Extensions of the vector-valued Hausdorff–Young inequalities

Oscar Dominguez, Mark Veraar

Research output: Contribution to journalArticleScientificpeer-review

Abstract

In this paper we study the vector-valued analogues of several inequalities for the Fourier transform. In particular, we consider the inequalities of Hausdorff–Young, Hardy–Littlewood, Paley, Pitt, Bochkarev and Zygmund. The Pitt inequalities include the Hausdorff–Young and Hardy–Littlewood inequalities and state that the Fourier transform is bounded from Lp(Rd, | · | βp) into Lq(Rd, | · | -γq) under certain condition on p, q, β and γ. Vector-valued analogues are derived under geometric conditions on the underlying Banach space such as Fourier type and related geometric properties. Similar results are derived for Td and Zd by a transference argument. We prove sharpness of our results by providing elementary examples on ℓp-spaces. Moreover, connections with Rademacher (co)type are discussed as well.

Original languageEnglish
Number of pages53
JournalMathematische Zeitschrift
DOIs
Publication statusPublished - 2021

Keywords

  • (co)type
  • Banach space
  • Bochkarev
  • Fourier type
  • Hardy–Littlewood
  • Hausdorff–Young
  • Limiting interpolation
  • Paley
  • Pitt
  • Vector-valued Fourier transform
  • Zygmund

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