Hardy type inequalities for two-parameter Vilenkin-Fourier coefficients

Péter Simon; Ferenc Weisz

Displaying similar documents to “Hardy type inequalities for two-parameter Vilenkin-Fourier coefficients”

Two-parameter Hardy-Littlewood inequalities

Ferenc Weisz (1996)

Studia Mathematica

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The inequality (*) $(\sum_{| n | = 1}^{\infty} \sum_{| m | = 1}^{\infty} {| n m |}^{p - 2} {| f ̂ (n, m) |}^{p})^{1 / p} \leq C_{p} ∥ ƒ ∥_{H_{p}}$ (0 < p ≤ 2) is proved for two-parameter trigonometric-Fourier coefficients and for the two-dimensional classical Hardy space $H_{p}$ on the bidisc. The inequality (*) is extended to each p if the Fourier coefficients are monotone. For monotone coefficients and for every p, the supremum of the partial sums of the Fourier series is in $L_{p}$ whenever the left hand side of (*) is finite. From this it follows that under the same condition the two-dimensional trigonometric-Fourier...

Inequalities relative to two-parameter Vilenkin-Fourier coefficients

Ferenc Weisz (1991)

Studia Mathematica

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Regularity of some nonlinear quantities on superharmonic functions in local Herz-type Hardy spaces.

Dashan Fan, Shanzhen Lu, Dachun Yang (1998)

Publicacions Matemàtiques

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In this paper, the authors introduce a kind of local Hardy spaces in R associated with the local Herz spaces. Then the authors investigate the regularity in these local Hardy spaces of some nonlinear quantities on superharmonic functions on R. The main results of the authors extend the corresponding results of Evans and Müller in a recent paper.

Note on Fourier series

L. S. Bosanquet, A. C. Offord (1935)

Compositio Mathematica

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$(H_{p}, L_{p})$ -type inequalities for the two-dimensional dyadic derivative

Ferenc Weisz (1996)

Studia Mathematica

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It is shown that the restricted maximal operator of the two-dimensional dyadic derivative of the dyadic integral is bounded from the two-dimensional dyadic Hardy-Lorentz space $H_{p, q}$ to $L_{p, q}$ (2/3 < p < ∞, 0 < q ≤ ∞) and is of weak type $(L_{1}, L_{1})$ . As a consequence we show that the dyadic integral of a ∞ function $f \in L_{1}$ is dyadically differentiable and its derivative is f a.e.

Strong convergence theorems for two-parameter Walsh-Fourier and trigonometric-Fourier series

Ferenc Weisz (1996)

Studia Mathematica

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The martingale Hardy space $H_{p} ({[0, 1)}^{2})$ and the classical Hardy space $H_{p} (^{2})$ are introduced. We prove that certain means of the partial sums of the two-parameter Walsh-Fourier and trigonometric-Fourier series are uniformly bounded operators from $H_{p}$ to $L_{p}$ (0 < p ≤ 1). As a consequence we obtain strong convergence theorems for the partial sums. The classical Hardy-Littlewood inequality is extended to two-parameter Walsh-Fourier and trigonometric-Fourier coefficients. The dual inequalities are also verified...

Two-parameter Hardy-Littlewood inequality and its variants

Chang-Pao Chen, Dah-Chin Luor (2000)

Studia Mathematica

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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: ${| | s * | |}_{p, μ} \leq C_{p, α, β} {Σ_{j = 0}^{\infty} Σ_{k = 0}^{\infty} {(j ̅)}^{p - α - 2} {(k ̅)}^{p - β - 2} {| c_{j k} |}^{p}}^{1 / p}$ , 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...

$L^{p}$ -behaviour of the integral means of analytic functions

M. Mateljević, M. Pavlović (1984)

Studia Mathematica

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On the boundedness of multipliers, commutators and the second derivatives of Green’s operators on $H^{1}$ and $B M O$

Der-Chen Chang, Song-Ying Li (1999)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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The local versions of $H^{p} (ℝ^{n})$ spaces at the origin

Shan Lu, Da Yang (1995)

Studia Mathematica

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Let 0 < p ≤ 1 < q < ∞ and α = n(1/p - 1/q). We introduce some new Hardy spaces $H K ̇_{q}^{α, p} (ℝ^{n})$ which are the local versions of $H^{p} (ℝ^{n})$ spaces at the origin. Characterizations of these spaces in terms of atomic and molecular decompositions are established, together with their φ-transform characterizations in M. Frazier and B. Jawerth’s sense. We also prove an interpolation theorem for operators on $H K ̇_{q}^{α, p} (ℝ^{n})$ and discuss the $H K ̇_{q}^{α, p} (ℝ^{n})$ -boundedness of Calderón-Zygmund operators. Similar results can also be obtained...