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Orthogonality Principle for Bilinear Littlewood–Paley Decompositions

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Abstract

We explore Littlewood–Paley like decompositions of bilinear Fourier multipliers. Grafakos and Li (Am. J. Math. 128(1):91–119 2006) showed that a bilinear symbol supported in an angle in the positive quadrant is bounded from \(L^p\times L^q\) into \(L^r\) if its restrictions to dyadic annuli are bounded bilinear multipliers in the local \(L^2\) case \(p\ge 2\), \(q\ge 2\), \(r= 1/(p^{-1}+q^{-1})\le 2\). We show that this range of indices is sharp and also discuss similar results for multipliers supported near axis and negative diagonal.

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Acknowledgments

The author was supported by the ERC CZ Grant LL1203 of the Czech Ministry of Education. The author is a researcher in the University Centre for Mathematical Modelling, Applied Analysis and Computational Mathematics (Math MAC).

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Authors and Affiliations

  1. Department of Mathematical Analysis, Charles University, Sokolovská 83, 186 00, Prague 8, Czech Republic

    Petr Honzík

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  1. Petr Honzík
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Correspondence to Petr Honzík.

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Communicated by Rodolfo H. Torres.

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Honzík, P. Orthogonality Principle for Bilinear Littlewood–Paley Decompositions. J Fourier Anal Appl 20, 1171–1178 (2014). https://doi.org/10.1007/s00041-014-9350-5

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  • DOI: https://doi.org/10.1007/s00041-014-9350-5

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