Lebesgue points, Hölder continuity and Sobolev functions

University essay from Matematiska institutionen

Author: John Karlsson; [2009]

Keywords: Lebesgue point; Hausdorff dimension; Hausdorff measure; Hölder continuity; Maximal function; Poincaré inequality; Sobolev space; Uniform continuity;

Abstract: This paper deals with Lebesgue points and studies properties of the set of Lebesgue points for various classes of functions. We consider continuous functions, L1 functions and Sobolev functions. In the case of uniformly continuous functions and Hölder continuous functions we develop a characterization in terms of Lebesgue points. For Sobolev functions we study the dimension of the set of non-Lebesgue points.

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