Besov‐Morrey spaces and Triebel‐Lizorkin‐Morrey spaces for nondoubling measures

Abstract

We define Morrey type Besov-Triebel spaces with the underlying measure non-doubling. After defining the function spaces, we investigate boundedness property of some class of the singular integral operators.

Characterizations of Morrey type Besov and Triebel–Lizorkin spaces with variable exponents

“…Many results paralleled with the theory of standard Besov and Triebel-Lizorkin spaces have been obtained and some new applications have also been given; see, for example, [1,2,[26][27][28][29][30][35][36][37][38][39][40]42,48]. Actually, Mazzucato in [27] established some decomposition of Morrey type Besov spaces, which are called Besov-Morrey spaces, in terms of smooth wavelets, molecules concentrated on dyadic cubes, and atoms supported on dyadic cubes.…”

Continuous embeddings of Besov-Morrey function spaces

“…Kozono and Yamazaki [20] and Mazzucato [21] used these spaces in the theory of Navier-Stokes equations. Some properties of the spaces including the wavelet characterisations were described in the papers by Sawano [22][23][24], Sawano and Tanaka [25,26], Tang and Xu [27]. The most systematic and general approach can certainly be found in the very recent book [28] of Yuan et al, we also recommend this monograph for further up-to-date references on this subject.…”

“…We follow the ideas of Tang and Xu [27], where a somewhat different definition is proposed. The ideas were developed by Sawano and Tanaka [22,23,25,26].…”

Generalized Morrey spaces and trace operator

“…Najafov also considered the embedding results for Sobolev-Morrey spaces in [38]. Sawano and Tanaka considered the complex interpolation of Morrey spaces, Besov-Morrey spaces in [61] but there was a mistake in [61,Proposition 5.3]. The method introduced in the book [2] was not used in [61].…”

“…Sawano and Tanaka considered the complex interpolation of Morrey spaces, Besov-Morrey spaces in [61] but there was a mistake in [61,Proposition 5.3]. The method introduced in the book [2] was not used in [61]. Yuan, Sickel and Yang overcame this problem in [97].…”