We investigate some convergence and divergence properties of the logarithmic means of quadratic partial sums of double Fourier series of functions, in measure and in the L Lebesgue norm.

The weak type inequality for the Walsh system

Ushangi Goginava — 2008

Studia Mathematica

The main aim of this paper is to prove that the maximal operator $σ^{}$ is bounded from the Hardy space $H_{1 / 2}$ to weak- $L_{1 / 2}$ and is not bounded from $H_{1 / 2}$ to $L_{1 / 2}$ .

Norm convergence of Fejér means of two-dimensional Walsh-Fourier series

Ushangi Goginava — 2011

Banach Center Publications

The main aim of this paper is to prove that there exists a martingale $f \in H_{1 / 2}$ such that the Fejér means of the two-dimensional Walsh-Fourier series of f is not uniformly bounded in the space weak- $L_{1 / 2}$ .

On the maximal operator of Walsh-Kaczmarz-Fejér means

Ushangi Goginava; Károly Nagy — 2011

Czechoslovak Mathematical Journal

In this paper we prove that the maximal operator ${\tilde{σ}}^{κ, *} f : = sup_{n \in ℙ} \frac{| σ_{n}^{κ} f |}{{log}^{2} (n + 1)},$ where $σ_{n}^{κ} f$ is the $n$ -th Fejér mean of the Walsh-Kaczmarz-Fourier series, is bounded from the Hardy space $H_{1 / 2} (G)$ to the space $L_{1 / 2} (G) .$

Maximal operators of Fejér means of double Vilenkin-Fourier series

István Blahota; György Gát; Ushangi Goginava — 2007

Colloquium Mathematicae

The main aim of this paper is to prove that the maximal operator $σ ₀ * : = s u p ₙ | σ_{n, n} |$ of the Fejér means of the double Vilenkin-Fourier series is not bounded from the Hardy space $H_{1 / 2}$ to the space weak- $L_{1 / 2}$ .

Relations between some classes of functions of generalized bounded variation

Amiran Gogatishvili; Ushangi Goginava; George Tephnadze — 2014

Banach Center Publications

We prove inclusion relations between generalizing Waterman's and generalized Wiener's classes for functions of two variable.

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