Frame decomposition and radial maximal semigroup characterization of Hardy spaces associated to operators

Xuan Thinh Duong, Ji Li, Liang Song, Lixin Yan

Research output: Contribution to journalArticleResearchpeer-review

Abstract

Let L be the generator of an analytic semigroup whose kernels satisfy Gaussian upper bounds and Hölder's continuity. Also assume that L has a bounded holomorphic functional calculus on L2 (Rn ). In this paper, we construct a frame decomposition for the functions belonging to the Hardy space HL1 (Rn ) associated to L, and for functions in the Lebesgue spaces Lp , 1<p<∞. We then show that the corresponding HL1 (R n )-norm (resp. Lp (Rn )-norm) of a function f in terms of the frame coefficients is equivalent to the HL1 (Rn )-norm (resp. Lp (Rn )-norm) of f. As an application of the frame decomposition, we establish the radial maximal semigroup characterization of the Hardy space HL1 (Rn ) under the extra condition of Gaussian upper bounds on the gradient of the heat kernels of L.

LanguageEnglish
Pages45-85
Number of pages41
JournalJournal of Approximation Theory
Volume243
DOIs
Publication statusPublished - 1 Jul 2019

Fingerprint

Hardy Space
Semigroup
Lp-norm
Decomposition
Decompose
Operator
Upper bound
Norm
Analytic Semigroup
Functional Calculus
Hölder Continuity
Lebesgue Space
Heat Kernel
Generator
kernel
Gradient
Coefficient

Keywords

  • Frame decomposition
  • Functional calculus
  • Gaussian estimate
  • Hardy space
  • Heat semigroup
  • Radial maximal function

Cite this

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Frame decomposition and radial maximal semigroup characterization of Hardy spaces associated to operators. / Duong, Xuan Thinh; Li, Ji; Song, Liang; Yan, Lixin.

In: Journal of Approximation Theory, Vol. 243, 01.07.2019, p. 45-85.

Research output: Contribution to journalArticleResearchpeer-review

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