Generalized Lebesgue points for Haj{\l} 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 Haj{\l} 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--Haj\l asz, Orlicz--Haj\l asz and variable exponent Haj\l asz functions.