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Sharpness of Complex Interpolation on \(\alpha \)-Modulation Spaces

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Abstract

In this paper, we solve a long standing problem on the modulation spaces, \(\alpha \)-modulation spaces and Besov spaces. We establish sharp conditions for the complex interpolation between these function spaces. We show that no \(\alpha \)-modulation space \(M_{p,q}^{s,\alpha }\) can be regarded as the interpolation space between \(M_{p_1,q_1}^{s_1,\alpha _1}\) and \(M_{p_2,q_2}^{s_2,\alpha _2}\), unless \(\alpha _1\) is equal to \(\alpha _2\), essentially. Especially, our results show that the \(\alpha \)-modulation spaces can not be obtained by complex interpolation between modulation spaces and Besov spaces.

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Acknowledgments

The authors would like to sincerely appreciate Prof. H.G. Feichtinger for reading a preliminary version of the manuscript and making valuable comments. The authors are also thankful to the anonymous referee for having read the paper very carefully and giving very detailed comments, which made the present paper more valuable. This work was supported by the National Natural Foundation of China (Nos. 11371295, 11471041 and 11471288).

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Authors and Affiliations

  1. School of Mathematical Sciences, Xiamen University, Xiamen, 361005, People’s Republic of China

    Weichao Guo & Huoxiong Wu

  2. Department of Mathematics, University of Wisconsin-Milwaukee, Milwaukee, WI, 53201, USA

    Dashan Fan

  3. Department of Mathematics, Zhejiang University, Hangzhou, 310027, People’s Republic of China

    Guoping Zhao

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  1. Weichao Guo
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  2. Dashan Fan
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  3. Huoxiong Wu
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  4. Guoping Zhao
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Corresponding author

Correspondence to Guoping Zhao.

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Communicated by Hartmut Führ.

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Guo, W., Fan, D., Wu, H. et al. Sharpness of Complex Interpolation on \(\alpha \)-Modulation Spaces. J Fourier Anal Appl 22, 427–461 (2016). https://doi.org/10.1007/s00041-015-9424-z

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  • DOI: https://doi.org/10.1007/s00041-015-9424-z

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