Fejér means of two-dimensional Fourier transforms on H p ( ℝ × ℝ )

Ferenc Weisz

Colloquium Mathematicae (1999)

  • Volume: 82, Issue: 2, page 155-166
  • ISSN: 0010-1354

Abstract

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The two-dimensional classical Hardy spaces H p ( ℝ × ℝ ) are introduced and it is shown that the maximal operator of the Fejér means of a tempered distribution is bounded from H p ( ℝ × ℝ ) to L p ( ℝ 2 ) (1/2 < p ≤ ∞) and is of weak type ( H 1 ( ℝ × ℝ ) , L 1 ( ℝ 2 ) ) where the Hardy space H 1 ( ℝ × ℝ ) is defined by the hybrid maximal function. As a consequence we deduce that the Fejér means of a function f ∈ H 1 ( ℝ × ℝ ) ⊃ L l o g L ( ℝ 2 ) converge to f a.e. Moreover, we prove that the Fejér means are uniformly bounded on H p ( ℝ × ℝ ) whenever 1/2 < p < ∞. Thus, in case f ∈ H p ( ℝ × ℝ ) , the Fejér means converge to f in H p ( ℝ × ℝ ) norm (1/2 < p < ∞). The same results are proved for the conjugate Fejér means.

How to cite

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Weisz, Ferenc. "Fejér means of two-dimensional Fourier transforms on $H_p(ℝ × ℝ)$." Colloquium Mathematicae 82.2 (1999): 155-166. <http://eudml.org/doc/210754>.

@article{Weisz1999,
abstract = {The two-dimensional classical Hardy spaces $H_p(ℝ × ℝ)$ are introduced and it is shown that the maximal operator of the Fejér means of a tempered distribution is bounded from $H_p(ℝ × ℝ)$ to $L_p(ℝ^2)$ (1/2 < p ≤ ∞) and is of weak type $(H^\{#\}_1 (ℝ × ℝ), L_1(ℝ^2))$ where the Hardy space $H^#_1(ℝ × ℝ)$ is defined by the hybrid maximal function. As a consequence we deduce that the Fejér means of a function f ∈ $H_1^#(ℝ × ℝ)$ ⊃ $LlogL(ℝ^2)$ converge to f a.e. Moreover, we prove that the Fejér means are uniformly bounded on $H_p(ℝ × ℝ)$ whenever 1/2 < p < ∞. Thus, in case f ∈ $H_p(ℝ × ℝ)$, the Fejér means converge to f in $H_p(ℝ × ℝ)$ norm (1/2 < p < ∞). The same results are proved for the conjugate Fejér means.},
author = {Weisz, Ferenc},
journal = {Colloquium Mathematicae},
keywords = {p-atom; Hardy spaces; atomic decomposition; interpolation; Fejér means; maximal operator},
language = {eng},
number = {2},
pages = {155-166},
title = {Fejér means of two-dimensional Fourier transforms on $H_p(ℝ × ℝ)$},
url = {http://eudml.org/doc/210754},
volume = {82},
year = {1999},
}

TY - JOUR
AU - Weisz, Ferenc
TI - Fejér means of two-dimensional Fourier transforms on $H_p(ℝ × ℝ)$
JO - Colloquium Mathematicae
PY - 1999
VL - 82
IS - 2
SP - 155
EP - 166
AB - The two-dimensional classical Hardy spaces $H_p(ℝ × ℝ)$ are introduced and it is shown that the maximal operator of the Fejér means of a tempered distribution is bounded from $H_p(ℝ × ℝ)$ to $L_p(ℝ^2)$ (1/2 < p ≤ ∞) and is of weak type $(H^{#}_1 (ℝ × ℝ), L_1(ℝ^2))$ where the Hardy space $H^#_1(ℝ × ℝ)$ is defined by the hybrid maximal function. As a consequence we deduce that the Fejér means of a function f ∈ $H_1^#(ℝ × ℝ)$ ⊃ $LlogL(ℝ^2)$ converge to f a.e. Moreover, we prove that the Fejér means are uniformly bounded on $H_p(ℝ × ℝ)$ whenever 1/2 < p < ∞. Thus, in case f ∈ $H_p(ℝ × ℝ)$, the Fejér means converge to f in $H_p(ℝ × ℝ)$ norm (1/2 < p < ∞). The same results are proved for the conjugate Fejér means.
LA - eng
KW - p-atom; Hardy spaces; atomic decomposition; interpolation; Fejér means; maximal operator
UR - http://eudml.org/doc/210754
ER -

References

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