A new framework for generalized Besov-type and Triebel-Lizorkin-type spaces

By Yiyu Liang, Dachun Yang, Wen Yuan, Yoshihiro Sawano, Tino Ullrich. 2012

109 citations recorded by the source corpus.

In this paper, the authors propose a new framework under which a theory of generalized Besov-type and Triebel-Lizorkin-type function spaces is developed. Many function spaces appearing in harmonic analysis fall under the scope of this new framework. Among others, the boundedness of the Hardy-Littlewood maximal operator or the related vector-valued maximal function on any of these function spaces is not required to construct these generalized scales of smoothness spaces. Instead of this, a key idea used in this framework is an application of the Peetre maximal function. This idea originates from recent findings in the abstract coorbit space theory obtained by Holger Rauhut and Tino Ullrich. Under this new setting, the authors establish the boundedness of pseudo-differential operators based on atomic and molecular characterizations and also the boundedness of the Fourier multipliers. The characterizations of these function spaces by means of differences and oscillations are also established. As further applications of this new framework, the authors reexamine and polish some existing results for many different scales of function spaces.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/3155c97772040b6b653a70a51fbd7695baae96ef

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In Applied Mathematics

In Mathematical Physics

Citations in the corpus

Besov-Type Spaces with Variable Smoothness and Integrability (2015, unknown): “(iv) If q, s are constants and φ is as in (ii), then B s(·),φ p(·),q(·)(R ) = B p(·),q(R ), which was investigated in [43].”
Generalized Morrey spaces and trace operator (2016, methodology): “However, our result will be sharper; in the earlier work [30] the authors depended on the Peetre maximal operator but in the present paper we do not have to rely upon this maximal operator.”
Triebel-Lizorkin-Type Spaces with Variable Exponents (2015, unknown): “(iii) When q, s are constant exponents and φ(Q) := |Q|τ with τ ∈ [0,∞) for all cubes Q, then F s(·),φ p(·),q(·)(R ) = F s,τ p(·),q(R ), which was investigated in [47].”
Interpolation of Morrey-Campanato and related smoothness spaces (2014, unknown): “Concerning the mapping Φ : f 7→ {λk}k ∪ {λi,j,k}i,j,k the following is known (see [42]).”