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Journal of Function Spaces and Applications
VolumeΒ 2012, Article IDΒ 582726, 15 pages
http://dx.doi.org/10.1155/2012/582726
Research Article

Convergence in Measure of Logarithmic Means of Quadratical Partial Sums of Double Walsh-Kaczmarz-Fourier Series

1Institute of Mathematics, Faculty of Exact and Natural Sciences, Tbilisi State University, Chavchavadze Street 1, 0128 Tbilisi, Georgia
2Institute of Mathematics and Computer Sciences, College of NyΓ­regyhΓ‘za, P.O. Box 166, NyΓ­regyhΓ‘za 4400, Hungary

Received 31 January 2011; Accepted 3 April 2011

Academic Editor: AnnaΒ Kaminska

Copyright Β© 2012 Ushangi Goginava and KΓ‘roly Nagy. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The main aim of this paper is to prove that the logarithmic means of quadratical partial sums of the double Walsh-Kaczmarz series does not improve the convergence in measure. In other words, we prove that for any Orlicz space, which is not a subspace of 𝐿 log+ 𝐿(𝐼2), the set of the functions the logarithmic means of quadratical partial sums of the double Walsh-Kaczmarz series of which converge in measure is of first Baire category.

1. Definitions and Notations

We denote by 𝐿0=𝐿0(𝐼2) the Lebesgue space of functions that are measurable and finite almost everywhere on 𝐼2=[0,1)Γ—[0,1). mes(𝐴) is the Lebesgue measure of the set π΄βŠ‚πΌ2. The constants appearing in this paper are denoted by 𝑐.

Let 𝐿Φ=𝐿Φ(𝐼2) be the Orlicz space [1] generated by Young’s function Ξ¦; that is, Ξ¦ is convex, continuous, even function such that Ξ¦(0)=0 andlim𝑒→+∞Φ(𝑒)𝑒=+∞,lim𝑒→0Ξ¦(𝑒)𝑒=0.(1.1)

This space is endowed with the norm‖𝑓‖𝐿Φ(𝐼2)ξ‚»ξ€œ=infπ‘˜>0∢𝐼2Ξ¦ξ‚΅||||𝑓(π‘₯,𝑦)π‘˜ξ‚Άξ‚Όπ‘‘π‘₯𝑑𝑦≀1.(1.2) In particular case, if Ξ¦(𝑒)=𝑒ln(1+𝑒), 𝑒>0, then the corresponding space is denoted by 𝐿log+𝐿(𝐼2).

We denote the set of nonnegative integers by 𝐍. By a dyadic interval in 𝐼∢=[0,1) we mean one of the form [𝑝/2𝑛,(𝑝+1)/2𝑛) for some π‘βˆˆπ, 0≀𝑝<2𝑛. Given π‘›βˆˆπ and π‘₯∈[0,1), let 𝐼𝑛(π‘₯) denote the dyadic interval of length 2βˆ’π‘› which contains the point π‘₯.

Every point π‘₯∈𝐼 can be written in the following form:π‘₯=βˆžξ“π‘˜=0π‘₯π‘˜2π‘˜+1ξ€·π‘₯=∢0,π‘₯1,…,π‘₯𝑛,…,π‘₯π‘˜βˆˆ{0,1}.(1.3) In that case when there are two different forms we choose the one for which limπ‘˜β†’βˆžπ‘₯π‘˜=0. We use the notation𝑒𝑗1∢=2𝑗+1=ξ€·0,…,0,π‘₯𝑗=1,0,….(1.4)

It is well-known that [2, 3]𝐼𝑛π‘₯0,…,π‘₯π‘›βˆ’1ξ€ΈβˆΆ=𝐼𝑛𝑝(π‘₯)=2𝑛,𝑝+12𝑛,(1.5) where βˆ‘π‘=π‘›βˆ’1𝑗=0π‘₯𝑗2π‘›βˆ’1βˆ’π‘—.

Let π‘Ÿ0(π‘₯) be a function defined byπ‘Ÿ0⎧βŽͺ⎨βŽͺβŽ©ξ‚ƒ1(π‘₯)=1,ifπ‘₯∈0,2,1βˆ’1,ifπ‘₯∈2,π‘Ÿ,10(π‘₯+1)=π‘Ÿ0(π‘₯).(1.6) The Rademacher functions are defined byπ‘Ÿπ‘›(π‘₯)=π‘Ÿ0(2𝑛[π‘₯),𝑛β‰₯0,π‘₯∈0,1).(1.7)

Let 𝑀0,𝑀1,… represent the Walsh functions, that is, 𝑀0(π‘₯)=1, and if βˆ‘π‘›=βˆžπ‘–=0𝑛𝑖2𝑖 is a positive integer with π‘›π‘–βˆˆ{0,1}, then𝑀𝑛(π‘₯)∢=βˆžξ‘π‘–=0ξ€·π‘Ÿπ‘–ξ€Έ(π‘₯)𝑛𝑖.(1.8) The order of the 𝑛 is denoted by |𝑛|∢=max{π‘—βˆˆπβˆΆπ‘›π‘—β‰ 0}.

The Walsh-Kaczmarz functions are given by πœ…0∢=1, and for 𝑛β‰₯1πœ…π‘›(π‘₯)∢=π‘Ÿ|𝑛|(π‘₯)|𝑛|βˆ’1ξ‘π‘˜=0ξ€·π‘Ÿ|𝑛|βˆ’1βˆ’π‘˜ξ€Έ(π‘₯)π‘›π‘˜.(1.9) For 𝐴∈𝐍 and π‘₯∈𝐼, Skvorcov [4] defined a transformation πœπ΄βˆΆπΌβ†’πΌ by𝜏𝐴(π‘₯)∢=π΄βˆ’1ξ“π‘˜=0π‘₯π΄βˆ’π‘˜βˆ’12βˆ’(π‘˜+1)+βˆžξ“π‘—=𝐴π‘₯𝑗2βˆ’(𝑗+1).(1.10) By the definition of 𝜏𝐴, we have (see [4])πœ…π‘›(π‘₯)=π‘Ÿ|𝑛|(π‘₯)π‘€π‘›βˆ’2|𝑛|ξ€·πœ|𝑛|ξ€Έ(π‘₯)(π‘›βˆˆπ,π‘₯∈𝐼).(1.11)

The Dirichlet kernels are defined by𝐷𝛼𝑛(π‘₯)∢=π‘›βˆ’1ξ“π‘˜=0π›Όπ‘˜(π‘₯),(1.12) where π›Όπ‘˜=π‘€π‘˜ for all π‘˜βˆˆπ or πœ…π‘˜ for all π‘˜βˆˆπ.

It is well-known that [2, 4]π·πœ…π‘›(π‘₯)=𝐷2|𝑛|(π‘₯)+𝑀2|𝑛|(π‘₯)π·π‘€π‘›βˆ’2|𝑛|ξ€·πœ|𝑛|ξ€Έ,𝐷(π‘₯)(1.13)2𝑛(π‘₯)∢=𝐷𝑀2𝑛(π‘₯)=π·πœ…2𝑛(⎧βŽͺ⎨βŽͺ⎩2π‘₯)=𝑛1,ifπ‘₯∈0,2𝑛,10,ifπ‘₯∈2𝑛.,1(1.14)

The 𝑛th FejΓ©r means of the Walsh-(Kaczmarz-)Fourier series of function 𝑓 are given byπœŽπ›Όπ‘›1(𝑓,π‘₯)∢=𝑛𝑛𝑗=1𝑆𝛼𝑗(𝑓,π‘₯),(1.15) where𝑆𝛼𝑛(𝑓,π‘₯)=π‘›βˆ’1ξ“π‘˜=0𝑓𝛼(π‘˜)π›Όπ‘˜(π‘₯)(1.16) is the 𝑛th partial sum of the Fourier series and ξπ‘“π›Όβˆ«(𝑛)∢=𝐼𝑓𝛼𝑛(π‘›βˆˆπ) is said to be the 𝑛th Walsh-(Kaczmarz-)Fourier coefficient of the function 𝑓.

The NΓΆrlund logarithmic (simply we say logarithmic) means and kernels of one-dimensional Walsh-(Kaczmarz-)Fourier series are defined as follows:𝑑𝛼𝑛1(𝑓,π‘₯)=π‘™π‘›π‘›βˆ’1ξ“π‘˜=1π‘†π›Όπ‘˜(𝑓,π‘₯)π‘›βˆ’π‘˜,𝐹𝛼𝑛1(𝑑)=π‘™π‘›π‘›βˆ’1ξ“π‘˜=1π·π›Όπ‘˜(𝑑)π‘›βˆ’π‘˜,(1.17) where 𝑙𝑛=βˆ‘π‘›βˆ’1π‘˜=11/π‘˜.

The Kronecker product (π›Όπ‘š,π‘›βˆΆπ‘›,π‘šβˆˆπ) of two Walsh(-Kaczmarz) systems is said to be the two-dimensional Walsh(-Kaczmarz) system. That is,π›Όπ‘š,𝑛(π‘₯,𝑦)=π›Όπ‘š(π‘₯)𝛼𝑛(𝑦).(1.18)

If π‘“βˆˆπΏ(𝐼2), then the number ξπ‘“π›Όβˆ«(π‘š,𝑛)∢=𝐼2π‘“π›Όπ‘š,𝑛(𝑛,π‘šβˆˆπ) is said to be the (π‘š,𝑛)th Walsh-(Kaczmarz-)Fourier coefficient of 𝑓. The rectangular partial sums of double Fourier series with respect to both system are defined byπ‘†π›Όπ‘š,𝑛(𝑓,π‘₯,𝑦)=π‘šβˆ’1𝑖=0π‘›βˆ’1𝑗=0𝑓𝛼(𝑖,𝑗)𝛼𝑖(π‘₯)𝛼𝑗(𝑦).(1.19)

The logarithmic means and kernels of quadratical partial sums of double Walsh-(Kaczmarz)-Fourier series are given by𝐭𝛼𝑛1(𝑓,π‘₯,𝑦)=π‘™π‘›π‘›βˆ’1ξ“π‘˜=1π‘†π›Όπ‘˜,π‘˜(𝑓,π‘₯,𝑦)π‘›βˆ’π‘˜,𝐅𝛼𝑛1(π‘₯,𝑦)=π‘™π‘›π‘›βˆ’1ξ“π‘˜=1π·π›Όπ‘˜(π‘₯)π·π›Όπ‘˜(𝑦)π‘›βˆ’π‘˜.(1.20)

It is evident that𝐭𝛼𝑛(ξ€œπ‘“,π‘₯,𝑦)βˆ’π‘“(π‘₯,𝑦)=𝐼2[]𝐅𝑓(π‘₯βŠ•π‘‘,π‘¦βŠ•π‘ )βˆ’π‘“(π‘₯,𝑦)𝛼𝑛(𝑑,𝑠)𝑑𝑑𝑑𝑠,(1.21) where βŠ• denotes the dyadic addition [3]. The 𝑛th Marcinkiewicz kernels are given by𝒦𝛼𝑛1(π‘₯,𝑦)∢=π‘›π‘›ξ“π‘˜=1π·π›Όπ‘˜(π‘₯)π·π›Όπ‘˜(𝑦).(1.22)

In 1948 Ε neΔ­der [5] showed that the inequalitylimsupπ‘›β†’βˆžπ·πœ…π‘›(π‘₯)log𝑛β‰₯𝐢>0(1.23) holds a.e. for the Walsh-Kaczmarz Dirichlet kernel. This inequality shows that the behavior of the Walsh-Kaczmarz system is worse than the behavior of the Walsh system in the Paley enumeration. This β€œspreadness” property of the kernel makes it easier to construct examples of divergent Fourier series [6]. On the other hand, Schipp [7] and Young [8] in 1974 proved that the Walsh-Kaczmarz system is a convergence system. Skvorcov in 1981 [4] showed that the FejΓ©r means with respect to the Walsh-Kaczmarz system converge uniformly to 𝑓 for any continuous functions 𝑓. For any integrable functions, GΓ‘t [9] proved that the FejΓ©r means with respect to the Walsh-Kaczmarz system converge a.e. to the function. Recently, GΓ‘t’s result was generalized by Simon [10, 11]. The a.e. convergence of the Walsh-Kaczmarz-Marcinkiewicz means of integrable functions was discussed by the second author [12].

The partial sums 𝑆𝑀𝑛(𝑓) of the Walsh-Fourier series of a function π‘“βˆˆπΏ(𝐼), 𝐼=[0,1) converge in measure on 𝐼 [2]. The condition π‘“βˆˆπΏlog+𝐿(𝐼2) provides convergence in measure on 𝐼2 of the rectangular partial sums 𝑆𝑀𝑛,π‘š(𝑓) of double Walsh-Fourier series [13]. The first example of a function from classes wider than 𝐿log+𝐿(𝐼2) with 𝑆𝑀𝑛,𝑛(𝑓) divergent in measure on 𝐼2 was obtained in [14, 15]. Moreover, in [16] Tkebuchava proved that in each Orlicz space wider than 𝐿log+𝐿(𝐼2) the set of functions where quadratic Walsh-Fourier sums converge in measure on 𝐼2 is of first Baire category (see Goginava [17] for Walsh-Kaczmarz-Fourier series). In the paper [18] it was showed that the NΓΆrlund logarithmic means π‘‘πœ…π‘›π‘“ of one-dimensional Walsh-Kaczmarz-Fourier series are of weak type (1,1), and this fact implies that π‘‘πœ…π‘›π‘“ converge in measure on 𝐼 for all functions π‘“βˆˆπΏ(𝐼) and π‘‘πœ…π‘›,π‘šπ‘“ converge in measure on 𝐼2 for all functions π‘“βˆˆπΏlog+𝐿(𝐼2).

At last, we note that the Walsh-NΓΆrlund logarithmic means are closer to the partial sums than to the classical logarithmic means or the FejΓ©r means. Namely, it was proved that there exists a function in a certain class of functions and a set with positive measure, such that the Walsh-NΓΆrlund logarithmic means of the function diverge on the set [19].

For results with respect to logarithmic means of cubical and rectangular partial sums of two-dimensional Walsh-Fourier series, see [17, 19–22].

In the present paper we investigate convergence in measure of logarithmic means of quadratical partial sums1π‘™π‘›π‘›βˆ’1𝑖=1𝑆𝑖,𝑖(𝑓,π‘₯,𝑦)π‘›βˆ’π‘–(1.24) of double Walsh-Kaczmarz series and prove Theorem 2.1 that is, for any Orlicz space, which is not a subspace of 𝐿log+𝐿(𝐼2), the set of the functions where logarithmic means converges in measure is of first Baire category. From this result it follows that (Corollary 2.2) in classes wider than 𝐿log+𝐿(𝐼2) there exists functions 𝑓 for which logarithmic means π­πœ…π‘›(𝑓) of quadratical partial sums of double Walsh-Kaczmarz series diverge in measure.

Thus, in question of convergence in measure logarithmic means of quadratical partial sums of double Walsh-Kaczmarz series differ from the Marcinkiewicz means and are like the usual quadratical partial sums of double Walsh-Fourier series.

2. Main Results

The main results of this paper are presented in the following proposition.

Theorem 2.1. Let 𝐿Φ(𝐼2) be an Orlicz space, such that 𝐿Φ𝐼2ξ€ΈΜΈβŠ†πΏlog+𝐿𝐼2ξ€Έ.(2.1)
Then the set of the functions in the Orlicz space 𝐿Φ(𝐼2) with logarithmic means of quadratical partial sums of double Walsh-Kaczmarz series convergent in measure on 𝐼2 is of first Baire category in 𝐿Φ(𝐼2).

Corollary 2.2. Let πœ‘βˆΆ[0,∞[β†’[0,∞[ be a nondecreasing function satisfying for π‘₯β†’+∞ the condition πœ‘(π‘₯)=π‘œ(π‘₯logπ‘₯).(2.2)
Then there exists a function π‘“βˆˆπΏ(𝐼2) such that
(a)ξ€œπΌ2πœ‘ξ€·||||𝑓(π‘₯,𝑦)𝑑π‘₯𝑑𝑦<∞,(2.3)(b)logarithmic means of quadratical partial sums of double Walsh-Kaczmarz series of 𝑓 diverge in measure on 𝐼2.

3. Auxiliary Results

It is well-known [2, 3] for the Dirichlet kernel function that||𝐷𝑀𝑛||<1(π‘₯)π‘₯(3.1) for any 0<π‘₯<1. Then for these π‘₯’s we also get||𝐹𝑀𝑛||<1(π‘₯)π‘₯,(3.2) where π‘›βˆˆπ is a nonnegative integer. It is also well-known for the Walsh-Paley-Dirichlet kernel functions the following lower pointwise estimates holds. Let 𝑝𝐴=22𝐴+β‹―+22+20 (𝐴∈𝐍). Then for any 2βˆ’2π΄βˆ’1≀π‘₯<1 and 𝐴∈𝐍, we have||𝐷𝑀𝑝𝐴||β‰₯1(π‘₯)4π‘₯.(3.3)

Since this inequality plays a prominent role in the proofs of some divergence results concerning the partial sums of the Fourier series, then it seems that it would be useful to get a similar inequality also for the logarithmic kernels. In [20] the first author, GΓ‘t, and Tkebuchava proved the inequality||𝐹𝑀𝑝𝐴||(π‘₯)β‰₯𝑐log(1/π‘₯)π‘₯log𝑝𝐴(3.4) for all 1β‰€π΄βˆˆπ, and π‘₯∈(2βˆ’2π΄βˆ’1,1)⧡𝐽. We have the exceptional set 𝐽, such that it is β€œrare around zero.” For 𝑑=𝑑0,𝑑0+1,…,2𝐴,𝑑0=inf{π‘‘βˆΆβŒŠ(𝑙𝑝[𝑑/2]βˆ’1/16)βˆ’215βŒ‹>1} set Μƒπ‘‘βˆΆ=⌊(𝑙𝑝[𝑑/2]βˆ’1/16)βˆ’215βŒ‹ (where βŒŠπ‘’βŒ‹ denotes the lower integral part of 𝑒), and we take a β€œsmall part’’ of the interval 𝐼𝑑⧡𝐼𝑑+1=[2βˆ’π‘‘βˆ’1,2βˆ’π‘‘). This way the intervals are defined as follows:Δ𝑑1∢=2𝑑+1,12𝑑+1+12𝑑+̃𝑑ξƒͺ.(3.5) The exceptional set 𝐽 is given by𝐽∢=βˆžξšπ‘‘=𝑑0Δ𝑑.(3.6)

For the logarithmic kernels π…πœ…π‘π΄ of quadratical partial sums of double Walsh-Kaczmarz-Fourier series, we prove an analogue result. To do this we need the following lemma of the first author, GΓ‘t and Tkebuchava [22].

Lemma 3.1 (see [22]). Let 𝑠,𝑑=0,1,…,2𝐴, 𝑠≀𝑑, and (π‘₯,𝑦)∈(𝐽𝑠×𝐽𝑑)⧡(𝑄(𝑠,𝑑)𝑑+̃𝑠βˆͺ(Δ𝑠×Δ𝑑)). Then one has ||𝐅𝑀𝑝𝐴||2(π‘₯,𝑦)β‰₯𝑐𝑑+𝑠𝑠log𝑝𝐴,(3.7) where π½π‘‘βˆΆ=𝐼𝑑⧡𝐼𝑑+1 and 𝑄𝐡(𝑠,𝑑)∢=π‘₯𝑠+1+β‹―+π‘₯π‘‘βˆ’11≀1π‘₯𝑑=0β‹―1π‘₯π΅βˆ’1=0𝐼𝐡0,…,0,1,π‘₯𝑠+1,…,π‘₯π΅βˆ’1×𝐼𝐡0,…,0,1,𝑦𝑑+1=π‘₯𝑑+1,…,π‘¦π΅βˆ’1=π‘₯π΅βˆ’1ξ€Έ.(3.8)

Now, for the Walsh-Kaczmarz logarithmic kernels π…πœ…π‘π΄, we prove the following.

Lemma 3.2. Let π‘₯∈𝐼2𝐴(1,π‘₯1,…,π‘₯π‘‘βˆ’2,π‘₯π‘‘βˆ’1=1,π‘₯𝑑,…,π‘₯π‘ βˆ’1,π‘₯𝑠=1,π‘₯𝑠+1=1,0,…,0)=βˆΆπΌπ‘‘,𝑠2𝐴,β€‰β€‰π‘¦βˆˆπΌ2𝐴(1,𝑦1,…,π‘¦π‘‘βˆ’2,π‘¦π‘‘βˆ’1=0,𝑦𝑑=1,𝑦𝑑+1𝐼=1,0,…,0)=βˆΆπ‘‘2𝐴 for 𝑠,𝑑=2,3,…,𝐴 and 𝑑≀𝑠. Then ||π…πœ…π‘π΄||(π‘₯,𝑦)β‰₯𝑐24π΄βˆ’π‘‘βˆ’π‘ .(3.9)

Proof. Set (π‘₯,𝑦)βˆˆπΌπ‘‘,𝑠2𝐴×𝐼𝑑2𝐴. Let 𝐆𝛼𝑝𝐴(π‘₯,𝑦)∢=𝑙𝑝𝐴𝐅𝛼𝑝𝐴(π‘₯,𝑦),𝐺𝛼𝑝𝐴(π‘₯)∢=𝑙𝑝𝐴𝐹𝛼𝑝𝐴(π‘₯)(3.10) for 𝛼=𝑀 or πœ…. Now, we write that π†πœ…π‘π΄(π‘₯,𝑦)=22𝐴𝑗=1π·πœ…π‘—(π‘₯)π·πœ…π‘—(𝑦)𝑝𝐴+βˆ’π‘—π‘π΄βˆ’1𝑗=22𝐴+1π·πœ…π‘—(π‘₯)π·πœ…π‘—(𝑦)π‘π΄βˆ’π‘—βˆΆ=𝐼+𝐼𝐼.(3.11)
First, we discuss 𝐼𝐼. We use equality  (1.13)𝐼𝐼=π‘π΄βˆ’1βˆ’1𝑗=1π·πœ…π‘—+22𝐴(π‘₯)π·πœ…π‘—+22𝐴(𝑦)π‘π΄βˆ’1βˆ’π‘—=π‘™π‘π΄βˆ’1𝐷22𝐴(π‘₯)𝐷22𝐴(𝑦)+𝐷22𝐴(π‘₯)π‘Ÿ2𝐴(𝑦)πΊπ‘€π‘π΄βˆ’1ξ€·πœ2𝐴(𝑦)+𝐷22𝐴(𝑦)π‘Ÿ2𝐴(π‘₯)πΊπ‘€π‘π΄βˆ’1ξ€·πœ2𝐴(π‘₯)+π‘Ÿ2𝐴(π‘₯)π‘Ÿ2𝐴(𝑦)π†π‘€π‘π΄βˆ’1ξ€·πœ2𝐴(π‘₯),𝜏2𝐴.(𝑦)(3.12)
If (π‘₯,𝑦)βˆˆπΌπ‘‘,𝑠2𝐴×𝐼𝑑2𝐴, then (see (1.14))𝐷22𝐴(π‘₯)=𝐷22𝐴𝜏(𝑦)=0,2𝐴(π‘₯)=0,…,0,π‘₯𝑠+1=1,π‘₯𝑠=1,π‘₯π‘ βˆ’1,…,π‘₯𝑑,π‘₯π‘‘βˆ’1=1,π‘₯π‘‘βˆ’2,…,π‘₯1,π‘₯0=1,π‘₯2𝐴,𝜏,…2𝐴(𝑦)=0,…,0,𝑦𝑑+1=1,𝑦𝑑=1,π‘¦π‘‘βˆ’1=0,π‘¦π‘‘βˆ’2,…,𝑦1,𝑦0=1,𝑦2𝐴.,…(3.13) Moreover, by Lemma 3.1, we have ||||=||π†πΌπΌπ‘€π‘π΄βˆ’1ξ€·πœ2𝐴(π‘₯),𝜏2𝐴||(𝑦)β‰₯𝑐(2π΄βˆ’π‘ )24π΄βˆ’π‘‘βˆ’π‘ .(3.14)
Now, we discuss 𝐼 by the help of  (1.13)𝐼=2π΄βˆ’12𝑙=0𝑙+1βˆ’1𝑗=2π‘™π·πœ…π‘—(π‘₯)π·πœ…π‘—(𝑦)𝑝𝐴+π·βˆ’π‘—22𝐴(π‘₯)𝐷22𝐴(𝑦)π‘π΄βˆ’1=2π΄βˆ’12𝑙=0π‘™βˆ’1𝑗=0π·πœ…π‘—+2𝑙(π‘₯)π·πœ…π‘—+2𝑙(𝑦)π‘π΄βˆ’π‘—βˆ’2𝑙=2π΄βˆ’12𝑙=0π‘™βˆ’1𝑗=0𝐷2𝑙(π‘₯)𝐷2𝑙(𝑦)π‘π΄βˆ’π‘—βˆ’2𝑙+2π΄βˆ’1𝑙=0𝐷2𝑙(π‘₯)2π‘™βˆ’1𝑗=0π‘Ÿπ‘™(𝑦)π·π‘€π‘—ξ€·πœπ‘™ξ€Έ(𝑦)π‘π΄βˆ’π‘—βˆ’2𝑙+2π΄βˆ’1𝑙=0𝐷2𝑙(𝑦)2π‘™βˆ’1𝑗=0π‘Ÿπ‘™(π‘₯)π·π‘€π‘—ξ€·πœπ‘™(ξ€Έπ‘₯)π‘π΄βˆ’π‘—βˆ’2𝑙+2π΄βˆ’12𝑙=0π‘™βˆ’1𝑗=0π‘Ÿπ‘™(π‘₯)π‘Ÿπ‘™(𝑦)π·π‘€π‘—ξ€·πœπ‘™(𝐷π‘₯)π‘€π‘—ξ€·πœπ‘™(𝑦)π‘π΄βˆ’π‘—βˆ’2𝑙.(3.15) Since π‘₯0=𝑦0=1, 𝐷2𝑙(π‘₯)=𝐷2𝑙(𝑦)=0 for all 𝑙β‰₯1, thus, 1𝐼=𝑝𝐴+βˆ’12π΄βˆ’1𝑙=1π‘Ÿπ‘™(π‘₯)π‘Ÿπ‘™(𝑦)2π‘™βˆ’1𝑗=1π·π‘€π‘—ξ€·πœπ‘™ξ€Έπ·(π‘₯)π‘€π‘—ξ€·πœπ‘™ξ€Έ(𝑦)π‘π΄βˆ’π‘—βˆ’2𝑙1=βˆΆπ‘π΄+βˆ’12π΄βˆ’1𝑙=1𝐼𝑙.(3.16)
We use Abel’s transformation for 𝐼𝑙 (𝑙β‰₯1)𝐼𝑙=π‘Ÿπ‘™(π‘₯)π‘Ÿπ‘™(𝑦)2π‘™βˆ’2𝑗=1ξ‚΅1π‘π΄βˆ’π‘—βˆ’2π‘™βˆ’1π‘π΄βˆ’π‘—βˆ’2π‘™ξ‚Άβˆ’1π‘—π’¦π‘€π‘—ξ€·πœπ‘™(π‘₯),πœπ‘™ξ€Έ(𝑦)+π‘Ÿπ‘™(π‘₯)π‘Ÿπ‘™ξ€·2(𝑦)π‘™ξ€Έπ’¦βˆ’1𝑀2π‘™βˆ’1ξ€·πœπ‘™(π‘₯),πœπ‘™(𝑦)π‘π΄βˆ’2𝑙+1+1=∢𝐼1𝑙+𝐼2𝑙.(3.17) We write ||𝐼1𝑙||≀𝑐224π΄π‘™βˆ’2𝑗=1𝑗||π’¦π‘€π‘—ξ€·πœπ‘™(π‘₯),πœπ‘™ξ€Έ||,||𝐼(𝑦)2𝑙||≀𝑐22𝐴2𝑙||π’¦βˆ’1𝑀2π‘™βˆ’1ξ€·πœπ‘™(π‘₯),πœπ‘™(ξ€Έ||.𝑦)(3.18) Since |𝐷𝑖|≀𝑖, we have 𝑛||𝒦𝑀𝑛||≀|||||𝑛𝑖=0𝐷𝑀𝑖𝐷𝑀𝑖|||||≀𝑐𝑛3.(3.19) For 𝑙≀𝑠+2, we obtain that ||𝐼1𝑙||2≀𝑐4𝑙24𝐴,||𝐼2𝑙||2≀𝑐3𝑙22𝐴.(3.20) Moreover, 𝑠+2𝑙=1||𝐼1𝑙||≀𝑐24𝐴𝑠+2𝑙=124𝑙2≀𝑐4𝑠24𝐴<𝑐,𝑠+2𝑙=1||𝐼2𝑙||≀𝑐22𝐴𝑠+2𝑙=123𝑙≀𝑐23𝑠22𝐴≀𝑐2𝐴.(3.21) Let 𝑠+2<𝑙<2𝐴. Then we have πœπ‘™ξ€·(π‘₯)=0,…,0,1,1,π‘₯π‘ βˆ’1,…,π‘₯π‘‘βˆ’1=1,…,π‘₯1,1,0,…,0,π‘₯2𝐴,𝜏,…𝑙(𝑦)=0,…,0,1,1,π‘¦π‘‘βˆ’1=0,π‘¦π‘‘βˆ’2,…,𝑦1,1,0,…,0,𝑦2𝐴,,…(3.22) that is, πœπ‘™(π‘₯)βˆˆπΌπ‘™βˆ’π‘ βˆ’3β§΅πΌπ‘™βˆ’π‘ βˆ’2, πœπ‘™(𝑦)βˆˆπΌπ‘™βˆ’π‘‘βˆ’3β§΅πΌπ‘™βˆ’π‘‘βˆ’2.
Now, we introduce the following notation:π’¦π‘€π‘Ž,𝑏(π‘₯,𝑦)∢=π‘Ž+π‘βˆ’1𝑗=π‘Žπ·π‘€π‘—(π‘₯)𝐷𝑀𝑗(𝑦),(3.23) and 𝑛(𝑠)βˆ‘βˆΆ=βˆžπ‘–=𝑠𝑛𝑖2𝑖 (𝑛,π‘ βˆˆπ). A simple consideration gives 𝑛𝒦𝑀𝑛(π‘₯,𝑦)=|𝑛|𝑠=0𝑛𝑠𝒦𝑛(𝑠+1),2𝑠(π‘₯,𝑦)+𝐷𝑀𝑛(π‘₯)𝐷𝑀𝑛(𝑦).(3.24) It is known [23] that if π‘‘β‰€π‘˜ and (π‘₯,𝑦)∈(𝐼𝑑⧡𝐼𝑑+1)Γ—(πΌπ‘˜β§΅πΌπ‘˜+1), then ||𝒦𝑀𝑛(𝑠+1),2𝑠||(π‘₯,𝑦)≀𝑐2𝑠+𝑑+π‘˜.(3.25) Using this inequality, we have that ||𝐼1𝑙||≀𝑐24π΄π‘™βˆ’12π‘˜=0π‘˜+1βˆ’1𝑗=2π‘˜π‘—||π’¦π‘€π‘—ξ€·πœπ‘™(π‘₯),πœπ‘™ξ€Έ||≀𝑐(𝑦)24π΄π‘™βˆ’12π‘˜=0π‘˜+1βˆ’1𝑗=2π‘˜ξƒ©π‘˜ξ“πœˆ=0|||𝒦𝑀𝑗(𝜈+1),2πœˆξ€·πœπ‘™(π‘₯),πœπ‘™(ξ€Έ|||+||𝐷𝑦)π‘—ξ€·πœπ‘™(𝐷π‘₯)π‘—ξ€·πœπ‘™(ξ€Έ||ξƒͺ≀𝑐𝑦)24π΄π‘™βˆ’12π‘˜=0π‘˜+1βˆ’1𝑗=2π‘˜ξƒ©π‘˜ξ“πœˆ=02𝜈+2π‘™βˆ’π‘‘βˆ’π‘ +𝑗2ξƒͺ2≀𝑐4π‘™βˆ’π‘‘βˆ’π‘ 24𝐴2+𝑐3𝑙24𝐴.(3.26) Moreover, 2π΄βˆ’1𝑙=𝑠+3||𝐼1𝑙||2≀𝑐8π΄βˆ’π‘‘βˆ’π‘ 24𝐴2+𝑐6𝐴24𝐴≀𝑐24π΄βˆ’π‘‘βˆ’π‘ .(3.27) Analogously, ||𝐼2𝑙||≀𝑐22π΄ξƒ©π‘™βˆ’1ξ“πœˆ=02𝜈+2π‘™βˆ’π‘‘βˆ’π‘ +22𝑙ξƒͺ2≀𝑐3π‘™βˆ’π‘‘βˆ’π‘ 22𝐴2+𝑐2𝑙22𝐴,(3.28)2π΄βˆ’1𝑙=𝑠+3||𝐼2𝑙||≀𝑐24π΄βˆ’π‘‘βˆ’π‘ .(3.29) Taking into account (3.11), we get ||π…πœ…π‘π΄||β‰₯||||βˆ’||𝐼||(π‘₯,𝑦)𝐼𝐼𝑙𝑝𝐴β‰₯𝑐1||||βˆ’||𝐼||𝐼𝐼2𝐴.(3.30) Therefore by (3.14)–(3.29), it follows that ||π…πœ…π‘π΄||(π‘₯,𝑦)β‰₯𝑐⋅𝑐1𝑠1βˆ’ξ‚22𝐴4π΄βˆ’π‘‘βˆ’π‘ βˆ’Μƒπ‘22𝐴4π΄βˆ’π‘‘βˆ’π‘ (3.31) with suitable constants 𝑐,𝑐1,̃𝑐>0. It is clear that for 𝐴 large enough we have ||π…πœ…π‘π΄||(π‘₯,𝑦)β‰₯π‘βˆ—24π΄βˆ’π‘‘βˆ’π‘ ,(3.32) where π‘βˆ—>0 is an absolute constant.

We apply the reasoning of [24] formulated as the following proposition in particular case.

Theorem A. Let 𝐻∢𝐿1(𝐼2)→𝐿0(𝐼2) be a linear continuous operator, which commutes with family of translations β„°, that is, for all πΈβˆˆβ„° for all π‘“βˆˆπΏ1(𝐼2)𝐻𝐸𝑓=𝐸𝐻𝑓. Let ‖𝑓‖𝐿1(𝐼2)=1 and πœ†>1. Then for any 1β‰€π‘Ÿβˆˆβ„• under condition, mes{(π‘₯,𝑦)∈𝐼2∢|𝐻𝑓|>πœ†}β‰₯1/π‘Ÿ, there exist 𝐸1,…,πΈπ‘Ÿ,πΈξ…ž1,…,πΈξ…žπ‘Ÿβˆˆβ„° and πœ€π‘–=Β±1, 𝑖=1,…,π‘Ÿ, such that ⎧βŽͺ⎨βŽͺ⎩mes(π‘₯,𝑦)∈𝐼2∢|||||π»ξƒ©π‘Ÿξ“π‘–=1πœ€π‘–π‘“ξ€·πΈπ‘–π‘₯,πΈξ…žπ‘–π‘¦ξ€Έξƒͺ|||||⎫βŽͺ⎬βŽͺ⎭β‰₯1>πœ†8.(3.33)

Theorem B. Let {π»π‘š}βˆžπ‘š=1 be a sequence of linear continuous operators, acting from Orlicz space 𝐿Φ(𝐼2) into the space 𝐿0(𝐼2). Suppose that there exist the sequence of functions {πœ‰π‘˜}βˆžπ‘˜=1 from unit bull 𝑆Φ(0,1) of space 𝐿Φ(𝐼2), and sequences of integers {π‘šπ‘˜}βˆžπ‘˜=1 and {πœˆπ‘˜}βˆžπ‘˜=1 increasing to infinity such that πœ€0=infπ‘˜ξ€½mes(π‘₯,𝑦)∈𝐼2∢||π»π‘šπ‘˜πœ‰π‘˜||(π‘₯,𝑦)>πœˆπ‘˜ξ€Ύ>0.(3.34)
Then 𝐡—the set of functions 𝑓 from space 𝐿Φ(𝐼2), for which the sequence {π»π‘šπ‘“} converges in measure to an a.e. finite functionβ€”is of first Baire category in space 𝐿Φ(𝐼2).

Theorem GGT. Let 𝐿Φ be an Orlicz space, and let πœ‘βˆΆ[0,∞)β†’[0,∞) be measurable function with condition πœ‘(π‘₯)=π‘œ(Ξ¦(π‘₯)) as π‘₯β†’βˆž. Then there exists Orlicz space πΏπœ”, such that πœ”(π‘₯)=π‘œ(Ξ¦(π‘₯))as π‘₯β†’βˆž, and πœ”(π‘₯)β‰₯πœ‘(π‘₯) for π‘₯β‰₯𝑐β‰₯0.

The proof of Theorems B and GGT can be found in [20].

4. Proof of the Theorem

Proof of Theorem 2.1. By Theorem B the proof of Theorem 2.1 will be complete if we show that there exist sequences of integers {π΄π‘˜βˆΆπ‘˜β‰₯1} and {πœˆπ‘˜βˆΆπ‘˜β‰₯1} increasing to infinity, and a sequence of functions {πœ‰π‘˜βˆΆπ‘˜β‰₯1} from the unit bull 𝑆Φ(0,1) of Orlicz space 𝐿Φ(𝐼2), such that for all π‘˜ξ‚†mes(π‘₯,𝑦)∈𝐼2∢|||π­πœ…π‘π΄π‘˜ξ€·πœ‰π‘˜ξ€Έ|||,π‘₯,𝑦>πœˆπ‘˜ξ‚‡β‰₯18.(4.1)
First, we prove that there exists 𝑐>0 such thatmes(π‘₯,𝑦)∈𝐼2∢|||π­πœ…π‘π΄π‘˜ξ€·π·22𝐴+1βŠ—π·22𝐴+1ξ€Έ|||;π‘₯,𝑦>𝑐23𝐴𝐴>𝑐23𝐴.(4.2)
It is easy to show thatπ­πœ…π‘π΄π‘˜ξ€·π·22𝐴+1βŠ—π·22𝐴+1ξ€Έ;π‘₯,𝑦=𝑆22𝐴+1,22𝐴+1ξ€·π…πœ…π‘π΄ξ€Έ;π‘₯,𝑦=π…πœ…π‘π΄(π‘₯,𝑦).(4.3) Hence, from Lemma 3.2, we can write |||π­πœ…π‘π΄π‘˜ξ€·π·22𝐴+1βŠ—π·22𝐴+1ξ€Έ|||=||𝐅;π‘₯,π‘¦πœ…π‘π΄||(π‘₯,𝑦)β‰₯𝑐24π΄βˆ’π‘‘βˆ’π‘ ,(4.4) for (π‘₯,𝑦)βˆˆπΌπ‘‘,𝑠2𝐴×𝐼𝑑2𝐴.
Since𝑐24π΄βˆ’π‘‘βˆ’π‘ >𝑐23𝐴(4.5) for 𝑑+𝑠<𝐴 and 0≀𝑑≀𝑠<𝐴, from (4.4) we conclude that ξ€½mes(π‘₯,𝑦)∈𝐼2∢||π­πœ…π‘π΄ξ€·π·22𝐴+1βŠ—π·22𝐴+1ξ€Έ||;π‘₯,𝑦>𝑐23𝐴β‰₯𝑐𝐴[]𝑠=𝐴/2π΄βˆ’π‘ ξ“π‘‘=0𝐼mes𝑑,𝑠2𝐴×𝐼𝑑2𝐴β‰₯𝑐𝐴[]𝑠=𝐴/2π΄βˆ’π‘ ξ“π‘‘=02𝑠22𝐴2𝑑22𝐴β‰₯𝑐𝐴[]𝑠=𝐴/22𝑠22𝐴2π΄βˆ’π‘ 22𝐴β‰₯𝑐𝐴23𝐴.(4.6) Hence, (4.2) is proved.
From the condition of the theorem, we write [1]liminfπ‘’β†’βˆžΞ¦(𝑒)𝑒log𝑒=0.(4.7) Consequently, there exists a sequence of integers {π΄π‘˜}βˆžπ‘˜=1 increasing to infinity, such that limπ‘˜β†’βˆžΞ¦ξ€·24π΄π‘˜ξ€Έ2βˆ’4π΄π‘˜π΄π‘˜βˆ’1Ξ¦ξ€·2=0,4π΄π‘˜ξ€Έ24π΄π‘˜β‰₯4,βˆ€π‘˜.(4.8)
From (4.2) we writemes(π‘₯,𝑦)∈𝐼2∢|||π­πœ…π‘π΄π‘˜ξ€·π·2π‘˜2𝐴+1βŠ—π·2π‘˜2𝐴+1ξ€Έ|||;π‘₯,𝑦>𝑐23π΄π‘˜ξ‚‡π΄>π‘π‘˜23π΄π‘˜.(4.9) Then by the virtue of Theorem A, there exists 𝑒1,…,π‘’π‘Ÿ,π‘’ξ…ž1,…,π‘’ξ…žπ‘Ÿβˆˆ[0,1] and πœ€1,…,πœ€π‘Ÿ=Β±1 such that ⎧βŽͺ⎨βŽͺ⎩mes(π‘₯,𝑦)∈𝐼2∢|||||π‘Ÿξ“π‘–=1πœ€π‘–π­πœ…π‘π΄π‘˜ξ€·π·2π‘˜2𝐴+1βŠ—π·2π‘˜2𝐴+1,π‘’π‘–βŠ•π‘₯,π‘’ξ…žπ‘–ξ€Έ|||||βŠ•π‘¦>23π΄π‘˜βŽ«βŽͺ⎬βŽͺ⎭>18,(4.10) where π‘Ÿ=[23π΄π‘˜/(π‘π΄π‘˜)]+1.
Setπœ‰π‘˜2(π‘₯,𝑦)=4π΄π‘˜βˆ’1Ξ¦ξ€·24π΄π‘˜ξ€Έπ‘€π‘˜(π‘₯,𝑦),(4.11) where π‘€π‘˜1(π‘₯,𝑦)=π‘Ÿπ‘Ÿξ“π‘–=1πœ€π‘–π·2π‘˜2𝐴+1ξ€·π‘’π‘–ξ€Έπ·βŠ•π‘₯2π‘˜2𝐴+1ξ€·π‘’ξ…žπ‘–ξ€ΈβŠ•π‘¦.(4.12) Denote πœˆπ‘˜=27π΄π‘˜βˆ’1ξ€·2π‘ŸΞ¦4π΄π‘˜ξ€Έ.(4.13) It is easy to show that mes(π‘₯,𝑦)∈𝐼2∢|||π­πœ…π‘π΄π‘˜ξ€·πœ‰π‘˜ξ€Έ|||,π‘₯,𝑦>πœˆπ‘˜ξ‚‡βŽ§βŽͺ⎨βŽͺ⎩=mes(π‘₯,𝑦)∈𝐼2∢|||||π‘Ÿξ“π‘–=1πœ€π‘–π­πœ…π‘π΄π‘˜ξ€·π·2π‘˜2𝐴+1βŠ—π·2π‘˜2𝐴+1,π‘’π‘–βŠ•π‘₯,π‘’ξ…žπ‘–ξ€Έ|||||βŠ•π‘¦>23π΄π‘˜βŽ«βŽͺ⎬βŽͺ⎭>18.(4.14) Hence, (4.1) is proved.
Sinceβ€–β€–π‘€π‘˜β€–β€–πΏβˆž(𝐼2)≀24π‘šπ‘˜+2,β€–β€–π‘€π‘˜β€–β€–πΏ1(𝐼2)≀1.(4.15) Moreover, ||||ξ€œπΌ2||||≀1𝑓𝑔2ξ‚Έξ€œπΌ2Ξ¦ξ€·2||𝑓||ξ€Έξ‚Ή,+1Ξ¦(𝑒)𝑒<Ξ¦ξ€·π‘’ξ…žξ€Έπ‘’ξ…žξ€·0<𝑒<π‘’ξ…žξ€Έ.(4.16) From (4.8) we can write β€–β€–πœ‰π‘˜β€–β€–πΏΞ¦(𝐼2)≀12ξƒ¬ξ€œ1+𝐼2Φ24π΄π‘˜||π‘€π‘˜||(π‘₯,𝑦)Ξ¦ξ€·24π΄π‘˜ξ€Έξƒͺ≀1𝑑π‘₯𝑑𝑦2ξƒ¬ξ€œ1+𝐼2Ξ¦ξ€·24π΄π‘˜24π΄π‘˜+2/ξ€·Ξ¦ξ€·24π΄π‘˜ξ€Έξ€Έξ€Έ24π΄π‘˜ξ€·24π΄π‘˜+2/ξ€·Ξ¦ξ€·24π΄π‘˜2ξ€Έξ€Έξ€Έ4π΄π‘˜||π‘€π‘˜(||π‘₯,𝑦)Ξ¦ξ€·24π΄π‘˜ξ€Έξƒ­β‰€1𝑑π‘₯𝑑𝑦2ξƒ¬ξ€œ1+𝐼2Ξ¦ξ€·24π΄π‘˜ξ€Έ24π΄π‘˜24π΄π‘˜||π‘€π‘˜||(π‘₯,𝑦)Ξ¦ξ€·24π΄π‘˜ξ€Έξƒ­π‘‘π‘₯𝑑𝑦≀1.(4.17)
Hence, πœ‰π‘˜βˆˆπ‘†Ξ¦(0,1), and Theorem 2.1 is proved.

The validity of Corollary 2.2 follows immediately from Theorems 2.1 and GGT.

References

  1. M. A. Krasnosel'skii and Y. B. Rutickii, Convex Functions and Orlicz Space, P. Noorhoff, Groningen, The Netherlands, 1961.
  2. B. I. Golubov, A. V. Efimov, and V. A. Skvorcov, Series and Transformations of Walsh, Nauka, Moscow, Russia, 1987.
  3. F. Schipp, W. R. Wade, and P. Simon, Walsh Series, Introduction to Dyadic Harmonic Analysis, Adam Hilger Ltd, Bristol, UK, 1990.
  4. V. A. Skvorcov, β€œOn Fourier series with respect to the Walsh-Kaczmarz system,” Analysis Mathematica, vol. 7, no. 2, pp. 141–150, 1981. View at Publisher Β· View at Google Scholar
  5. A. A. Ε neΔ­der, β€œOn series of Walsh functions with monotonic coefficients,” Izvestiya Akademii Nauk SSSR, vol. 12, pp. 179–192, 1948. View at Google Scholar
  6. L. A. BalaΕ‘ov, β€œSeries with respect to the Walsh system with monotone coefficients,” SibirskiΔ­ MatematičeskiΔ­ Ε½urnal, vol. 12, pp. 25–39, 1971 (Russian). View at Google Scholar
  7. F. Schipp, β€œPointwise convergence of expansions with respect to certain product systems,” Analysis Mathematica, vol. 2, no. 1, pp. 65–76, 1976. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  8. W. S. Young, β€œOn the a.e. convergence of Walsh-Kaczmarz-Fourier series,” Proceedings of the American Mathematical Society, vol. 44, pp. 353–358, 1974. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  9. G. GΓ‘t, β€œOn (C, 1) summability of integrable functions with respect to the Walsh-Kaczmarz system,” Studia Mathematica, vol. 130, no. 2, pp. 135–148, 1998. View at Google Scholar Β· View at Zentralblatt MATH
  10. P. Simon, β€œOn the Cesaro summability with respect to the Walsh-Kaczmarz system,” Journal of Approximation Theory, vol. 106, no. 2, pp. 249–261, 2000. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  11. P. Simon, β€œ(C,Ξ±) summability of Walsh-Kaczmarz-Fourier series,” Journal of Approximation Theory, vol. 127, no. 1, pp. 39–60, 2004. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  12. K. Nagy, β€œOn the two-dimensional Marcinkiewicz means with respect to Walsh-Kaczmarz system,” Journal of Approximation Theory, vol. 142, no. 2, pp. 138–165, 2006. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  13. L. V. Zhizhiashvili, β€œSome problems of multidimensional harmonic analysis,” TGU, Tbilisi, 1996.
  14. R. Getsadze, β€œOn the divergence in measure of multiple Fourier seties,” Some Problems of Functions Theory, vol. 4, pp. 84–117, 1988 (Russian). View at Google Scholar
  15. S. V. Konyagin, β€œOn a subsequence of Fourier-Walsh partial sums,” RossiΔ­skaya Akademiya Nauk, vol. 54, no. 4, pp. 69–75, 1993. View at Publisher Β· View at Google Scholar
  16. G. Tkebuchava, β€œSubsequences of partial sums of multiple Fourier and Fourier-Walsh series,” Bulletin of the Georgian Academy of Sciences, vol. 169, no. 2, pp. 252–253, 2004. View at Google Scholar
  17. U. Goginava, β€œConvergence in measure of partial sums of double Fourier series with respect to the Walsh-Kaczmarz system,” Journal of Mathematical Analysis and Approximation Theory, vol. 2, no. 2, pp. 160–167, 2007. View at Google Scholar Β· View at Zentralblatt MATH
  18. U. Goginava and K. Nagy, β€œWeak type inequality for logarithmic means of Walsh-Kaczmarz-Fourier series,” Real Analysis Exchange, vol. 35, no. 2, pp. 445–461, 2010. View at Google Scholar Β· View at Zentralblatt MATH
  19. G. GΓ‘t and U. Goginava, β€œOn the divergence of NΓΆrlund logarithmic means of Walsh-Fourier series,” Acta Mathematica Sinica, vol. 25, no. 6, pp. 903–916, 2009. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  20. G. GΓ‘t, U. Goginava, and G. Tkebuchava, β€œConvergence in measure of logarithmic means of double Walsh-Fourier series,” Georgian Mathematical Journal, vol. 12, no. 4, pp. 607–618, 2005. View at Google Scholar Β· View at Zentralblatt MATH
  21. G. GΓ‘t, U. Goginava, and G. Tkebuchava, β€œConvergence of logarithmic means of multiple Walsh-Fourier series,” Analysis in Theory and Applications, vol. 21, no. 4, pp. 326–338, 2005. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  22. G. GΓ‘t, U. Goginava, and G. Tkebuchava, β€œConvergence in measure of logarithmic means of quadratical partial sums of double Walsh-Fourier series,” Journal of Mathematical Analysis and Applications, vol. 323, no. 1, pp. 535–549, 2006. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  23. K. Nagy, β€œSome convergence properties of the Walsh-Kaczmarz system with respect to the Marcinkiewicz means,” Rendiconti del Circolo Matematico di Palermo. Serie II, no. 76, pp. 503–516, 2005. View at Google Scholar Β· View at Zentralblatt MATH
  24. A. M. Garsia, Topics in Almost Everywhere Convergence, vol. 4, Markham Publishing Co., Chicago, Ill, USA, 1970.