Abstract
It is well known that Lipschitz spaces on the torus are an algebra. It is no more the case in the non compact situation because of the behavior at infinity. This is a companion article to Bonami et al. (J Math Pures Appl (9) 131:130–170, 2019), where pointwise multipliers on Lipschitz spaces on \({\mathbb {R}}^n\) are characterized for non-integer values of the parameter. In this article, the authors first establish two equivalent characterizations of a modified Zygmund space, and then characterize the pointwise multipliers on Lipschitz spaces on \({\mathbb {R}}^n\) for the integer values of the parameter, in particular, for the Zygmund class, via the intersection of the Lebesgue space \(L^\infty ({\mathbb {R}}^n)\) and the modified Zygmund space. This result can be used to show that the bilinear decomposition of the pointwise product of the Hardy space and its dual, in the integer values of the parameter, obtained in the aforementioned reference is sharp in the dual space sense.
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Acknowledgements
The work of Guido Weiss, particularly [10, 11, 24, 25], was a source of inspiration for us. We all are very deeply indebted to him. From Aline Bonami: First I would like to express here my gratitude to Guido as a mathematician. All along the years mathematical discussions and collaboration with him were extremely fruitful for me. But one cannot speak of Guido without talking of his particular talent to foster friendship around him. I am proud that he considers me as part of the vast family of mathematician friends that he gathered around him, those who see Washington University as a second home. Thank you Guido! From Dachun Yang: I visited Guido from September 1994 to August 1995 and, till now, I still remember his smile and those jokes given in the seminar during that period. His kindness will stay with me for my whole life. The authors would also like to thank both referees for their carefully reading and many helpful comments which indeed improve the presentation of this article.
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This article is dedicated to Professor Guido Weiss.
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This project is supported by the National Natural Science Foundation of China (Grant Nos. 11761131002, 11771446, 11971058, 11671185 and 11871100).
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Bonami, A., Liu, L., Yang, D. et al. Pointwise Multipliers of Zygmund Classes on
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J Geom Anal 31, 8879–8902 (2021). https://doi.org/10.1007/s12220-020-00453-8
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DOI: https://doi.org/10.1007/s12220-020-00453-8
Keywords
- Pointwise multiplier
- Campanato space
- Zygmund class
- Musielak–Orlicz Campanato space
- Bilinear decomposition