Several dimensional θ-summability and Hardy spaces

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16 Citations (Scopus)

Abstract

The d-dimensional Hardy spaces Hp(Td1 × . . . × Tdk) (d = d1 + . . . + dk) and a general summability method of Fourier series and Fourier transforms are introduced with the help of integrable functions θj having integrable Fourier transforms. Under some conditions on θj we show that the maximal operator of the θ-means of a distribution is bounded from Hp (Td1 × . . . × Tdk) to Lp (Td) where p0 < p < ∞ and p0 < 1 is depending only on the functions θj. By an interpolation theorem we get that the maximal operator is also of weak type (H#i1, L1) (i = 1, . . . , k) where the Hardy space H#i1 is defined by a hybrid maximal function and H#i1 = L1 if k = 1. As a consequence we obtain that the θ-means of a function f ∈ H#i1 ⊃ L(log L)k-1 converge a. e. to the function in question. If k = 1 then we get this convergence result for all f ∈ L1. Moreover, we prove that the θ-means are uniformly bounded on the spaces Hp (Td1 × . . . × Tdk) whenever p0 < p < ∞, thus the θ-means converge to f in Hp (Td1 × . . . × Tdk) norm. The same results are proved for the conjugate θ-means and for d-dimensional Fourier transforms, too. Some special cases of the θ-summation are considered, such as the Weierstrass, Picar, Bessel, Fejér, Riemann, de La Vallée-Poussin, Rogosinski and Riesz summations.

Original languageEnglish
Pages (from-to)159-180
Number of pages22
JournalMathematische Nachrichten
Volume230
DOIs
Publication statusPublished - Jan 1 2001

Keywords

  • Atomic decomposition
  • Fourier transforms
  • Hardy spaces
  • Interpolation
  • p-atom
  • θ-summation

ASJC Scopus subject areas

  • Mathematics(all)

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