Weak type inequalities for the ℓ1-summability of higher dimensional Fourier transforms

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Abstract

Under some weak conditions on θ, it was verified in [21, 17] that the maximal operator of the ℓ 1-θ-means of a tempered distribution is bounded from H p(ℝd) to L p(ℝd) for all d/(d + α) <p ≤ ∞, where 0 <α ≤ 1 depends only on θ. In this paper, we prove that the maximal operator is bounded from H d/(d+α)(ℝd) to the weak L d/(d+α)(ℝd) space. The analogous result is given for Fourier series, as well. Some special cases of the ℓ 1-θ-summation are considered, such as the Weierstrass, Picard, Bessel, Fejér, de La Vallée-Poussin, Rogosinski and Riesz summations.

Original languageEnglish
Pages (from-to)297-320
Number of pages24
JournalAnalysis Mathematica
Volume39
Issue number4
DOIs
Publication statusPublished - Dec 2013

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Weak Type Inequality
Maximal Operator
Summability
Summation
Fourier transform
High-dimensional
Tempered Distribution
D-space
Friedrich Wilhelm Bessel
Fourier series

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

Weak type inequalities for the ℓ1-summability of higher dimensional Fourier transforms. / Weisz, F.

In: Analysis Mathematica, Vol. 39, No. 4, 12.2013, p. 297-320.

Research output: Contribution to journalArticle

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