Generalized Lebesgue points for Hajł asz functions

Abstract

Let $X$ be a quasi-Banach function space over a doubling metric measure space $\mathcal{P}$. Denote by ${\alpha }_X$ the generalized upper Boyd index of $X$. We show that if ${\alpha }_X<\infty$ and $X$ has absolutely continuous quasinorm, then quasievery point is a generalized Lebesgue point of a quasicontinuous Hajł asz function $u\in {\dot{M}}^{s,X}$. Moreover, if ${\alpha }_X<(Q+s)/Q$, then quasievery point is a Lebesgue point of $u$. As an application we obtain Lebesgue type theorems for Lorentz--Hajłasz, Orlicz--Hajłasz and variable exponent Hajłasz functions.