - y|} \rho_n (x - y) dxdy = \int_0^1\;| f'|. $$ \noindent We do not know whether a similar conclusion holds when $N \geq 2$ (even for a special sequence of mollifiers), i.e., whether $$ \lim\limits_{n\to\infty}\int\limits_\Omega\int\limits_\Omega \frac{|f(x)-f(y)|^p}{|x-y|^p}\rho_n (x-y)dxdy=K_{1,N}\int\limits_\Omega |\nabla f|. $$ \endremark \medskip Here are some simple consequences of the above results (and their proofs), where $K$ denotes various constants depending only on $p$ and $N$. \medskip \noindent \proclaim{Corollary 1} Assume $f \in W^{1,p}(\Omega)$ with $1 \leq p < \infty$. Then $$ \lim\limits_{n \to\infty} \int\limits_{\Omega}\;\;\frac{|f(x) - f(y)|^p}{|x - y|^p} \rho_n (x-y)\; dy = K\;\; |\nabla f (x)|^p \text{ in }L^1(\Omega). $$ \endproclaim \noindent \proclaim{Corollary 2} Assume $f \in L^p (\Omega ), 1 < p < \infty$. Then $$ \lim\limits_{\varepsilon\to 0}\;\; \varepsilon \; \|f\|_{W^{1 -\varepsilon,p}}^p \;\; = K \;\; \|f\|_{W^{1,p}}^p. $$ \endproclaim \noindent \remark{Remark 2} In the special case where $p=2$ and $\Omega =\Bbb R^N$ a similar conclusion follows from the result of Masja and Nagel [5] using the Fourier characterization of $H^s$. \endremark \medskip \noindent \proclaim{Corollary 3} Assume $f \in L^p(\Omega ), 1 < p < \infty$. Then $$ \lim\limits_{\varepsilon\to 0}\;\;\varepsilon^{-N}\;\iint\limits_{|x-y|<\varepsilon}\frac{|f(x) - f(y)|^p}{|x-y|^p}\;\;dxdy = K\|f\|_{W^{1,p}}^p. $$ \endproclaim \noindent \proclaim{Corollary 4} Assume $f \in L^p (\Omega ), 1 < p < \infty$. Then $$ \lim\limits_{\varepsilon\to 0}\frac{1}{|\log\varepsilon|}\;\iint\limits_{|x-y|>\varepsilon}\;\frac{|f(x)-f(y)|^p}{|x-y|^{N+p}} dxdy = K\|f\|_{W^{1,p}}^p. $$ \endproclaim \noindent \remark{Remark 3} P. Mironescu and I. Shafrir [7] have studied related limits, e.g., when $N=1$ and $f \in BV(0,1)$, $$ \lim\limits_{\varepsilon\to 0}\frac{1}{|\log\varepsilon |}\;\iint\limits_{|x-y|>\varepsilon}\;\frac{|f(x)-f(y)|^2}{|x-y|^2} dxdy. $$ \endremark \medskip The case where $f \in BV (\Omega)$ is not fully satisfactory; we have only partial results, for example \medskip \noindent \proclaim{Corollary 5} Assume $f \in L^1(\Omega )$. Then $$ \align C_1 \|f\|_{BV} &\leq \liminf_{\varepsilon \to 0} \varepsilon\,\int\limits_{\Omega}\int\limits_{\Omega}\frac{|f(x) - f(y)|}{|x-y|^{N+1-\varepsilon}} dxdy\\ &\leq \limsup_{\varepsilon\to 0}\varepsilon\;\int\limits_{\Omega}\int\limits_{\Omega}\frac{|f(x) -f(y)|}{|x-y|^{N+1-\varepsilon}} dxdy \leq C_2 \|f\|_{BV}. \endalign $$ \endproclaim \medskip \noindent \remark{Remark 4} In particular when $f=\chi_A$ is the characteristic function of a measurable set $A\subset\Omega$ having finite perimeter, then $$ \|\chi_A\|_{BV} \leq C \liminf\limits_{\varepsilon \to 0} \varepsilon \int\limits_{\Omega\setminus A}\int\limits_{A} \frac{dxdy}{|x-y|^{N+1-\varepsilon}}. $$ \noindent Combining this with the Sobolev inequality yields $$ \big( |A| |\Omega\setminus A|\big)^{(N-1)/N} \leq C\, \liminf\limits_{\varepsilon\to 0} \varepsilon \int\limits_{\Omega\setminus A}\int\limits_{A}\frac{dxdy}{|x-y|^{N+1-\varepsilon}}. $$ \noindent In particular, if $A$ is a measurable subset of $\Omega\subset\Bbb R^N, N \geq 1$, such that $$ \int\limits_{\Omega\setminus A}\int\limits_{A}\quad \frac{dxdy}{|x-y|^{N+1}} < \infty, $$ \noindent then either $|A|= 0$ or $|\Omega\setminus A| = 0$. This fact was already established in Bourgain, Brezis and Mironescu [2] (Appendix B) with a different proof (see also Bourgain, Brezis and Mironescu [3] and Brezis [5]).\endremark \medskip \noindent {\bf 2. Proofs} \smallskip \noindent {\bf Proof of Theorem 1}. By standard extension we may always assume that $f \in W^{1,p}(\Bbb R^N)$ and then there is some constant $C = C(p,\Omega )$ such that $$ \big( \int\limits_{\Bbb R^N} |f(x+h)-f(x)|^p{dx}\big)^{1/p} \leq |h| \|f\|_{W^{1,p}(\Bbb R^N)} \leq C |h| \|f\|_{W^{1,p}(\Omega)},\tag 2 $$ \noindent for all $f \in W^{1,p}$ and $h \in \Bbb R^N$ (see, e.g., Brezis [4], Proposition IX.3). By (2), we obtain $$ \align \int\limits_{\Omega} \int\limits_{\Omega} \frac{|f(x)-f(y)|^p}{|x-y|^p} \rho (x-y) dxdy &\leq \underset {\Bbb R^N}\to \int \frac{\rho (h)}{|h|^p} \underset {\Bbb R^N}\to\int |f(x+h) - f(x)|^pdxdh\\ &\leq C^p\|f\|_{W^{1,p}}^p \underset {\Bbb R^N}\to\int \rho (h)dh = C^p \|f\|_{W^{1,p}}^p \|\rho\|_{L^1}.\endalign $$ \noindent {\bf Proof of Theorem 2}. For $f \in L^p$, let $$ F_n (x,y)\;\;=\;\;\frac{|f(x) - f(y)|}{|x-y|} \rho_n^{1/p} (x-y). $$ \noindent Assuming first that $f \in W^{1,p}$, we have to prove that $$ \lim\limits_{n \to\infty}\quad \|F_n\|_{L^p}^p\ = \ K \|f\|_{W^{1,p}}^p,\tag 3 $$ \noindent for some $K = K_{p,N}$. \smallskip By Theorem 1, we have, for any $n$ and $f, g\in W^{1,p}$, $$\big|\| F_n\|_{L^p}-\|G_n\|_{L^p}\big|\leq \| F_n - G_n\|_{L^p} \leq C \| f - g\|_{W^{1,p}}, \tag 4 $$ \noindent for some constant $C$ independent of $n, f$ and $g$. Therefore it suffices to establish (3) for $f$ in some dense subset of $W^{1,p}$, e.g., for $f \in C^2 (\bar\Omega)$. \medskip Fix some $f \in C^2 (\bar\Omega)$. Then $$ \frac{|f(x)-f(y)|}{|x-y|} = \bigg|(\nabla f)(x)\cdot\frac{x-y}{|x-y|}\bigg| + O(|x-y|). $$ For each fixed $x \in\Omega$, let $R=$ dist $(x,\partial\Omega )$. We have $$ \align \int\limits_{\Omega}\frac{|f(x) - f(y)|^p}{|x-y|^p} \rho_n (x-y) dy &= \\ \int\limits_{B(x,R)}\frac{|f(x) - f(y)|^p}{|x-y|^p} \rho_n (x-y) dy\,&+ \int\limits_{\Omega\setminus B(x,R)}\frac{|f(x) -f(y)|^p}{|x-y|^p} \rho_n(x-y)dy.\tag 5\endalign $$ Clearly, the last integral in (5) tends to 0 as $n\to \infty$. On the other hand, $$ \aligned &\int\limits_{B(x,R)}\frac{|f(x) - f(y)|^p}{|x-y|^p} \rho_n (x-y)dy\\ &=\overset{R}\to{\underset{0}\to{\int}} \rho_n(r) \int\limits_{|y-x|=r} \bigg(\bigg|(\nabla f)(x)\cdot\frac{x-y}{|x-y|}\bigg|^p +O (|x-y|^p)\bigg) d\sigma dr\\ &=\underset{0}\to{\int^R} \rho_n(r) \int\limits_{|\omega|=r}\bigg(\bigg|(\nabla f)(x)\cdot\frac{\omega}{|\omega|}\bigg|^p + O (r^p)\bigg)d\sigma dr\\ &=K|\nabla f(x)|^p\underset{0}\to{\int^R} |S^{N-1}| r^{N-1} \rho_n(r) dr + O(\underset{0}\to{\int^R} r^{N+p-1} \rho_n (r) dr),\endaligned $$ where $K=K_{p,1}=1$ for all $p\ge 1$ and for $N\ge 2$, $p\ge 1$, $$K = K_{p,N} = \frac{1}{|S^{N-1}|}\int\limits_{\omega \in S^{N-1}} |\omega\cdot e|^p d\sigma ; $$ here $e$ is any unit vector in $\Bbb R^N$. \medskip Therefore, $$ \int\limits_{\Omega}\frac{|f(x) - f(y)|^p}{|x-y|^p} \rho_n(x-y)dy\longrightarrow K |\nabla f (x)|^p, \quad \forall x \in\Omega. \tag 6 $$ \noindent If $L$ is such that $|f(x)-f(y)| \leq L |x-y|,\;\;\forall x, y \in\Omega$, then $$ \int\limits_{\Omega}\frac{|f(x)-f(y)|^p}{|x-y|^p} \rho_n (x-y) dy \leq L^p,\quad \forall x \in\Omega. \tag 7 $$ \noindent Hence, for $f\in C^2(\bar\Omega)$, (3) follows by dominated convergence from (6) and (7). \medskip In order to complete the proof of Theorem 2, it suffices to prove that, if $f \in L^p$ and $$ A_p= \liminf_{n\to\infty}\bigg[\int\limits_{\Omega}\int\limits_{\Omega}\frac {|f(x)-f(y)|^p}{|x-y|^p}\rho_n(x-y)dxdy\bigg]^{1/p} <\infty, \tag 8 $$ \noindent then $f \in W^{1,p}$. We will use the following \proclaim{Lemma 1} Assume $f\in L^1(\Bbb R^N)$, $\varphi\in C^\infty_0(\Bbb R^N)$, $\rho\in L^1(\Bbb R^N)$, $\rho$ radial, $\rho\ge 0$ and let $e$ be any unit vector in $\Bbb R^N$. Then $$ \bigg|\int\limits_{x\in\Bbb R^N}f(x)dx\int\limits_{(y-x)\cdot e\ge 0}\frac{\varphi (y)-\varphi (x)}{|y-x|} \rho (y-x)dy\bigg|\le \int\limits_{\Bbb R^N}\int\limits_{\Bbb R^N}\frac{|f(x)-f(y)|}{|x-y|}|\varphi (y)|\rho (x-y)dxdy. $$ \endproclaim \medskip Note that the integral on the l.h.s. makes sense and is finite since $|\varphi (y)-\varphi (x)|\le L|x-y| $; the integral on the r.h.s. makes sense but is possibly infinite. \medskip \noindent {\bf Proof.} For any $\delta >0 $ set $$ \rho_\delta (t)=\,\,\,\cases 0\quad \text{ if }t<\delta \\ \rho (t)\,\,\text{ if }t>\delta . \endcases $$ \noindent It suffices to prove the lemma when $\rho$ is replaced by $\rho_\delta$ and then pass to the limit as $\delta\to 0$. Note that the two functions $$ |f(x)||\varphi (y)|\frac{\rho_\delta (y-x)}{|y-x|}\,\text{ and }\, |f(x)||\varphi (x)|\frac{\rho_\delta (y-x)}{|y-x|} $$ \noindent are integrable on $\Bbb R^N\times\Bbb R^N$; therefore we have $$ \align I & =\int\limits_{x\in\Bbb R^N}f(x)dx\int\limits_{(y-x)\cdot e\ge 0}\frac{\varphi (y)-\varphi (x)}{|y-x|} \rho_\delta (y-x)dy\\ & =\iint\limits_{(y-x)\cdot e\ge 0}f(x)\varphi (y)\frac{\rho_\delta (y-x)}{|y-x|}dxdy- \iint\limits_{(y-x)\cdot e\ge 0}f(x)\varphi (x)\frac{\rho_\delta (y-x)}{|y-x|}dxdy\\ & =I_1-I_2. \endalign $$ \noindent Changing $x$ into $y$ and $y$ into $x$ in $I_2$ yields $$ \align I_2 & =\iint\limits_{(x-y)\cdot e\ge 0}f(y)\varphi (y)\frac{\rho_\delta (x-y)}{|x-y|}dxdy\\ & =\iint\limits_{(x-y)\cdot e\le 0}f(y)\varphi (y)\frac{\rho_\delta (x-y)}{|x-y|}dxdy, \endalign $$ \noindent where the last equality holds since $\rho_\delta$ is radial. Hence we obtain $$ \align I & =\iint\limits_{(y-x)\cdot e\ge 0}\varphi (y)\frac{f(x)-f(y)}{|y-x|}\rho_\delta (y-x)dxdy\\ & \le \int\limits_{\Bbb R^N}\int\limits_{\Bbb R^N}\frac{|f(x)-f(y)||}{|y-x|}|\varphi (y)|\rho_\delta (x-y)dxdy, \endalign $$ \noindent which is the desired conclusion. \bigskip {\bf Proof of Theorem 2 completed.} Let $\varphi\in C^\infty_0(\Omega )$ (extended by $0$ outside $\Omega$) and let $e$ be a unit vector in $\Bbb R^N$. As above, for every $x\in\Bbb R^N$, $$ \int\limits_{(y-x)\cdot e\ge 0}\frac{\varphi (y)-\varphi (x)}{|y-x|} \rho_n(y-x)dy\,\,\overset{n\to\infty}\to\longrightarrow \,\, K\nabla\varphi (x)\cdot e \tag 9 $$ \noindent where $$ K=\frac{1}{2|S^{N-1}|}\int\limits_{\omega\in S^{N-1}}|\omega\cdot e|d\sigma =\frac{1}{2}K_{1,N} $$ \noindent depends only on $N$. \medskip Applying Lemma 1 with $f$ replaced by $\bar f$, $$ \bar f (x)=\,\, \cases & f(x),\,\, \text{ if }x\in\Omega\\ & 0,\quad \hskip 3mm\text{ if } x\notin\Omega, \endcases $$ \noindent we obtain $$ \align J_n & =\bigg|\int\limits_{\Omega}f(x)dx\int\limits_{(y-x)\cdot e\ge 0}\frac{\varphi (y)-\varphi (x)}{|y-x|} \rho_n(y-x)dy\bigg|\\ & \le \int\limits_{\Bbb R^N}dx \int\limits_{supp\,\varphi}\frac{|f(x)-f(y)|}{|x-y|}|\varphi (y)|\rho_n (x-y)dy\tag {10}\\ & \le \int\limits_{\Omega}dx \int\limits_{\Omega}\frac{|f(x)-f(y)|}{|x-y|}|\varphi (y)|\rho_n (x-y)dy+ \int\limits_{\Bbb R^N\setminus\Omega}dx \int\limits_{suppp\,\varphi}|f(y)||\varphi (y)|\frac{\rho_n(x-y)}{|x-y|}dy\\ & =J_{1,n}+J_{2,n}. \endalign $$ \noindent By H\" older we have $$ J_{1,n}\le \bigg( \int\limits_{\Omega}\int\limits_{\Omega}\frac{|f(x)-f(y)|^p}{|x-y|^p} \rho_n (x-y) \bigg)^{1/p} \|\varphi\|_{L^{p'}} $$ \noindent and $$ J_{2,n}\le\frac{1}{d}\|\varphi\|_{L^{p'}}\| f\|_{L^{p}} \int\limits_{|\xi |>d}\rho_n (\xi )d\xi\hskip 24mm $$ \noindent where $d=$ dist $(\Bbb R^N\setminus\Omega ,\, supp \varphi )$, so that $J_{2,n}\to 0$ as $n\to\infty$. Passing to the limit in (10) as $n\to\infty$ yields $$ K\bigg|\int\limits_{\Omega}f(x)\big(\nabla\varphi (x)\cdot e\big)\bigg|\le A_p\|\varphi\|_{L^{p'}}, $$ \noindent where $A_p$ is defined in (8). Choosing $e=e_i$, $i=1,\, 2,\, ...,N$, we obtain $$ \bigg|\int\limits_{\Omega}f\frac{\partial\varphi}{\partial x_i}\bigg|\le \frac{A_p}{K}\|\varphi\|_{L^{p'}} $$ \noindent and consequently $f\in W^{1,p}$. The proof of Theorem 2 is complete. \medskip \noindent {\bf Proof of Corollary 1}. The conclusion is clear when $f\in C^2(\bar\Omega)$. For a general $ f\in W^{1,p}$, the statement follows by density using (4). \bigskip The proof of Theorem 3 is the same as the first part of the proof of Theorem 2, since smooth functions are dense in $W^{1,1}$. \bigskip \noindent {\bf Proof of Theorem 3'}. The last inequality in (1) is proved as in Theorem 1. The first inequality in (1) is proved as in the second part of the proof of Theorem 2 (using duality). \medskip In fact, a more precise computation in Lemma 1 yields $$ \align & \bigg|\,\,\iint\limits_{(y-x)\cdot e\ge 0}\frac{\varphi (y)-\varphi (x)}{|y-x|}\rho (y-x)dxdy\bigg|+ \bigg|\,\,\iint\limits_{(y-x)\cdot e\le 0}\frac{\varphi (y)-\varphi (x)}{|y-x|}\rho (y-x)dxdy \bigg|\\ & \le\int\limits_{\Bbb R^N}\int\limits_{\Bbb R^N}\frac{|f (x)-f (y)}{|x-y|}|\varphi (y)|\rho (x-y)dxdy. \endalign $$ \noindent If we proceed as above we then obtain $$ K_{1,N}\bigg|\int\limits_{\Omega}f(x)\big( \nabla\varphi(x)\cdot e \big)dx\bigg|\le A_p\|\varphi\|_{L^{p'}}. $$ \noindent In particular, when $p=1$, $N=1$ and $\Omega =(0,1)$, we find $$ \big|\int_0^1 f(x)\varphi' (x)dx\big|\le\liminf\limits_{n\to\infty} \int_0^1\int_0^1 \frac{|f(x)-f(y)|}{|x-y|}\rho_n (x-y)dxdy,\quad\forall\varphi\in C^\infty_0\text{ with }|\varphi |\le 1, $$ \noindent i.e., $$ \| f\|_{BV}\le\liminf\limits_{n\to\infty}\int_0^1\int_0^1 \frac{|f(x)-f(y)|}{|x-y|}\rho_n (x-y)dxdy. \tag 11 $$ \noindent On the other hand, for every $f\in BV(0,1)$ we have, as in the proof of Theorem 1, $$ \limsup\limits_{n\to\infty}\int_0^1\int_0^1 \frac{|f(x)-f(y)|}{|x-y|}\rho_n (x-y)dxdy \le\| f\|_{BV}. \tag 12 $$ \noindent Combining (11) and (12) we see that for every $f\in L^1(0,1)$, $$ \lim\limits_{n\to\infty}\int_0^1\int_0^1 \frac{|f(x)-f(y)|}{|x-y|}\rho_n (x-y)dxdy =\| f\|_{BV} \tag 13 $$ \noindent which is the content of Remark 1 for $N=1$. \bigskip \bigskip \noindent {\bf 3. The case of a sequence $(f_n)$} \medskip In the previous sections $f$ was a fixed function. Throughout this section we assume that $(f_n)$ is a sequence of functions in $L^p(\Omega )$ satisying the uniform estimate $$ \int\limits_{\Omega} \int\limits_{\Omega} \frac{|f_n(x) -f_n(y)|^p}{|x-y|^p} \rho_n (x-y)dxdy \leq C_0, \tag 14 $$ \medskip \noindent where $\Omega$ is a smooth bounded domain in $\Bbb R^N, 1\leq p < \infty$, and $(\rho_n)$ is a sequence of radial mollifiers. Without loss of generality, we may also assume the normalization condition $$ \int\limits_{\Omega} f_n(x)dx =0,\quad \forall n.\tag 15 $$ \medskip \noindent \proclaim{Theorem 4} Assume (14), (15) and $$ \text{for each }n,\text{ the function } t\in(0,\infty) \mapsto\rho_n(t)\text{ is non-increasing }.\tag 16 $$ Then the sequence $(f_n)$ is relatively compact in $L^p(\Omega )$ and (up to a subsequence) we may assume that $f_n \to f$ in $L^p(\Omega )$. Moreover, \medskip \noindent a) if $ 1< p < \infty$, then $f \in W^{1,p}(\Omega )$ and $\|f\|_{W^{1,p}}^p \leq C (p, \Omega) C_0,$ \smallskip \noindent b) if $p = 1$, then $f \in BV(\Omega )$ and $\|f\|_{BV} \leq C (\Omega) C_0$. \endproclaim \medskip \noindent \remark{Remark 5} In view of Theorems 2 and 3, the additional assumption (16) may seem artificial. Actually, it is possible to slightly weaken (16); for example we may assume $$ \rho_n(t) \geq C_1\rho_n(s), \quad \forall\, n,\quad \forall t \leq s, \tag 17 $$ \noindent for some $C_1$ independent of $n, t, s$. \medskip \noindent However, the conclusions of Theorem 4 fail for {\bf general} $\rho_n's$. We shall give below a counterexample where the sequence $(f_n)$ need not be relatively compact in $L^p$ (Counterexample 2).\endremark \medskip Here are two examples of interest \medskip \noindent \proclaim{Corollary 6} For $1 \leq p < \infty$, let $(f_\varepsilon)$ be a family of functions in $L^p(\Omega )$ such that $$ \iint\limits_{|x-y| < \varepsilon}\frac{|f_\varepsilon(x) - f_\varepsilon(y)|^p}{|x-y|^p} dxdy \leq C_0 \varepsilon^N. $$ \noindent Then, up to a subsequence, $(f_\varep)$ converges in $L^p(\Omega )$ to some $f\in W^{1,p}(\Omega )$\hfill\break \noindent ( for $10$) to some $f \in W^{1,p}(\Omega )$.\endproclaim \medskip \noindent {\bf Proof of Theorem 4.} The heart of the proof consists of showing that $(f_n)$ is relatively compact in $L^p$. The rest is done as in the second part of the proof of Theorem 2. \medskip Without loss of generality, we may assume that $\Omega = \Bbb R^N$ and that supp $f_n \subset B$, a ball in $\Bbb R^N$ of diameter 1. This can be achieved by extending each function $f_n$ by reflection across the boundary in a neighborhood of $\partial\Omega$. Using the monotonicity assumption (16), we see that assumption (14) still holds. \medskip In order to prove compactness in $L^p$, we rely on the following variant of the Riesz-Fr\'echet-Kolmogorov theorem(which can be proved combining the arguments in Brezis [4], Th\'eor\`eme IV.25 and Corollaire IV.27) : let, for $\delta > 0, \varPhi_\delta$ be the mollifier $$ \varPhi_\delta = \frac{1}{|B_\delta(0)|} \chi_{B_\delta(0)}. $$ A sequence $(f_n)$ is relatively compact in $L^p(\Omega)$ if and only if $$ \|f_n\|_{L^p} \leq C \tag 18 $$ and $$ \lim\limits_{\delta \to 0}\,\,(\limsup\limits_{n\to\infty} \,\|f_n -f_n\ast \varPhi_\delta\|_{L^p}) = 0.\tag 19 $$ For each $n$ and $t > 0$, let $$ \align F_n(t) & =\int\limits_{\omega\in S^{N-1}}\,\,\int\limits_{\Bbb R^N} |f_n(x+t\omega) - f_n(x)|^p dx d\sigma \\ & =\frac{1}{t^{N-1}} \int\limits_{|h|=t}\,\,\int\limits_{\Bbb R^N} |f_n(x+h) - f_n(x)|^p dx d\sigma.\\ \endalign $$ Using the triangle inequality, we obtain $$ F_n (2t) \leq 2^p F_n(t). \tag 20 $$ In terms of $F_n$, assumption (14) can be expressed as $$ \int^1_0 t^{N-1}\frac{F_n(t)}{t^p}\,\rho_n(t) dt \leq C_0.\tag 21 $$ We claim that $$ \int|f_n(x)|^pdx\, \leq \, C \int^1_0 t^{N-1} F_n(t)dt \tag 22 $$ \noindent and $$ \int |f_n(x)-(f_n \ast \Phi_\delta)(x)|^p dx \leq C \delta^{-N} \int^\delta_0 t^{N-1}F_n(t)dt, \tag 23 $$ \noindent for some $C$ independent of $n$ and $\delta$. \medskip We prove for example (23): \medskip $$ \align \int |f_n(x)-(f_n \ast \Phi_\delta)(x)|^p dx &= \int \bigg| f_n(x)-\frac{1}{|B_1|\delta^N} \int\limits_{|y-x|<\delta} f_n(y)dy\bigg|^pdx \\ &= \frac{1}{(|B_1|\delta^N)^p} \int \, \bigg|\int\limits_{|y-x|< \delta}(f_n(x)-f_n(y))dy\bigg|^p dx \\ &\le \frac{1}{|B_1|}\delta^{-N} \iint\limits_{|y-x|<\delta} |f_n(x)-f_n(y)|^p dxdy \\ &= \frac{1}{|B_1|} \delta^{-N} \int\limits_{|h|<\delta} (\int|f_n(x+h)-f_n(x)|^p dx)dh \\ &= C \delta^{-N}\int^\delta_0t^{N-1}F_n(t)dt. \endalign $$ The proof of (22) is similar, since $$ f_n(x) = f_n(x) - \frac{1}{|B|}\int\limits_{B} f_n(y)dy. $$ We are going to establish below the key inequality $$ \delta^{-N} \int^\delta_0 t^{N-1} \frac{F_n(t)}{t^p} dt \le C \bigg(\int^\delta_0 t^{N-1} \frac{F_n(t)}{t^p} \rho_n(t)dt\bigg)\bigg/ \bigg(\int\limits_{|x|<\delta} \rho_n(x)dx\bigg).\tag 24 $$ Assuming (24) has been proved, we proceed as follows : since $$ \lim\limits_{n\to\infty} \int\limits_{|x|<\delta}\rho_n(x)dx =1, $$ \noindent by combining (21) with (24) we find $$ \delta^{-N} \int^\delta_0 t^{N-1} \frac{F_n(t)}{t^p} dt \le C \quad\text{ for } n \ge n_\delta.\tag 25 $$ \noindent In particular, we have $$ \delta^{-N} \int^\delta_0 t^{N-1} F_n(t)dt \le C \delta^p \quad\text{ for } n \geq n_\delta . \tag 26 $$ Inequalities (18), (19) --and thus the conclusion of Theorem 4-- follow from (22), (23) and (26). \medskip It remains to establish inequality (24). Note that it is a particular case $ (g(t)=\frac{F_n(t)}{t^p}, h(t) = \rho_n(t))$ of the following variant of an inequality due to Chebyshev: \medskip \noindent \proclaim{Lemma 2} Let $g,h: (0,\delta)\to\Bbb R_+$. Assume that $g(t) \leq g(t/2), t\in(0,\delta)$, and that $h$ is non-increasing. \medskip \noindent Then, for some $C=C(N) >0$, $$ \int^\delta_0t^{N-1}g(t)h(t)dt \geq C\delta^{-N}\int^\delta_0t^{N-1}g(t)dt\int^\delta_0t^{N-1}h(t)dt. $$ \endproclaim \medskip \noindent {\bf Proof of Lemma 2.} It suffices to consider the case $\delta=1$; the general case follows by scaling. We have $$ \align \int^1_0 t^{N-1} g(t) h(t)dt &= \sum\limits_{j\ge 0} \,\,\,\int^{1/2^j}_{1/2^{j+1}} t^{N-1} g(t)h(t)dt \\ &= \sum\limits_{j\ge 0} \frac{1}{2^{Nj}} \int^1_{1/2} s^{N-1} g\bigg(\frac{s}{2^j}\bigg) h\bigg(\frac{s}{2^j}\bigg)ds \\ &= \int^1_{1/2} s^{N-1}\,\sum\limits_{j\ge 0} \frac{1}{2^{Nj}} g\bigg(\frac{s}{2^j}\bigg) h\bigg(\frac{s}{2^j}\bigg)ds, \tag 27 \endalign $$ \noindent and a similar equality holds for $\overset 1 \to{\underset 0\to\int} t^{N-1} g (t) dt$. We recall the classical Chebyshev inequality: if $G,H: X \to \Bbb R, \mu$ a positive measure on $X$ and $$(G(x) -G(y)) (H(x)-H(y)) \geq 0, \quad \forall x, y \in X,$$ \noindent then $$ \int\limits_{X} GHd\mu \geq \frac{1}{\mu (X)} \quad \int\limits_{X} Gd\mu \,\, \int\limits_{X} Hd\mu. $$ \noindent In particular, if $\alpha_j\geq0$ and the sequences $(a_j), (b_j)$ have the same monotonicity, then $$ \sum\alpha_j a_j b_j \geq\frac{1}{\sum\alpha_j}\,\sum\alpha_j a_j\,\,\sum\alpha_j b_j.\tag 28 $$ \noindent Since for each $s \in(1/2,1),$ the sequences $(g(\frac{s}{2^j}))$ and $(h(\frac{s}{2^j}))$ are non-decreasing, (28) with $\alpha_j =\frac{1}{2^{Nj}}$ yields $$ \sum\limits_{j\ge 0}\frac{1}{2^{Nj}} g\bigg(\frac{s}{2^j}\bigg)h\bigg(\frac{s}{2^j}\bigg)\,\ge \,C \sum\limits_{j\geq0}\frac{1}{2^{Nj}} g\bigg(\frac{s}{2^j}\bigg)\sum\limits_{j\ge 0}\frac{1}{2^Nj}h\bigg(\frac{s}{2_j}\bigg).\tag 29 $$ \noindent Now clearly, for each $s \in(1/2,1)$ and each $j\geq 1$, $$ \frac{1}{2^{Nj}}h\bigg(\frac{s}{2^j}\bigg) \geq \frac{1}{2^{Nj}}h\bigg(\frac{1}{2^j}\bigg)\geq C\int^{1/2^{j-1}}_{1/2^j} t^{N-1} h(t)dt, $$ \noindent for some $C$ depending only on $N$, so that $$ \sum\limits_{j\geq 0}\frac{1}{2^{Nj}}h\bigg(\frac{s}{2^j}\bigg)\geq\,C\int^1_0 t^{N-1}h(t)dt.\tag 30 $$ \noindent It follows from (29) and (30) that $$ \sum\limits_{j\geq 0}\frac{1}{2^{Nj}}g\bigg(\frac{s}{2^j}\bigg)h\bigg(\frac{s}{2^j}\bigg)\geq C \int^1_0 t^{N-1}h(t)dt \sum\limits_{j\geq 0}\frac{1}{2^{Nj}}g\bigg(\frac{s}{2^j}\bigg).\tag 31 $$ \noindent Inserting (31) into (27), we find $$ \align \int^1_0 t^{N-1}g(t)h(t)dt &\geq C \int^1_0 t^{N-1}h(t)dt \int^1_{1/2}s^{N-1}\sum\limits_{j\geq 0}\frac{1}{2^{Nj}}g\bigg(\frac{s}{2^j}\bigg)ds\\ &=C \int^1_0 t^{N-1} h(t)dt \,\,\int^1_0 t^{N-1}g(t)dt. \endalign $$ \noindent The proof of Theorem 4 is complete. \medskip Returning to Corollary 7, we still have to prove that, for any fixed $\delta > 0$, we have, for small $\varep >0$, $$ \|f_\varep\|_{W^{1-\delta,p}} \leq C. $$ Considering the same functions $F_\varep(t)$ as above (relative to the parameter $\varep$ instead of $n$) we have to prove that $$ \int^1_0 \frac{F_\varep(t)}{t^{(1-\delta)p+1}} dt \le C, \text{ for small } \varep >0,\tag 32 $$ \noindent under the assumption $$ \varep\,\int^1_{0}\frac{F_\varep(t)}{t^{(1-\varep)p+1}} dt \, \leq C.\tag 33 $$ \noindent The proof of (32) is similar to that of Lemma 2, so we just sketch it. We start by rewriting (32) and (33) as $$ \int^1_0\frac{1}{t^{1-\delta p}}\frac{F_\varep(t)}{t^p}dt \leq C \tag34 $$ \noindent and $$ \int^1_0\frac{1}{t^{1-\delta p}}\frac{F_\varep(t)}{t^p} \frac{\varep}{t^{(\delta -\varep )p}}dt \leq C.\tag 35 $$ \noindent We apply Lemma 2 with $\delta =1$, $N=\delta p$, $g(t) = \frac{F_\varep(t)}{t^p}$, $h (t) = \frac{\varep}{t^{(\delta -\varep)p}}$, and take $ 0<\varep < \delta$. We find $$ \int^1_0\frac{1}{t^{1-\delta p}}\frac{F_\varep(t)}{t^p}\,\frac{\varep}{t^{(\delta -\varep )p}}dt\geq C \int^1_0\frac{1}{t^{1-\delta p}}\frac{F_\varep (t)}{t^p}dt \tag 36 $$ \noindent for some $C$ depending on $\delta$ and $p$, but not on $\varep$. \medskip \noindent \remark{Remark 6} If we renorm the $W^{s,p}(\Omega)$ spaces by $$ |f|_{W^{s,p}}^p \quad =\,\,\cases(1-s) \|f\|_{W^{s,p}}^p, \,\,&\,0 < s < 1\\ \|f\|_{W^{1,p}}^p, &\,s = 1,\endcases $$ \noindent the above computation yields $$ |f|_{W^{\sigma,p}}\,\,\leq\,\,C |f|_{W^{s,p}},\quad 0 < \sigma < s\leq 1 $$ \noindent for some constant $C$ {\bf independent of $s$ and $\sigma$}.\endremark \medskip \noindent {\bf Counterexample 1:} a sequence $(f_n)$ unbounded in $L^p$ and a sequence of radial mollifiers$(\rho_n)$ such that $$ \int\limits_{\Omega}\int\limits_{\Omega}\frac{|f_n(x)-f_n(y)|^p}{|x-y|^p}\rho_n(x-y)dxdy \, \leq C. \tag 37 $$ We take $\Omega = (0,1)$. Fix some function $f \in L_{loc}^p (\Bbb R),$ non-constant, periodic of period 1, such that $$ \int^1_0 f(x)dx=0\,\, (\text{ e.g. },f(x)= \text{ sin }(2\pi x)). $$ \noindent Define $g_n(x)=f(nx)$, so that $\|g_n\|_{L^p(\Omega )}^p = \int^1_0|f(x)|^p dx = C$. \medskip \noindent Clearly, $\int^1_0|g_n(x \pm\frac{1}{n})-g_n(x)|^p dx = 0$. Since the translations are continuous in $L^p$, we may find some $ 0 < \delta_n < \frac{1}{2n}$ such that $\overset {1}\to{\underset {0}\to{\int}}|g_n(x+h)-g_n(x)|^p dx \leq \frac{1}{n^{2p}}$ for $|h\pm\frac{1}{n}| < \delta_n$. \medskip \noindent Let $\rho_n = \frac{1}{4\delta_n}(\chi_{(\frac{1}{n}- \delta_n,\frac{1}{n}+ \delta_n)}\,\,+\,\,\chi_{(-\frac{1}{n} -\delta_n,-\frac{1}{n} + \delta_n)})$. Then clearly $$ \int\limits_{\Omega}\int\limits_{\Omega}\frac{|g_n(x)-g_n(y)|^p}{|x-y|^p}\rho_n(x-y)dxdy\,\,\leq\,\,\frac{C}{n^p}. $$ \noindent Finally, the functions $f_n = ng_n$ satisfy the desired inequality (37) and $\| f_n\|_{L^p(\Omega )}\sim n$. \medskip \noindent {\bf Counterexample 2:} the sequence $(g_n)$ constructed above is bounded in $L^p$, is not relatively compact in $L^p$, and yet it satisfies $$ \int\limits_{\Omega}\int\limits_{\Omega}\frac{|g_n(x)-g_n(y)|^p}{|x-y|^p} \rho_n(x-y)dx dy \,\,\leq C. $$ \magnification=\magstep1 \Refs \widestnumber\no{999} \ref \no [1] \by R. A. Adams \paper Sobolev spaces \jour Acad. Press \vol \issue \yr 1975 \pages \endref \ref \no [2] \by J. Bourgain, H. Brezis and P. Mironescu \paper Lifting in Sobolev spaces \jour J. d'Analyse \vol 80 \issue \yr 2000 \pages 37--86 \endref \ref \no [3] \by J. Bourgain, H. Brezis and P. Mironescu \paper \jour in preparation \vol \issue \yr \pages \endref \ref \no [4] \by H. 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