Even Fourier multipliers and martingale transforms in infinite dimensions

Ivan S. Yaroslavtsev

Research output: Contribution to journalArticleScientificpeer-review

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Abstract

In this paper we show sharp lower bounds for norms of even homogeneous Fourier multipliers in L(Lp(Rd;X)) for 1<p<∞and for a UMD Banach space X in terms of the range of the corresponding symbol. For example, if the range contains a1,…,aN∈C, then the norm of the multiplier exceeds ‖a1R1 2+⋯+aNRN 2L(Lp(RN;X)), where Rn is the corresponding Riesz transform. We also provide sharp upper bounds of norms of Bañuelos–Bogdan type multipliers in terms of the range of the functions involved. The main tools that we exploit are A-weak differential subordination of martingales and UMDp A constants, which are introduced here.


Original languageEnglish
Pages (from-to)1290-1309
Number of pages20
JournalIndagationes Mathematicae
Volume29
Issue number5
DOIs
Publication statusPublished - 2018

Keywords

  • Even Fourier multipliers
  • Martingale transforms
  • UMD Banach spaces
  • Weak differential subordination

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