Relations among Besov-type spaces, Triebel–Lizorkin-type spaces and generalized Carleson measure spaces

Abstract

In this article, 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 Lin and Wang [C.-C. Lin and K.Wang, Equivalency between the generalized Carleson measure spaces and Triebel–Lizorkin-type spaces, Taiwanese J. Math. 15 (2011), pp. 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 Yuan et al. [W. Yuan, W. Sickel and D. Yang, Morrey and Campanato Meet Besov, Lizorkin and Triebel, Lecture Notes in Mathematics 2005, Springer-Verlag, Berlin, 2010]. In conclusion, the Triebel–Lizorkin-type space and the Besov-type space become the classical Besov spaces, when the fourth parameter is sufficiently large.

Keywords:

• Besov-type space
• Triebel–Lizorkin-type space
• generalized Carleson measure space

AMS Subject Classifications::

• Primary: 42B35
• Secondary: 46E35

Acknowledgements

The authors would like to thank Professor Marcin Bownik and Professor Winfried Sickel for some helpful discussions on the subject of this article. The authors also wish to express their sincere thanks to the referees for their very carefully reading and valuable remarks which improved the presentation of this article. D. Yang is supported by the National Natural Science Foundation (Grant No. 11171027) of China and Program for Changjiang Scholars and Innovative Research Team in University of China, and W. Yuan (corresponding) is supported by the National Natural Science Foundation (Grant No. 11101038) of China.