Sobolev spaces revisited
Haïm Brézis, Andreas Seeger, Jean Van Schaftingen, Po‐Lam Yung
Abstract
We describe a recent, one-parameter family of characterizations of Sobolev and BV functions on \mathbb{R}^n n, using sizes of superlevel sets of suitable difference quotients. This provides an alternative point of view to the BBM formula by Bourgain, Brezis, and Mironescu, and complements in the case of BV some results of Cohen, Dahmen, Daubechies, and DeVore about the sizes of wavelet coefficients of such functions. An application towards Gagliardo–Nirenberg interpolation inequalities is then given. We also establish a related one-parameter family of formulae for the L^p norm of functions in L^p(\mathbb{R}^n) .
Topics & Concepts
MathematicsSobolev spaceSobolev inequalityInterpolation spacePure mathematicsFunctional analysisChemistryBiochemistryGeneAdvanced Numerical Methods in Computational MathematicsNonlinear Partial Differential Equations