Cesàro summability of one- and two-dimensional trigonometric-Fourier series

Ferenc Weisz

Colloquium Mathematicae (1997)

  • Volume: 74, Issue: 1, page 123-133
  • ISSN: 0010-1354

Abstract

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We introduce p-quasilocal operators and prove that if a sublinear operator T is p-quasilocal and bounded from L ∞ to L ∞ then it is also bounded from the classical Hardy space H p ( T ) to L p (0 < p ≤ 1). As an application it is shown that the maximal operator of the one-parameter Cesàro means of a distribution is bounded from H p ( T ) to L p (3/4 < p ≤ ∞) and is of weak type ( L 1 , L 1 ) . We define the two-dimensional dyadic hybrid Hardy space H 1 ♯ ( T 2 ) and verify that the maximal operator of the Cesàro means of a two-dimensional function is of weak type ( H 1 ♯ ( T 2 ) , L 1 ) . So we deduce that the two-parameter Cesàro means of a function f ∈ H 1 ♯ ( T 2 ) ⊃ L l o g L converge a.e. to the function in question.

How to cite

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Weisz, Ferenc. "Cesàro summability of one- and two-dimensional trigonometric-Fourier series." Colloquium Mathematicae 74.1 (1997): 123-133. <http://eudml.org/doc/210495>.

@article{Weisz1997,
abstract = {We introduce p-quasilocal operators and prove that if a sublinear operator T is p-quasilocal and bounded from $L_∞$ to $L_∞$ then it is also bounded from the classical Hardy space $H_p(T)$ to $L_p$ (0 < p ≤ 1). As an application it is shown that the maximal operator of the one-parameter Cesàro means of a distribution is bounded from $H_p(T)$ to $L_p$ (3/4 < p ≤ ∞) and is of weak type $(L_1,L_1)$. We define the two-dimensional dyadic hybrid Hardy space $H_\{1\}^\{♯\}(T^2)$ and verify that the maximal operator of the Cesàro means of a two-dimensional function is of weak type $(H_\{1\}^\{♯\}(T^2),L_1)$. So we deduce that the two-parameter Cesàro means of a function $f ∈ H_1^\{♯\}(T^2) ⊃ Llog L$ converge a.e. to the function in question.},
author = {Weisz, Ferenc},
journal = {Colloquium Mathematicae},
keywords = {p-atom; Hardy spaces; Cesàro summability; atomic decomposition; p-quasilocal operator; interpolation; trigonometric Fourier series; Cesàro maximal operator; Hardy-Lorentz spaces; double Fourier series},
language = {eng},
number = {1},
pages = {123-133},
title = {Cesàro summability of one- and two-dimensional trigonometric-Fourier series},
url = {http://eudml.org/doc/210495},
volume = {74},
year = {1997},
}

TY - JOUR
AU - Weisz, Ferenc
TI - Cesàro summability of one- and two-dimensional trigonometric-Fourier series
JO - Colloquium Mathematicae
PY - 1997
VL - 74
IS - 1
SP - 123
EP - 133
AB - We introduce p-quasilocal operators and prove that if a sublinear operator T is p-quasilocal and bounded from $L_∞$ to $L_∞$ then it is also bounded from the classical Hardy space $H_p(T)$ to $L_p$ (0 < p ≤ 1). As an application it is shown that the maximal operator of the one-parameter Cesàro means of a distribution is bounded from $H_p(T)$ to $L_p$ (3/4 < p ≤ ∞) and is of weak type $(L_1,L_1)$. We define the two-dimensional dyadic hybrid Hardy space $H_{1}^{♯}(T^2)$ and verify that the maximal operator of the Cesàro means of a two-dimensional function is of weak type $(H_{1}^{♯}(T^2),L_1)$. So we deduce that the two-parameter Cesàro means of a function $f ∈ H_1^{♯}(T^2) ⊃ Llog L$ converge a.e. to the function in question.
LA - eng
KW - p-atom; Hardy spaces; Cesàro summability; atomic decomposition; p-quasilocal operator; interpolation; trigonometric Fourier series; Cesàro maximal operator; Hardy-Lorentz spaces; double Fourier series
UR - http://eudml.org/doc/210495
ER -

References

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