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Journal of Fourier Analysis and Applications

, Volume 20, Issue 4, pp 801–815 | Cite as

\(L^p\)-Integrability, Dimensions of Supports of Fourier Transforms and Applications

  • K. S. Senthil RaaniEmail author
Article

Abstract

It is proved that there does not exist any non zero function in \(L^p({\mathbb R}^n)\) with \(1\le p\le 2n/\alpha \) if its Fourier transform is supported by a set of finite packing \(\alpha \)-measure where \(0<\alpha <n\). It is shown that the assertion fails for \(p>2n/\alpha \). The result is applied to prove \(L^p\) Wiener Tauberian theorems for \({\mathbb R}^n\) and \(M(2)\).

Keywords

Supports of Fourier Transform Hausdorff dimension Packing measure Salem sets Ahlfors–David regular sets Wiener Tauberian theorems 

Mathematics Subject Classification

Primary 42B10 Secondary 37F35 40E05 28A78 

Notes

Acknowledgments

The author wishes to thank both the referees for several suggestions and remarks which improved the presentation of the paper. The author wishes to thank Prof. Malabika Pramanik for useful discussions. The author is also grateful to Dr. E. K. Narayanan for his constant encouragement and for the many useful discussions during the course of this work. The author also wishes to thank UGC-CSIR for financial support. This work is supported in part by UGC Centre for Advanced Studies.

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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of ScienceBangaloreIndia

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