Two-parameter Hardy-Littlewood inequalities

Ferenc Weisz

Displaying similar documents to “Two-parameter Hardy-Littlewood inequalities”

Hardy type inequalities for two-parameter Vilenkin-Fourier coefficients

Péter Simon, Ferenc Weisz (1997)

Studia Mathematica

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Our main result is a Hardy type inequality with respect to the two-parameter Vilenkin system (*) $(\sum_{k = 1}^{\infty} \sum_{j = 1}^{\infty} {| f ̂ (k, j) |}^{p} {(k j)}^{p - 2})^{1 / p} \leq C_{p} ∥ f ∥_{H_{* *}^{p}}$ (1/2 < p≤2) where f belongs to the Hardy space $H_{* *}^{p} (G_{m} \times G_{s})$ defined by means of a maximal function. This inequality is extended to p > 2 if the Vilenkin-Fourier coefficients of f form a monotone sequence. We show that the converse of (*) also holds for all p > 0 under the monotonicity assumption.

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...

On the uniform convergence and L¹-convergence of double Walsh-Fourier series

Ferenc Móricz (1992)

Studia Mathematica

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In 1970 C. W. Onneweer formulated a sufficient condition for a periodic W-continuous function to have a Walsh-Fourier series which converges uniformly to the function. In this paper we extend his results from single to double Walsh-Fourier series in a more general setting. We study the convergence of rectangular partial sums in $L^{p}$ -norm for some 1 ≤ p ≤ ∞ over the unit square [0,1) × [0,1). In case p = ∞, by $L^{p}$ we mean $C_{W}$ , the collection of uniformly W-continuous functions f(x, y), endowed...

Almost everywhere convergence of Walsh Fourier series of $H^{1}$ -functions

N. Ladhawala, D. Pankratz (1976)

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.

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...

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...

Estimates of Fourier transforms in Sobolev spaces

V. Kolyada (1997)

Studia Mathematica

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We investigate the Fourier transforms of functions in the Sobolev spaces $W_{1}^{r_{1}, . . ., r_{n}}$ . It is proved that for any function $f \in W_{1}^{r_{1}, . . ., r_{n}}$ the Fourier transform f̂ belongs to the Lorentz space $L^{n / r, 1}$ , where $r = n {(\sum_{j = 1}^{n} 1 / r_{j})}^{- 1} \leq n$ . Furthermore, we derive from this result that for any mixed derivative $D^{s} f (f \in C_{0}^{\infty}, s = (s_{1}, . . ., s_{n}))$ the weighted norm $∥ {(D^{s} f)}^{\land} ∥_{L^{1} (w)} (w (ξ) = {| ξ |}^{- n})$ can be estimated by the sum of $L^{1}$ -norms of all pure derivatives of the same order. This gives an answer to a question posed by A. Pełczyński and M. Wojciechowski.

Inequalities relative to two-parameter Vilenkin-Fourier coefficients

Ferenc Weisz (1991)

Studia Mathematica

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

M. Mateljević, M. Pavlović (1984)

Studia Mathematica

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On (C,1) summability of integrable functions with respect to the Walsh-Kaczmarz system

G. Gát (1998)

Studia Mathematica

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Let G be the Walsh group. For $f \in L^{1} (G)$ we prove the a. e. convergence σf → f(n → ∞), where $σ_{n}$ is the nth (C,1) mean of f with respect to the Walsh-Kaczmarz system. Define the maximal operator $σ * f ≔ s u p_{n} | σ_{n} f | .$ We prove that σ* is of type (p,p) for all 1 < p ≤ ∞ and of weak type (1,1). Moreover, $∥ σ * f ∥_{1} \leq c ∥ | f | ∥_{H}$ , where H is the Hardy space on the Walsh group.