### The Maximal (C, ...) Operator of Two-Parameter Walsh-Fourier Series.

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Some duality results and some inequalities are proved for two-parameter Vilenkin martingales, for Fourier backwards martingales and for Vilenkin and Fourier coefficients.

The inequality (*) $({\sum}_{\left|n\right|=1}^{\infty}{\sum}_{\left|m\right|=1}^{\infty}{\left|nm\right|}^{p-2}{\left|f\u0302(n,m)\right|}^{p}{)}^{1/p}\le {C}_{p}\parallel \u0192{\parallel}_{{H}_{p}}$ (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 series...

We introduce p-quasilocal operators and prove that if a sublinear operator T is p-quasilocal and bounded from ${L}_{\infty}$ to ${L}_{\infty}$ then it is also bounded from the classical Hardy space ${H}_{p}\left(T\right)$ 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 ${H}_{p}\left(T\right)$ to ${L}_{p}$ (3/4 < p ≤ ∞) and is of weak type $({L}_{1},{L}_{1})$. We define the two-dimensional dyadic hybrid Hardy space ${H}_{1}^{\u266f}\left({T}^{2}\right)$ and verify that the maximal operator of the Cesàro means of a two-dimensional...

The two-dimensional classical Hardy spaces ${H}_{p}\left(\mathbb{R}\times \mathbb{R}\right)$ are introduced and it is shown that the maximal operator of the Fejér means of a tempered distribution is bounded from ${H}_{p}\left(\mathbb{R}\times \mathbb{R}\right)$ to ${L}_{p}\left({\mathbb{R}}^{2}\right)$ (1/2 < p ≤ ∞) and is of weak type $({H}_{1}^{}\left(\mathbb{R}\times \mathbb{R}\right),{L}_{1}\left({\mathbb{R}}^{2}\right))$ where the Hardy space ${H}_{1}\left(\mathbb{R}\times \mathbb{R}\right)$ is defined by the hybrid maximal function. As a consequence we deduce that the Fejér means of a function f ∈ ${H}_{1}^{(}\mathbb{R}\times \mathbb{R})$ ⊃ $LlogL\left({\mathbb{R}}^{2}\right)$ converge to f a.e. Moreover, we prove that the Fejér means are uniformly bounded on ${H}_{p}\left(\mathbb{R}\times \mathbb{R}\right)$ whenever 1/2 < p < ∞. Thus, in case f ∈ ${H}_{p}\left(\mathbb{R}\times \mathbb{R}\right)$, the Fejér means...

Hardy spaces consisting of adapted function sequences and generated by the q-variation and by the conditional q-variation are considered. Their dual spaces are characterized and an inequality due to Stein and Lepingle is extended.

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.

Characterizations of H₁, BMO and VMO martingale spaces generated by bounded Vilenkin systems via conjugate martingale transforms are studied.

Martingale Hardy spaces and BMO spaces generated by an operator T are investigated. An atomic decomposition of the space ${H}_{p}^{T}$ is given if the operator T is predictable. We generalize the John-Nirenberg theorem, namely, we prove that the $BM{O}_{q}$ spaces generated by an operator T are all equivalent. The sharp operator is also considered and it is verified that the ${L}_{p}$ norm of the sharp operator is equivalent to the ${H}_{p}^{T}$ norm. The interpolation spaces between the Hardy and BMO spaces are identified by the real method....

The martingale Hardy space ${H}_{p}\left({[0,1)}^{2}\right)$ 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 and a...

Elementary estimates for the Riesz kernel and for its derivative are given. Using these we show that the maximal operator of the Riesz means of a tempered distribution is bounded from ${H}_{p}\left(\mathbb{R}\right)$ to ${L}_{p}\left(\mathbb{R}\right)$ (1/(α+1) < p < ∞) and is of weak type (1,1), where ${H}_{p}\left(\mathbb{R}\right)$ is the classical Hardy space. As a consequence we deduce that the Riesz means of a function $\u2a0d\in {L}_{1}\left(\mathbb{R}\right)$ converge a.e. to ⨍. Moreover, we prove that the Riesz means are uniformly bounded on ${H}_{p}\left(\mathbb{R}\right)$ whenever 1/(α+1) < p < ∞. Thus, in case $\u2a0d\in {H}_{p}\left(\mathbb{R}\right)$, the Riesz means converge...

It is proved that the multi-dimensional maximal Fejér operator defined in a cone is bounded from the amalgam Hardy space $W({h}_{p},{\ell}_{\infty})$ to $W({L}_{p},{\ell}_{\infty})$. This implies the almost everywhere convergence of the Fejér means in a cone for all $f\in W(L\u2081,{\ell}_{\infty})$, which is larger than $L\u2081\left({\mathbb{R}}^{d}\right)$.

We investigate the bounded Ciesielski systems, which can be obtained from the spline systems of order (m,k) in the same way as the Walsh system arises from the Haar system. It is shown that the maximal operator of the Fejér means of the Ciesielski-Fourier series is bounded from the Hardy space ${H}_{p}$ to ${L}_{p}$ if 1/2 < p < ∞ and m ≥ 0, |k| ≤ m + 1. Moreover, it is of weak type (1,1). As a consequence, the Fejér means of the Ciesielski-Fourier series of a function f converges to f a.e. if f ∈ L₁ as n...

A strong summability result is proved for the Ciesielski-Fourier series of integrable functions. It is also shown that the strong maximal operator is of weak type (1,1).

Four basic results of Marcinkiewicz are presented in summability theory. We show that setting out from these theorems many mathematicians have reached several nice results for trigonometric, Walsh- and Ciesielski-Fourier series.

Some recent results on spline-Fourier and Ciesielski-Fourier series are summarized. The convergence of spline Fourier series and the basis properties of the spline systems are considered. Some classical topics, that are well known for trigonometric and Walsh-Fourier series, are investigated for Ciesielski-Fourier series, such as inequalities for the Fourier coefficients, convergence a.e. and in norm, Fejér and θ-summability, strong summability and multipliers. The connection between Fourier series...

Our main result is a Hardy type inequality with respect to the two-parameter Vilenkin system (*) $({\sum}_{k=1}^{\infty}{\sum}_{j=1}^{\infty}{\left|f\u0302(k,j)\right|}^{p}{\left(kj\right)}^{p-2}{)}^{1/p}\le {C}_{p}\parallel f{\parallel}_{{H}_{**}^{p}}$ (1/2 < p≤2) where f belongs to the Hardy space ${H}_{**}^{p}\left({G}_{m}\times {G}_{s}\right)$ 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.

The classical Hardy-Littlewood maximal operator is bounded not only on the classical Lebesgue spaces ${L}_{p}\left({\mathbb{R}}^{d}\right)$ (in the case $p>1$), but (in the case when $1/p(\xb7)$ is log-Hölder continuous and ${p}_{-}=inf\{p\left(x\right):x\in {\mathbb{R}}^{d}\}>1$) on the variable Lebesgue spaces ${L}_{p(\xb7)}\left({\mathbb{R}}^{d}\right)$, too. Furthermore, the classical Hardy-Littlewood maximal operator is of weak-type $(1,1)$. In the present note we generalize Besicovitch’s covering theorem for the so-called $\gamma $-rectangles. We introduce a general maximal operator ${M}_{s}^{\gamma ,\delta}$ and with the help of generalized $\Phi $-functions, the strong- and weak-type...

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