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On the Relationship Between Two Kinds of Besov Spaces with Smoothness Near Zero and Some Other Applications of Limiting Interpolation

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Abstract

Using limiting interpolation techniques we study the relationship between Besov spaces \(\mathbf B ^{0,-1/q}_{p,q}\) with zero classical smoothness and logarithmic smoothness \(-1/q\) defined by means of differences with similar spaces \(B^{0,b,d}_{p,q}\) defined by means of the Fourier transform. Among other things, we prove that \(\mathbf B ^{0,-1/2}_{2,2}=B^{0,0,1/2}_{2,2}\). We also derive several results on periodic spaces \(\mathbf B ^{0,-1/q}_{p,q}(\mathbb {T})\), including embeddings in generalized Lorentz–Zygmund spaces and the distribution of Fourier coefficients of functions of \(\mathbf B ^{0,-1/q}_{p,q}(\mathbb {T})\).

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Acknowledgments

The authors would like to thank the referees for their comments. The authors have been supported in part by the Spanish Ministerio de Economía y Competitividad (MTM2013-42220-P). O. Domínguez has also been supported by the FPU Grant AP2012-0779 of the Ministerio de Economía y Competitividad, and F. Cobos by UCM-BS (GR3/14-910348).

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Authors and Affiliations

  1. Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad Complutense de Madrid, Plaza de Ciencias 3, 28040, Madrid, Spain

    Fernando Cobos & Óscar Domínguez

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  1. Fernando Cobos
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  2. Óscar Domínguez
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Correspondence to Fernando Cobos.

Additional information

Communicated by Winfried Sickel.

To the memory of Professor Manuel Antonio Fugarolas.

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Cobos, F., Domínguez, Ó. On the Relationship Between Two Kinds of Besov Spaces with Smoothness Near Zero and Some Other Applications of Limiting Interpolation. J Fourier Anal Appl 22, 1174–1191 (2016). https://doi.org/10.1007/s00041-015-9454-6

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  • DOI: https://doi.org/10.1007/s00041-015-9454-6

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