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

Colloquium Mathematicae (1997)

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

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topWeisz, 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 -

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