Explicit interpolation bounds between hardy space and L2

H. Q. Bui, R. S. Laugesen

Research output: Contribution to journalArticle

Abstract

Every bounded linear operator that maps H1 to L1 and L2 to L2 is bounded from Lp to Lp for each p ∈ (1, 2), by a famous interpolation result of Fefferman and Stein. We prove Lp-norm bounds that grow like O(1=(p - 1)) as p 1. This growth rate is optimal, and improves significantly on the previously known exponential bound O(21/(p-1)). For p ∈ (2, ∞), we prove explicit Lp estimates on each bounded linear operator mapping L to bounded mean oscillation (BMO) and L2 to L 2. This BMO interpolation result implies the H1 result above, by duality. In addition, we obtain stronger results by working with dyadic H1 and dyadic BMO. The proofs proceed by complex interpolation, after we develop an optimal dyadic 'good lambda' inequality for the dyadic #-maximal operator.

Original languageEnglish (US)
Pages (from-to)158-168
Number of pages11
JournalJournal of the Australian Mathematical Society
Volume95
Issue number2
DOIs
StatePublished - Oct 1 2013

Keywords

  • BMO
  • Bounded mean oscillation
  • Interpolation of operators

ASJC Scopus subject areas

  • Mathematics(all)

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