Advanced Search

Query:

Fill in the form and click »Search«...

Format:

Display: entries per page entries

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

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Article Journal Journal

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Advanced Search

Zentralblatt MATH Berlin [Germany]