Orthogonality Principle for Bilinear Littlewood–Paley Decompositions

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• 作者：Petr Honzík (1)
  
• 关键词：Fourier multipliers ; Bilinear operators ; Littlewood ; Paley theory ; Primary 42B20 ; Secondary 42E30
• 刊名：Journal of Fourier Analysis and Applications
• 出版年：2014
• 出版时间：December 2014
• 年：2014
• 卷：20
• 期：6
• 页码：1171-1178
• 全文大小：194 KB
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  5. Grafakos, L., Li, X.: The disc as a bilinear multiplier. Am. J. Math. 128(1), 91-19 (2006) CrossRef
  6. Miyachi, A., Tomita, N.: Minimal smoothness conditions for bilinear Fourier multipliers. Rev. Mat. Iberoam. 29(2), 495-30 (2013) CrossRef
  7. Seeger, A.: Some inequalities for singular convolution operators in \(L^p\) -spaces. Trans. Am. Math. Soc. 308(1), 259-72 (1988)
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• 作者单位：Petr Honzík (1)
  
  1. Department of Mathematical Analysis, Charles University, Sokolovská 83, 186 00, Prague 8, Czech Republic
  
• ISSN：1531-5851

文摘

We explore Littlewood–Paley like decompositions of bilinear Fourier multipliers. Grafakos and Li (Am. J. Math. 128(1):91-19 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.