Generalized Lebesgue Points for Hajłasz Functions

Let X be a quasi-Banach function space over a doubling metric measure space P. Denote by αX the generalized upper Boyd index of X. We show that if αX<∞ and X has absolutely continuous quasinorm, then quasievery point is a generalized Lebesgue point of a quasicontinuous Hajłasz function u∈M˙s,X. Moreover, if αX<(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.