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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:
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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