Characterization of Sobolev and BV spaces. (English) Zbl 1236.46032
J. Funct. Anal. 261, No. 10, 2926-2958 (2011); corrigendum ibid. 266, No. 2, 1106-1114 (2014).
The main purpose of this paper is to characterize the spaces \(W^{1,p}(\Omega)\), \(1<p<\infty\), and \(\text{BV}(\Omega)\) for arbitrary domains \(\Omega\subset\mathbb{R}^N\). The authors answer the following question posed by H. Brézis [Russ. Math. Surv. 57, No. 4, 693–708 (2002); translation from Usp. Mat. Nauk 57, No. 4, 59–74 (2002; Zbl 1072.46020)]: If \(f\in L^p(\Omega)\) and
\[
\limsup_{\varepsilon\to 0}\;\int_\Omega\,\int_\Omega{|f(x)- f(y)|^p\over d_\Omega(x,y)^p} \widehat\rho_\varepsilon(d_\Omega(x,y))\,dx\,dy<+\infty,
\]
does this imply that \(f\) belongs to \(W^{1,p}(\Omega)\)? Here \(\rho_\varepsilon\) is a family of mollifiers satisfying certain conditions and \(\rho_\varepsilon(x)= \widehat\rho_\varepsilon(|x|)\), \(x\in\mathbb{R}^N\).
Reviewer: Ludvík Janoš (Claremont)
MSC:
| 46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
| 26B30 | Absolutely continuous real functions of several variables, functions of bounded variation |
Citations:
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