Maximal operators of (C,α)-means of cubic partial sums of d-dimensional Walsh-Fourier series

The boundednes of (C,α)-means of the cubic partial sums of d-dimensional Walsh-Fourier series is studied from the martingale Hardy space H _p into the space L _p .     

Relations between ΛBV and BV(p(n) ↑∞) Classes of Functions

The base radical class L _b(X), generated by a class X was introduced in [12]. It consists of those rings whose nonzero homomorphic images have nonzero accessible subrings in X. When X is homomorphically closed, L _b(X) is the lower radical class defined by X, but otherwise X may not be contained in L _b(X). We prove that for a hereditary radical class L with semisimple class S(R), L _b(S(R)) is the class of strongly R-semisimple rings if and only if R is supernilpotent or subidempotent. A number of further examples of radical classes of the form L _b(X) are discussed. This revised version was published online in June 2006 with corrections to the Cover Date.     

On the Fejér means of double Fourier series with respect to the Walsh-Kaczmarz system

The main aim of this paper is to prove that the maximal operator σ ₀ ^k*:= sup _n ∣σ _n,n ^k ∣ of the Fejér means of double Fourier series with respect to the Kaczmarz system is not bounded from the Hardy space H _1/2 to the space weak-L _1/2. Supported by the Hungarian NFSR (OTKA), grant no. M 36511/2001, T 048780.     

THE HARDY TYPE INEQUALITY FOR THE MAXIMAL OPERATOR OF THE ONE-DIMENSIONAL DYADIC DERIVATIVE

In this paper we prove that the maximal operator I of dyadic derivative is not bounded from the Hardy space H p [0, 1] to the Hardy space H p [0, 1], when 0 < p ≤ 1.     

On the maximal operator of (<Emphasis Type="Italic">C</Emphasis>, α)-means of Walsh–Kaczmarz–Fourier series

Simon [J. Approxim. Theory, 127, 39–60 (2004)] proved that the maximal operator σ^α,κ,* of the (C, α)-means of the Walsh–Kaczmarz–Fourier series is bounded from the martingale Hardy space H _p to the space L _p for p > 1 / (1 + α), 0 < α ≤ 1. Recently, Gát and Goginava have proved that this boundedness result does not hold if p ≤ 1 / (1 + α). However, in the endpoint case p = 1 / (1 + α ), the maximal operator σ^α,κ,* is bounded from the martingale Hardy space H _1/(1+α) to the space weak- L _1/(1+α). The main aim of this paper is to prove a stronger result, namely, that, for any 0 < p ≤ 1 / (1 + α), there exists a martingale f ∈ H _p such that the maximal operator σ^α,κ,* f does not belong to the space L _p .     

A note on strong summability of two-dimensional Walsh-Fourier series

The paper deals with the strong summability of Marcinkiewicz means with a variable power. Let $$H_n \left( {f,x,y,A_n } \right): = \tfrac{1} {n}\sum\nolimits_{l = 1}^n {\left( {e^{\left. {A_n } \right|\left. {S_{ll} \left( {f,x,y} \right) - f\left( {x,y} \right)} \right|^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} } - 1} \right)} .$$ It is shown that if A _n ↑ ∞ arbitrary slowly, there exists f ∈ C(I ²) such that lim _n→∞ H _n (f, 0, 0, A _n ) = +∞. At the same time, for every f ∈ C (I ²) there exists A _n (f) ↑ ∞ such that lim _n→∞ H _n (f, x, y, A _n ) = 0 uniformly on I ².     

The weak type inequality for the maximal operator of the Marcinkiewicz–Fejér means of the two-dimensional Walsh–Fourier series

The main aim of this paper is to prove that the maximal operator σ^* of the Marcinkiewicz–Fejér means of the two-dimensional Walsh–Fourier series is bounded from the Hardy space H_2/3 to the space weak-L_2/3.     

Convergence in measure of logarithmic means of multiple Fourier series

The maximal Orlicz space such that the mixed logarithmic means of rectangular partial sums of multiple Fourier series for the functions from this space converge in measure is found.     

APPROXIMATION PROPERTIES OF (C, α) MEANS OF DOUBLE WALSH-FOURIER SERIES

We study the rate of Zp approximation by Cesaro means of the quadratic partial sums of double Walsh-Fourier series offu nctions from Lp.