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

B. Bojarski — 2011

Bulletin of the Polish Academy of Sciences. Mathematics

For a function $f \in L_{l o c}^{p} (ℝ ⁿ)$ the notion of p-mean variation of order 1, $₁^{p} (f, ℝ ⁿ)$ is defined. It generalizes the concept of F. Riesz variation of functions on the real line ℝ¹ to ℝⁿ, n > 1. The characterisation of the Sobolev space $W^{1, p} (ℝ ⁿ)$ in terms of $₁^{p} (f, ℝ ⁿ)$ is directly related to the characterisation of $W^{1, p} (ℝ ⁿ)$ by Lipschitz type pointwise inequalities of Bojarski, Hajłasz and Strzelecki and to the Bourgain-Brezis-Mironescu approach.