Triebel-Lizorkin spaces on metric spaces via hyperbolic fillings

Abstract

We give a new characterization of (homogeneous) Triebel-Lizorkin spaceṡ F s p,q (Z) in the smoothness range 0 < s < 1 for a fairly general class of metric measure spaces Z. The characterization uses Gromov hyperbolic fillings of Z. This gives a short proof of the quasisymmetric invariance of these spaces in case Z is Q-Ahlfors regular and sp = Q > 1. We also obtain first results on complex interpolation for these spaces in the framework of doubling metric measure spaces.

Traces of Besov, Triebel-Lizorkin and Sobolev Spaces on Metric Spaces

“…Proof. The analogous results for the homogeneous versions of these spaces can be found in [2,Theorem 3.3] and [34,Theorem 3.2]. By Proposition 5.2, it thus suffices to verify that the sequence (T Z n (P f )) n≥0 converges to f in the quasinorm of L p (Z).…”

“…The construction referred to above as the hyperbolic filling of Z is roughly speaking a graph (X, E) such that if Z is nice enough and (X, E) is endowed with its natural path metric, (X, E) is hyperbolic in the sense of Gromov and its boundary at infinity coincides with Z. We refer to the introduction section of [2] for a detailed explanation of the motivation of this construction in the context of function spaces. We shall next explain the actual construction of (X, E).…”

“…We also impose a similar restriction on the parameter q for the spacesḞ s p,q (Z) because of certain technical reasons which are common in the study of these spaces; see e.g. the proofs in [2] or [36] for more information.…”

“…[14,3,13,8,26] and the references therein, although this list is by no means comprehensive. Especially the full scales of these spaces in the setting of doubling metric spaces were introduced in [26], and in this paper we shall work with the equivalent definitions given in [2,34] in terms of the "hyperbolic fillings" of the underlying metric space -the actual definitions are given in the next section.…”

Littlewood-Paley Characterizations of Hajłasz-Sobolev and Triebel-Lizorkin Spaces via Averages on Balls

“…Thus, these pointwise characterizations also lead to some new pointwise characterizations of (fractional) Haj lasz-Sobolev spaces in spirit of [7], which are different from those obtained in [12,13,17,31]. Recall that the pointwise characterizations of Besov and Triebel-Lizorkin spaces play important and key roles in the study for the invariance of these function spaces under quasi-conformal mappings; see, for example, [20,11,16,18,2].…”