Remarks on the Bourgain–Brezis–Mironescu Approach to Sobolev Spaces
Volume 59 / 2011
Bulletin Polish Acad. Sci. Math. 59 (2011), 65-75
MSC: 46E35.
DOI: 10.4064/ba59-1-8
Abstract
For a function $f\in L_{\rm loc}^p(\mathbb R^n)$ the notion of $p$-mean variation of order 1, $\mathsf{V}^{p}_{1} (f,\mathbb R^n)$ is defined. It generalizes the concept of F. Riesz variation of functions on the real line $\mathbb R^1$ to $\mathbb R^n$, $n>1$. The characterisation of the Sobolev space $W^{1,p}(\mathbb R^n)$ in terms of $\mathsf{V}^{p}_{1}(f,\mathbb R^n)$ is directly related to the characterisation of $W^{1,p}(\mathbb R^n)$ by Lipschitz type pointwise inequalities of Bojarski, Hajłasz and Strzelecki and to the Bourgain–Brezis–Mironescu approach.
