Relations among Besov-Type Spaces, Triebel-Lizorkin-Type Spaces and Generalized Carleson Measure Spaces

TL;DR

This paper investigates the relationships among Besov-type, Triebel-Lizorkin-type, and generalized Carleson measure spaces, providing counterexamples to previous claims of their equivalence and identifying conditions under which they coincide.

Contribution

It constructs counterexamples disproving the equivalence of these spaces for certain parameters and clarifies conditions where they coincide with Triebel-Lizorkin spaces.

Findings

Counterexamples show non-equivalence for certain parameters.

Conditions identified where spaces coincide with Triebel-Lizorkin spaces.

Answers a previously posed open question about space relationships.

Abstract

In this paper, the authors construct some counterexamples to show that the generalized Carleson measure space and the Triebel-Lizorkin-type space are not equivalent for certain parameters, which was claimed to be true in [Taiwanese J. Math. 15 (2011), 919-926]. Moreover, the authors show that for some special parameters, the generalized Carleson measure space, the Triebel-Lizorkin-type space and the Besov-type space coincide with certain Triebel-Lizorkin space, which answers a question posed in Remark 6.11(i) of [Lecture Notes in Mathematics 2005, Springer-Verlag, Berlin, 2010, xi+281 pp.].