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

The journal of Fourier analysis and applications [[Elektronische Ressource]]

### An application of two-parameter martingales in harmonic analysis

Studia Mathematica

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

### Two-parameter Hardy-Littlewood inequalities

Studia Mathematica

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

### Cesàro summability of one- and two-dimensional trigonometric-Fourier series

Colloquium Mathematicae

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 $\left({L}_{1},{L}_{1}\right)$. We define the two-dimensional dyadic hybrid Hardy space ${H}_{1}^{♯}\left({T}^{2}\right)$ and verify that the maximal operator of the Cesàro means of a two-dimensional...

### Fejér means of two-dimensional Fourier transforms on ${H}_{p}\left(ℝ×ℝ\right)$

Colloquium Mathematicae

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

### An extension of an inequality due to Stein and Lepingle

Colloquium Mathematicae

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.

### $\left({H}_{p},{L}_{p}\right)$-type inequalities for the two-dimensional dyadic derivative

Studia Mathematica

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 $\left({L}_{1},{L}_{1}\right)$. 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.

### Conjugate martingale transforms

Studia Mathematica

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

### Martingale operators and Hardy spaces generated by them

Studia Mathematica

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....

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

Studia Mathematica

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

### Riesz means of Fourier transforms and Fourier series on Hardy spaces

Studia Mathematica

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 ${H}_{p}\left(ℝ\right)$ to ${L}_{p}\left(ℝ\right)$ (1/(α+1) < p < ∞) and is of weak type (1,1), where ${H}_{p}\left(ℝ\right)$ is the classical Hardy space. As a consequence we deduce that the Riesz means of a function $⨍\in {L}_{1}\left(ℝ\right)$ converge a.e. to ⨍. Moreover, we prove that the Riesz means are uniformly bounded on ${H}_{p}\left(ℝ\right)$ whenever 1/(α+1) < p < ∞. Thus, in case $⨍\in {H}_{p}\left(ℝ\right)$, the Riesz means converge...

### Inequalities relative to two-parameter Vilenkin-Fourier coefficients

Studia Mathematica

### Multi-dimensional Fejér summability and local Hardy spaces

Studia Mathematica

It is proved that the multi-dimensional maximal Fejér operator defined in a cone is bounded from the amalgam Hardy space $W\left({h}_{p},{\ell }_{\infty }\right)$ to $W\left({L}_{p},{\ell }_{\infty }\right)$. This implies the almost everywhere convergence of the Fejér means in a cone for all $f\in W\left(L₁,{\ell }_{\infty }\right)$, which is larger than $L₁\left({ℝ}^{d}\right)$.

### On the Fejér means of bounded Ciesielski systems

Studia Mathematica

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...

### Strong summability of Ciesielski-Fourier series

Studia Mathematica

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).

### Some footprints of Marcinkiewicz in summability theory

Banach Center Publications

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.

### Results on spline-Fourier and Ciesielski-Fourier series

Banach Center Publications

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...

### Hardy type inequalities for two-parameter Vilenkin-Fourier coefficients

Studia Mathematica

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

### Weak- and strong-type inequality for the cone-like maximal operator in variable Lebesgue spaces

Czechoslovak Mathematical Journal

The classical Hardy-Littlewood maximal operator is bounded not only on the classical Lebesgue spaces ${L}_{p}\left({ℝ}^{d}\right)$ (in the case $p>1$), but (in the case when $1/p\left(·\right)$ is log-Hölder continuous and ${p}_{-}=inf\left\{p\left(x\right):x\in {ℝ}^{d}\right\}>1$) on the variable Lebesgue spaces ${L}_{p\left(·\right)}\left({ℝ}^{d}\right)$, too. Furthermore, the classical Hardy-Littlewood maximal operator is of weak-type $\left(1,1\right)$. 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...

### Summation of Fourier series.

Acta Mathematica Academiae Paedagogicae Nyí regyháziensis. New Series [electronic only]

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