Measure Density and Extension of Besov and Triebel–Lizorkin Functions

Abstract

Abstract. We show that a domain is an extension domain for a Haj lasz-Besov or for a Haj lasz-Triebel-Lizorkin space if and only if it satisfies a measure density condition. We use a modification of the Whitney extension where integral averages are replaced by median values, which allows us to handle also the case 0 < p < 1. The necessity of the measure density condition is derived from embedding theorems; in the case of Haj lasz-Besov spaces we apply an optimal Lorentz-type Sobolev embedding theorem which we prove using a new interpolation result. This interpolation theorem says that Haj lasz-Besov spaces are intermediate spaces between L p and Haj laszSobolev spaces. Our results are proved in the setting of a metric measure space, but most of them are new even in the Euclidean setting, for instance, we obtain a characterization of extension domains for classical Besov spaces B s p,q , 0 < s < 1, 0 < p < ∞, 0 < q ≤ ∞, defined via the L p -modulus of smoothness of a function.

Lower bound of measure and embeddings of Sobolev, Besov and Triebel–Lizorkin spaces

“…If we assume that the space (X, d, µ) is a Q-doubling space, Q > 1, then the above theorem deals with the case sp < Q for the function spaces M s p,q (X) and N s p,q (X). For the cases sp = Q and sp > Q, we can get the same result as Theorem 2.5 for M s p,q (X) and N s p,q (X) by mimicking the proof of the same and by using Corollary 3.12 of [HIT16]. In that case we assume that the space is connected and Q-doubling; the same result in a geodesic space is due to [Kar].…”

“…Now we will consider the spaces M s p,q (X), N s p,q (X) and prove a similar result as in [Gór17]. We will need the following lemma, which is proved using the idea of the proof of Lemma 3.10 of [HIT16].…”

“…Haj lasz-Besov spaces and Haj lasz-Trieble-Lizorkin spaces have been studied in [HIT16] and [Kar]. Recently, Górka in [Gór17] has proved that in a metric measure space (X, d, µ), where µ is a doubling measure, if the embedding M 1,p (X) ֒→ L p * (X) holds for any p * > p, then there exists b > 0 such that for any x ∈ X and 0 < r ≤ 1, the following inequality holds µ(B(x, r)) ≥ br Q ,…”

Generalized Lebesgue Points for Hajłasz Functions

“…Operator M γ = M γ ∞ and its variants have turned out to be useful in harmonic analysis and in the theory of function spaces, see for example [7], [8], [9], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [27], [28], [29], [32], [36], [37], [39]. Theorem 1.1.…”

A median approach to differentiation bases

“…where f : R n → [−∞, ∞] is a measurable function with |f (x)| < ∞ for almost every x ∈ R n . Medians and related maximal functions have turned out to be useful in harmonic analysis and function spaces, see [3], [4], [5], [8], [9], [11], [13], [14], [15], [16], [17], [18], [24], [26], [27], [28], [30], [31], [33]. The main advantage of a median over an integral average is that it applies also when the function is not necessarily locally integrable.…”