Lebesgue points for functions from generalized Sobolev classes Mpa(X) in the critical case

Abstract

Classical Lebesgue theorem states that for any integrable function almost every point (except the set of measure zero) is a Lebesgue point. The set of the points that are not Lebesgue points is called an exceptional set. One can estimate the «size» of the exceptional set for more regular functions (e. g. functions that belong to certain function space) using more refined than measure characteristics. The paper is devoted to the investigation of the properties of Lebesgue points for functions from Sobolev classes on general metric space in the critical case γ = α p, γ plays the role of the dimension of the space, α, p – smoothness and summability parameters. Estimates of the «size» of the exceptional set in terms of capacities and Hausdorff dimension are obtained. Exponential rate of convergence for Lebesgue points has been established. Similar results are known in subcritical case γ > α p as well.

Keywords

• analysis on metric measure spaces
• sobolev spaces
• fine properties of functions
• lebesgue points