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A new characterization of the Sobolev space. (English) Zbl 1059.46021
Let $$u \in L^{1}_{\text{loc}} (\mathbb R^{n})$$ and let $$B$$ be a ball in $$\mathbb R^{n}$$. 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 $M_R u (x) = \sup_{0 < r < R} |u|_{B(x,r)} ,\quad x \in \mathbb R^n.$ The main result of the paper is the following characterization of the Sobolev space $$W^{1,1} (\mathbb R^n)$$:
The function $$u$$ belongs to $$W^{1,1} (\mathbb R^n)$$ if and only if $$u \in L^1 (\mathbb R^n)$$ and there exist a non-negative function $$g \in L^1 (\mathbb R^n)$$ and a number $$\sigma \geq 1$$ such that $|u (x) - u (y)| \leq |x - y|\, (M_{\sigma |x - y|} g (x) + M_{\sigma |x - y|} g (y)) (1)$ 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 $$\sigma$$.)
The proof of this result consist of two main steps:
(i) To prove that $$(1)$$ implies the family of Poincaré type inequalities $|u - u_B|_B \leq C r \,g_{3 \sigma B} (2)$ for every ball $$B$$ of any radius $$r$$.
(ii) To show that the collection of inequalities $$(2)$$ implies that $$u \in W^{1,1} (\mathbb R^n)$$ and that $$|\nabla u| \leq c \,g$$ a.e.

##### MSC:
 4.6e+36 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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