Remarks on the Bourgain–Brezis–Mironescu Approach to Sobolev Spaces

B. Bojarski ¹

¹ Institute of Mathematics Polish Academy of Sciences 00-956 Warszawa, Poland

Bulletin of the Polish Academy of Sciences. Mathematics, Tome 59 (2011) no. 1, pp. 65-75

Résumé

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.

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DOI : 10.4064/ba59-1-8

Keywords: function loc mathbb notion p mean variation order nbsp mathsf mathbb defined generalizes concept nbsp riesz variation functions real line mathbb mathbb characterisation sobolev space mathbb terms mathsf mathbb directly related characterisation mathbb lipschitz type pointwise inequalities bojarski haj asz strzelecki bourgain brezis mironescu approach