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Zbl 1059.46021
Hajłasz, Piotr
A new characterization of the Sobolev space.
(English)
[J] Stud. Math. 159, No. 2, 263-275 (2003). ISSN 0039-3223; ISSN 1730-6337/e

Let $u \in L^{1}_{\text{loc}} (\Bbb R^{n})$ and let $B$ be a ball in $\Bbb R^{n}$. Denote by $u_B$ the integral average of a function $u$ over the ball $B$, that is, $$u_B = |B|^{-1} \int_B u \, dx,$$ where $|B|$ stands for Lebesgue measure of $B$. Moreover, for $R \in (0, \infty)$, let $M_R u$ be the restricted Hardy-Littlewood maximal function given by $$M_R u (x) = \sup_{0 < r < R} |u|_{B(x,r)} ,\quad x \in \Bbb R^n.$$ The main result of the paper is the following characterization of the Sobolev space $W^{1,1} (\Bbb R^n)$: \par The function $u$ belongs to $W^{1,1} (\Bbb R^n)$ if and only if $u \in L^1 (\Bbb R^n)$ and there exist a non-negative function $g \in L^1 (\Bbb R^n)$ and a number $\sigma \geq 1$ such that $$|u (x) - u (y)| \leq |x - y|\, (M_{\sigma |x - y|} g (x) + M_{\sigma |x - y|} g (y)) \leqno (1)$$ Moreover, if $(1)$ holds, then $| \nabla u| \leq C (n, \sigma) \,g$ a.e. (Here $C (n, \sigma)$ stands for a~non-negative constant which depends only on $n$ and $\sigma$.) \par The proof of this result consist of two main steps: \par (i) To prove that $(1)$ implies the family of Poincaré type inequalities $$|u - u_B|_B \leq C r \,g_{3 \sigma B} \leqno (2)$$ for every ball $B$ of any radius $r$. \par (ii) To show that the collection of inequalities $(2)$ implies that $u \in W^{1,1} (\Bbb R^n)$ and that $|\nabla u| \leq c \,g$ a.e.
[Bohumír Opic (Praha)]
MSC 2000:
*46E35 Sobolev spaces and generalizations

Keywords: Sobolev spaces, Lebesgue spaces, Poincaré type inequalities, restricted Hardy-Littlewood maximal functions

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