Variable exponent Triebel-Lizorkin-Morrey spaces

Abstract

We introduce variable exponent versions of Morreyfied Triebel-Lizorkin spaces. To that end, we prove an important convolution inequality which is a replacement for the Hardy-Littlewood maximal inequality in the fully variable setting. Using it we obtain characterizations by means of Peetre maximal functions and use them to show the independence of the introduced spaces from the admissible system used.

Decompositions with Atoms and Molecules for Variable Exponent Triebel–Lizorkin–Morrey Spaces

“…In the case of r > 1 we use r n/u(x)−n/p(x) ≤ 1 and Lemma 2.2 again to obtain Next we state the convolution inequality proved in [5,Thm. 3.3], where M u(·) p(·) (ℓ q(·) ) stands for the set of all sequences (f ν ) ν∈N0 of (complex or extended real-valued) measurable functions on R n such that…”

“…We shall also need the following lemmas, which we have also already considered in [5]. For the meaning of · | M u(·) p(·) (ℓ q(·) ) , see Definition 4.3 below.…”

“…p(·),q(·) (R n ), e.g. the fact that it is a quasi-normed space, can be seen in [5], to which we refer the reader.…”

“…In [5] we have introduced the Triebel-Lizorkin-Morrey spaces E s(·),u(·) p(·),q(·) (R n ) and proved there an important convolution inequality, which, in particular, allowed us to show that they satisfy a Peetre maximal function characterization, and afterwards concluded the independence of the introduced spaces from the admissible system used.…”

“…This paper is a continuation of [5], where we have introduced and presented some properties of the Triebel-Lizorkin-Morrey spaces E s(·),u(·) p(·),q(·) (R n ), which mix two recent trends in the literature, in this case starting from the Triebel-Lizorkin spaces F s p,q (R n ):…”