Two-parameter Hardy-Littlewood inequality and its variants

Chang-Pao Chen; Dah-Chin Luor

Studia Mathematica (2000)

  • Volume: 139, Issue: 1, page 9-27
  • ISSN: 0039-3223

Abstract

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Let s* denote the maximal function associated with the rectangular partial sums s m n ( x , y ) of a given double function series with coefficients c j k . The following generalized Hardy-Littlewood inequality is investigated: | | s * | | p , μ ≤ C p , α , β Σ j = 0 ∞ Σ k = 0 ∞ ( j ̅ ) p - α - 2 ( k ̅ ) p - β - 2 | c j k | p 1 / p , where ξ̅=max(ξ,1), 0 < p < ∞, and μ is a suitable positive Borel measure. We give sufficient conditions on c j k and μ under which the above Hardy-Littlewood inequality holds. Several variants of this inequality are also examined. As a consequence, the ||·||p,μ-convergence property of s m n ( x , y ) is established. These results generalize the work of Askey-Wainger [1], Balashov [2], Boas [3], Chen [5], [6], [8], [9], Marzug [15], Móricz [16]-[18], [19], Móricz-Schipp-Wade [20], Ram-Bhatia [22], Stechkin [24], Weisz [26]-[28], and Young [30].

How to cite

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Chen, Chang-Pao, and Luor, Dah-Chin. "Two-parameter Hardy-Littlewood inequality and its variants." Studia Mathematica 139.1 (2000): 9-27. <http://eudml.org/doc/216714>.

@article{Chen2000,
abstract = {Let s* denote the maximal function associated with the rectangular partial sums $s_\{mn\}(x,y)$ of a given double function series with coefficients $c_\{jk\}$. The following generalized Hardy-Littlewood inequality is investigated: $||s*||_\{p,μ\}≤C_\{p,α,β\} \{Σ_\{j=0\}^∞Σ_\{k=0\}^∞(j̅ )^\{p-α-2\}(k̅)^\{p-β-2\}|c_\{jk\}|^p \}^\{1/p\}$, where ξ̅=max(ξ,1), 0 < p < ∞, and μ is a suitable positive Borel measure. We give sufficient conditions on $c_\{jk\}$ and μ under which the above Hardy-Littlewood inequality holds. Several variants of this inequality are also examined. As a consequence, the ||·||p,μ-convergence property of $s_\{mn\}(x,y)$ is established. These results generalize the work of Askey-Wainger [1], Balashov [2], Boas [3], Chen [5], [6], [8], [9], Marzug [15], Móricz [16]-[18], [19], Móricz-Schipp-Wade [20], Ram-Bhatia [22], Stechkin [24], Weisz [26]-[28], and Young [30].},
author = {Chen, Chang-Pao, Luor, Dah-Chin},
journal = {Studia Mathematica},
keywords = {Hardy-Littlewood inequality; trigonometric system; Walsh system; Legendre system; maximal function},
language = {eng},
number = {1},
pages = {9-27},
title = {Two-parameter Hardy-Littlewood inequality and its variants},
url = {http://eudml.org/doc/216714},
volume = {139},
year = {2000},
}

TY - JOUR
AU - Chen, Chang-Pao
AU - Luor, Dah-Chin
TI - Two-parameter Hardy-Littlewood inequality and its variants
JO - Studia Mathematica
PY - 2000
VL - 139
IS - 1
SP - 9
EP - 27
AB - Let s* denote the maximal function associated with the rectangular partial sums $s_{mn}(x,y)$ of a given double function series with coefficients $c_{jk}$. The following generalized Hardy-Littlewood inequality is investigated: $||s*||_{p,μ}≤C_{p,α,β} {Σ_{j=0}^∞Σ_{k=0}^∞(j̅ )^{p-α-2}(k̅)^{p-β-2}|c_{jk}|^p }^{1/p}$, where ξ̅=max(ξ,1), 0 < p < ∞, and μ is a suitable positive Borel measure. We give sufficient conditions on $c_{jk}$ and μ under which the above Hardy-Littlewood inequality holds. Several variants of this inequality are also examined. As a consequence, the ||·||p,μ-convergence property of $s_{mn}(x,y)$ is established. These results generalize the work of Askey-Wainger [1], Balashov [2], Boas [3], Chen [5], [6], [8], [9], Marzug [15], Móricz [16]-[18], [19], Móricz-Schipp-Wade [20], Ram-Bhatia [22], Stechkin [24], Weisz [26]-[28], and Young [30].
LA - eng
KW - Hardy-Littlewood inequality; trigonometric system; Walsh system; Legendre system; maximal function
UR - http://eudml.org/doc/216714
ER -

References

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