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\def\varep{\varepsilon}
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\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  $1<p<\infty$) or  $f \in BV (\Omega )$
( for $ p=1)$.\endproclaim
\noindent
\proclaim{Corollary 7}  For $1<p<\infty$, let $f_\varep \in
W^{1-\varep,p}(\Omega )$.  Assume that 

$$ 
\varep \|f_\varep\|_{W^{1-\varep,p}}^p \leq C_0.
$$
\noindent
Then, up to a subsequence, $(f_\varep)$ converges in $L^p(\Omega )$ (and, in
fact, in $W^{1 - \delta,p}(\Omega )$, for all $\delta >0$) 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

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