Generalized Lebesgue Points for Hajłasz Functions

Abstract

Let X be a quasi-Banach function space over a doubling metric measure space P. Denote by α X the generalized upper Boyd index of X. We show that if α X < ∞ and X has absolutely continuous quasinorm, then quasievery point is a generalized Lebesgue point of a quasicontinuous Haj lasz function u ∈Ṁ s,X . Moreover, if α X < (Q + s)/Q, then quasievery point is a Lebesgue point of u. As an application we obtain Lebesgue type theorems for Lorentz-Haj lasz, Orlicz-Haj lasz and variable exponent Haj lasz functions.(1.3) lim r→0 m γ u (B(x, r)) = u(x),for every x ∈ P \ E and 0 < γ ≤ 1/2.

Approximation by Hölder Functions in Besov and Triebel–Lizorkin Spaces

“…where t > Q/(Q + s) and 0 < s ′ < s, is known to hold. The estimate (3.8) can be proven using a chaining argument and a Sobolev-Poincaré type inequality from [9], see [16,Lemma 4.2]. Somewhat surprisingly, with medians, a better estimate (3.7) follows by a completely elementary argument.…”

“…Property (8) above says that medians of small balls behave like integral averages of locally integrable functions on Lebesgue points. Recently, in [16], it was shown that for Haj lasz-Besov and Haj lasz-Triebel-Lizorkin functions, the limit in (8) exists outside a set of capacity zero. Note also that if u ∈ L p (A), p > 0, then by properties (7) and (8),…”

Lebesgue Points of Besov and Triebel–Lizorkin Spaces with Generalized Smoothness

“…As u has certain regularity, one would expect a smaller exceptional set than that of usual locally integrable functions. Inspired by [41,45], we introduce capacities related, respectively, to M φ p,q (X) and N φ p,q (X) to measure such exceptional sets.…”

“…Let u ∈ L 0 (X). Recall that a point x ∈ X is called a generalized Lebesgue point of u if (56) holds true for x and any γ ∈ (0, 1/2]; see, for instance [41,44,45]. If u is locally integrable, as was pointed by ( [46], p. 231), any Lebesgue point of u is a generalized Lebesgue point of u.…”

“…Note that functions in the Hajłasz-Besov or Hajłasz-Triebel-Lizorkin spaces with generalized smoothness might fail to be locally integrable when their index p or q is close to zero. To overcome this obstacle, similar to [41,44,45], we also consider a class of generalized Lebesgue points, which are defined via the γ-medians introduced in [46,47], instead of the classical integrals. As the main results of this article, we prove that the exceptional sets of (generalized) Lebesgue points of functions from the above spaces have zero capacity, where those capacities are defined by related spaces.…”