Inclusion of Hajłasz – Sobolev class Mpα(X) into the space of continuous functions in the critical case

Abstract

Let (X, d, µ) be a doubling metric measure space with doubling dimension γ, i. e. for any balls B(x, R) and B(x, r), r < R, following inequality holds µ(B(x, R)) ≤ aµ (R/r)γµ(B(x, r)) for some positive constants γ and aµ. Hajłasz – Sobolev space Mpα(X) can be defined upon such general structure. In the Euclidean case Hajłasz – Sobolev space coincides with classical Sobolev space when p > 1, α = 1. In this article we discuss inclusion of functions from Hajłasz – Sobolev space Mpα(X) into the space of continuous functions for p ≤ 1 in the critical case γ = α p. More precisely, it is shown that any function from Hajłasz – Sobolev class Mpα(X), 0 < p ≤ 1, α > 0, has a continuous representative in case of uniformly perfect space (X, d, µ).