On Pointwise ℓr -Sparse Domination in a Space of Homogeneous Type

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Abstract

We prove a general sparse domination theorem in a space of homogeneous type, in which a vector-valued operator is controlled pointwise by a positive, local expression called a sparse operator. We use the structure of the operator to get sparse domination in which the usual ℓ1-sum in the sparse operator is replaced by an ℓr-sum. This sparse domination theorem is applicable to various operators from both harmonic analysis and (S)PDE. Using our main theorem, we prove the A2-theorem for vector-valued Calderón–Zygmund operators in a space of homogeneous type, from which we deduce an anisotropic, mixed-norm Mihlin multiplier theorem. Furthermore, we show quantitative weighted norm inequalities for the Rademacher maximal operator, for which Banach space geometry plays a major role.

Original languageEnglish
Pages (from-to)1-40
Number of pages40
JournalJournal of Geometric Analysis
DOIs
Publication statusE-pub ahead of print - 2020

Keywords

  • Mihlin multiplier theorem
  • Muckenhoupt weight
  • Rademacher maximal operator
  • Singular integral operator
  • Space of homogeneous type
  • Sparse domination

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