Journal of Function Spaces

Journal of Function Spaces / 2012 / Article

Research Article | Open Access

Volume 2012 |Article ID 582726 | https://doi.org/10.1155/2012/582726

Ushangi Goginava, Kรกroly Nagy, "Convergence in Measure of Logarithmic Means of Quadratical Partial Sums of Double Walsh-Kaczmarz-Fourier Series", Journal of Function Spaces, vol. 2012, Article ID 582726, 15 pages, 2012. https://doi.org/10.1155/2012/582726

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

Academic Editor: Anna Kaminska
Received31 Jan 2011
Accepted03 Apr 2011
Published04 Jan 2012

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 ๐ฟโ€‰log+โ€‰๐ฟ(๐ผ2), 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 ๐ฟ0=๐ฟ0(๐ผ2) the Lebesgue space of functions that are measurable and finite almost everywhere on ๐ผ2=[0,1)ร—[0,1). mes(๐ด) is the Lebesgue measure of the set ๐ดโŠ‚๐ผ2. The constants appearing in this paper are denoted by ๐‘.

Let ๐ฟฮฆ=๐ฟฮฆ(๐ผ2) be the Orlicz space [1] generated by Youngโ€™s function ฮฆ; that is, ฮฆ is convex, continuous, even function such that ฮฆ(0)=0 andlim๐‘ขโ†’+โˆžฮฆ(๐‘ข)๐‘ข=+โˆž,lim๐‘ขโ†’0ฮฆ(๐‘ข)๐‘ข=0.(1.1)

This space is endowed with the normโ€–๐‘“โ€–๐ฟฮฆ(๐ผ2)๎‚ป๎€œ=inf๐‘˜>0โˆถ๐ผ2ฮฆ๎‚ต||||๐‘“(๐‘ฅ,๐‘ฆ)๐‘˜๎‚ถ๎‚ผ๐‘‘๐‘ฅ๐‘‘๐‘ฆโ‰ค1.(1.2) In particular case, if ฮฆ(๐‘ข)=๐‘ขln(1+๐‘ข), ๐‘ข>0, then the corresponding space is denoted by ๐ฟlog+๐ฟ(๐ผ2).

We denote the set of nonnegative integers by ๐. By a dyadic interval in ๐ผโˆถ=[0,1) we mean one of the form [๐‘/2๐‘›,(๐‘+1)/2๐‘›) for some ๐‘โˆˆ๐, 0โ‰ค๐‘<2๐‘›. Given ๐‘›โˆˆ๐ and ๐‘ฅโˆˆ[0,1), let ๐ผ๐‘›(๐‘ฅ) denote the dyadic interval of length 2โˆ’๐‘› which contains the point ๐‘ฅ.

Every point ๐‘ฅโˆˆ๐ผ can be written in the following form:๐‘ฅ=โˆž๎“๐‘˜=0๐‘ฅ๐‘˜2๐‘˜+1๎€ท๐‘ฅ=โˆถ0,๐‘ฅ1,โ€ฆ,๐‘ฅ๐‘›๎€ธ,โ€ฆ,๐‘ฅ๐‘˜โˆˆ{0,1}.(1.3) In that case when there are two different forms we choose the one for which lim๐‘˜โ†’โˆž๐‘ฅ๐‘˜=0. We use the notation๐‘’๐‘—1โˆถ=2๐‘—+1=๎€ท0,โ€ฆ,0,๐‘ฅ๐‘—๎€ธ=1,0,โ€ฆ.(1.4)

It is well-known that [2, 3]๐ผ๐‘›๎€ท๐‘ฅ0,โ€ฆ,๐‘ฅ๐‘›โˆ’1๎€ธโˆถ=๐ผ๐‘›๎‚ธ๐‘(๐‘ฅ)=2๐‘›,๐‘+12๐‘›๎‚ถ,(1.5) where โˆ‘๐‘=๐‘›โˆ’1๐‘—=0๐‘ฅ๐‘—2๐‘›โˆ’1โˆ’๐‘—.

Let ๐‘Ÿ0(๐‘ฅ) be a function defined by๐‘Ÿ0โŽงโŽชโŽจโŽชโŽฉ๎‚ƒ1(๐‘ฅ)=1,if๐‘ฅโˆˆ0,2๎‚,๎‚ƒ1โˆ’1,if๐‘ฅโˆˆ2๎‚,๐‘Ÿ,10(๐‘ฅ+1)=๐‘Ÿ0(๐‘ฅ).(1.6) The Rademacher functions are defined by๐‘Ÿ๐‘›(๐‘ฅ)=๐‘Ÿ0(2๐‘›[๐‘ฅ),๐‘›โ‰ฅ0,๐‘ฅโˆˆ0,1).(1.7)

Let ๐‘ค0,๐‘ค1,โ€ฆ represent the Walsh functions, that is, ๐‘ค0(๐‘ฅ)=1, and if โˆ‘๐‘›=โˆž๐‘–=0๐‘›๐‘–2๐‘– is a positive integer with ๐‘›๐‘–โˆˆ{0,1}, then๐‘ค๐‘›(๐‘ฅ)โˆถ=โˆž๎‘๐‘–=0๎€ท๐‘Ÿ๐‘–๎€ธ(๐‘ฅ)๐‘›๐‘–.(1.8) The order of the ๐‘› is denoted by |๐‘›|โˆถ=max{๐‘—โˆˆ๐โˆถ๐‘›๐‘—โ‰ 0}.

The Walsh-Kaczmarz functions are given by ๐œ…0โˆถ=1, and for ๐‘›โ‰ฅ1๐œ…๐‘›(๐‘ฅ)โˆถ=๐‘Ÿ|๐‘›|(๐‘ฅ)|๐‘›|โˆ’1๎‘๐‘˜=0๎€ท๐‘Ÿ|๐‘›|โˆ’1โˆ’๐‘˜๎€ธ(๐‘ฅ)๐‘›๐‘˜.(1.9) For ๐ดโˆˆ๐ and ๐‘ฅโˆˆ๐ผ, Skvorcov [4] defined a transformation ๐œ๐ดโˆถ๐ผโ†’๐ผ by๐œ๐ด(๐‘ฅ)โˆถ=๐ดโˆ’1๎“๐‘˜=0๐‘ฅ๐ดโˆ’๐‘˜โˆ’12โˆ’(๐‘˜+1)+โˆž๎“๐‘—=๐ด๐‘ฅ๐‘—2โˆ’(๐‘—+1).(1.10) By the definition of ๐œ๐ด, we have (see [4])๐œ…๐‘›(๐‘ฅ)=๐‘Ÿ|๐‘›|(๐‘ฅ)๐‘ค๐‘›โˆ’2|๐‘›|๎€ท๐œ|๐‘›|๎€ธ(๐‘ฅ)(๐‘›โˆˆ๐,๐‘ฅโˆˆ๐ผ).(1.11)

The Dirichlet kernels are defined by๐ท๐›ผ๐‘›(๐‘ฅ)โˆถ=๐‘›โˆ’1๎“๐‘˜=0๐›ผ๐‘˜(๐‘ฅ),(1.12) where ๐›ผ๐‘˜=๐‘ค๐‘˜ for all ๐‘˜โˆˆ๐ or ๐œ…๐‘˜ for all ๐‘˜โˆˆ๐.

It is well-known that [2, 4]๐ท๐œ…๐‘›(๐‘ฅ)=๐ท2|๐‘›|(๐‘ฅ)+๐‘ค2|๐‘›|(๐‘ฅ)๐ท๐‘ค๐‘›โˆ’2|๐‘›|๎€ท๐œ|๐‘›|๎€ธ,๐ท(๐‘ฅ)(1.13)2๐‘›(๐‘ฅ)โˆถ=๐ท๐‘ค2๐‘›(๐‘ฅ)=๐ท๐œ…2๐‘›(โŽงโŽชโŽจโŽชโŽฉ2๐‘ฅ)=๐‘›๎‚ƒ1,if๐‘ฅโˆˆ0,2๐‘›๎‚,๎‚ƒ10,if๐‘ฅโˆˆ2๐‘›๎‚.,1(1.14)

The ๐‘›th Fejรฉr means of the Walsh-(Kaczmarz-)Fourier series of function ๐‘“ are given by๐œŽ๐›ผ๐‘›1(๐‘“,๐‘ฅ)โˆถ=๐‘›๐‘›๎“๐‘—=1๐‘†๐›ผ๐‘—(๐‘“,๐‘ฅ),(1.15) where๐‘†๐›ผ๐‘›(๐‘“,๐‘ฅ)=๐‘›โˆ’1๎“๐‘˜=0๎๐‘“๐›ผ(๐‘˜)๐›ผ๐‘˜(๐‘ฅ)(1.16) is the ๐‘›th partial sum of the Fourier series and ๎๐‘“๐›ผโˆซ(๐‘›)โˆถ=๐ผ๐‘“๐›ผ๐‘›(๐‘›โˆˆ๐) is said to be the ๐‘›th Walsh-(Kaczmarz-)Fourier coefficient of the function ๐‘“.

The Nรถrlund logarithmic (simply we say logarithmic) means and kernels of one-dimensional Walsh-(Kaczmarz-)Fourier series are defined as follows:๐‘ก๐›ผ๐‘›1(๐‘“,๐‘ฅ)=๐‘™๐‘›๐‘›โˆ’1๎“๐‘˜=1๐‘†๐›ผ๐‘˜(๐‘“,๐‘ฅ)๐‘›โˆ’๐‘˜,๐น๐›ผ๐‘›1(๐‘ก)=๐‘™๐‘›๐‘›โˆ’1๎“๐‘˜=1๐ท๐›ผ๐‘˜(๐‘ก)๐‘›โˆ’๐‘˜,(1.17) where ๐‘™๐‘›=โˆ‘๐‘›โˆ’1๐‘˜=11/๐‘˜.

The Kronecker product (๐›ผ๐‘š,๐‘›โˆถ๐‘›,๐‘šโˆˆ๐) of two Walsh(-Kaczmarz) systems is said to be the two-dimensional Walsh(-Kaczmarz) system. That is,๐›ผ๐‘š,๐‘›(๐‘ฅ,๐‘ฆ)=๐›ผ๐‘š(๐‘ฅ)๐›ผ๐‘›(๐‘ฆ).(1.18)

If ๐‘“โˆˆ๐ฟ(๐ผ2), then the number ๎๐‘“๐›ผโˆซ(๐‘š,๐‘›)โˆถ=๐ผ2๐‘“๐›ผ๐‘š,๐‘›(๐‘›,๐‘šโˆˆ๐) is said to be the (๐‘š,๐‘›)th Walsh-(Kaczmarz-)Fourier coefficient of ๐‘“. The rectangular partial sums of double Fourier series with respect to both system are defined by๐‘†๐›ผ๐‘š,๐‘›(๐‘“,๐‘ฅ,๐‘ฆ)=๐‘šโˆ’1๎“๐‘–=0๐‘›โˆ’1๎“๐‘—=0๎๐‘“๐›ผ(๐‘–,๐‘—)๐›ผ๐‘–(๐‘ฅ)๐›ผ๐‘—(๐‘ฆ).(1.19)

The logarithmic means and kernels of quadratical partial sums of double Walsh-(Kaczmarz)-Fourier series are given by๐ญ๐›ผ๐‘›1(๐‘“,๐‘ฅ,๐‘ฆ)=๐‘™๐‘›๐‘›โˆ’1๎“๐‘˜=1๐‘†๐›ผ๐‘˜,๐‘˜(๐‘“,๐‘ฅ,๐‘ฆ)๐‘›โˆ’๐‘˜,๐…๐›ผ๐‘›1(๐‘ฅ,๐‘ฆ)=๐‘™๐‘›๐‘›โˆ’1๎“๐‘˜=1๐ท๐›ผ๐‘˜(๐‘ฅ)๐ท๐›ผ๐‘˜(๐‘ฆ)๐‘›โˆ’๐‘˜.(1.20)

It is evident that๐ญ๐›ผ๐‘›(๎€œ๐‘“,๐‘ฅ,๐‘ฆ)โˆ’๐‘“(๐‘ฅ,๐‘ฆ)=๐ผ2[]๐…๐‘“(๐‘ฅโŠ•๐‘ก,๐‘ฆโŠ•๐‘ )โˆ’๐‘“(๐‘ฅ,๐‘ฆ)๐›ผ๐‘›(๐‘ก,๐‘ )๐‘‘๐‘ก๐‘‘๐‘ ,(1.21) where โŠ• denotes the dyadic addition [3]. The ๐‘›th Marcinkiewicz kernels are given by๐’ฆ๐›ผ๐‘›1(๐‘ฅ,๐‘ฆ)โˆถ=๐‘›๐‘›๎“๐‘˜=1๐ท๐›ผ๐‘˜(๐‘ฅ)๐ท๐›ผ๐‘˜(๐‘ฆ).(1.22)

In 1948 ล neฤญder [5] showed that the inequalitylimsup๐‘›โ†’โˆž๐ท๐œ…๐‘›(๐‘ฅ)log๐‘›โ‰ฅ๐ถ>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 โ€œspreadnessโ€ 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 Fejรฉr means with respect to the Walsh-Kaczmarz system converge uniformly to ๐‘“ for any continuous functions ๐‘“. For any integrable functions, Gรกt [9] proved that the Fejรฉr means with respect to the Walsh-Kaczmarz system converge a.e. to the function. Recently, Gรกtโ€™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 ๐‘†๐‘ค๐‘›(๐‘“) of the Walsh-Fourier series of a function ๐‘“โˆˆ๐ฟ(๐ผ), ๐ผ=[0,1) converge in measure on ๐ผ [2]. The condition ๐‘“โˆˆ๐ฟlog+๐ฟ(๐ผ2) provides convergence in measure on ๐ผ2 of the rectangular partial sums ๐‘†๐‘ค๐‘›,๐‘š(๐‘“) of double Walsh-Fourier series [13]. The first example of a function from classes wider than ๐ฟlog+๐ฟ(๐ผ2) with ๐‘†๐‘ค๐‘›,๐‘›(๐‘“) divergent in measure on ๐ผ2 was obtained in [14, 15]. Moreover, in [16] Tkebuchava proved that in each Orlicz space wider than ๐ฟlog+๐ฟ(๐ผ2) the set of functions where quadratic Walsh-Fourier sums converge in measure on ๐ผ2 is of first Baire category (see Goginava [17] for Walsh-Kaczmarz-Fourier series). In the paper [18] it was showed that the Nรถrlund logarithmic means ๐‘ก๐œ…๐‘›๐‘“ of one-dimensional Walsh-Kaczmarz-Fourier series are of weak type (1,1), and this fact implies that ๐‘ก๐œ…๐‘›๐‘“ converge in measure on ๐ผ for all functions ๐‘“โˆˆ๐ฟ(๐ผ) and ๐‘ก๐œ…๐‘›,๐‘š๐‘“ converge in measure on ๐ผ2 for all functions ๐‘“โˆˆ๐ฟlog+๐ฟ(๐ผ2).

At last, we note that the Walsh-Nรถrlund logarithmic means are closer to the partial sums than to the classical logarithmic means or the Fejรฉr 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-Nรถrlund 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, 19โ€“22].

In the present paper we investigate convergence in measure of logarithmic means of quadratical partial sums1๐‘™๐‘›๐‘›โˆ’1๎“๐‘–=1๐‘†๐‘–,๐‘–(๐‘“,๐‘ฅ,๐‘ฆ)๐‘›โˆ’๐‘–(1.24) of double Walsh-Kaczmarz series and prove Theorem 2.1 that is, for any Orlicz space, which is not a subspace of ๐ฟlog+๐ฟ(๐ผ2), 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 ๐ฟlog+๐ฟ(๐ผ2) there exists functions ๐‘“ for which logarithmic means ๐ญ๐œ…๐‘›(๐‘“) 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 ๐ฟฮฆ(๐ผ2) be an Orlicz space, such that ๐ฟฮฆ๎€ท๐ผ2๎€ธฬธโІ๐ฟlog+๐ฟ๎€ท๐ผ2๎€ธ.(2.1)
Then the set of the functions in the Orlicz space ๐ฟฮฆ(๐ผ2) with logarithmic means of quadratical partial sums of double Walsh-Kaczmarz series convergent in measure on ๐ผ2 is of first Baire category in ๐ฟฮฆ(๐ผ2).

Corollary 2.2. Let ๐œ‘โˆถ[0,โˆž[โ†’[0,โˆž[ be a nondecreasing function satisfying for ๐‘ฅโ†’+โˆž the condition ๐œ‘(๐‘ฅ)=๐‘œ(๐‘ฅlog๐‘ฅ).(2.2)
Then there exists a function ๐‘“โˆˆ๐ฟ(๐ผ2) such that
(a)๎€œ๐ผ2๐œ‘๎€ท||||๎€ธ๐‘“(๐‘ฅ,๐‘ฆ)๐‘‘๐‘ฅ๐‘‘๐‘ฆ<โˆž,(2.3)(b)logarithmic means of quadratical partial sums of double Walsh-Kaczmarz series of ๐‘“ diverge in measure on ๐ผ2.

3. Auxiliary Results

It is well-known [2, 3] for the Dirichlet kernel function that||๐ท๐‘ค๐‘›||<1(๐‘ฅ)๐‘ฅ(3.1) for any 0<๐‘ฅ<1. Then for these ๐‘ฅโ€™s we also get||๐น๐‘ค๐‘›||<1(๐‘ฅ)๐‘ฅ,(3.2) where ๐‘›โˆˆ๐ is a nonnegative integer. It is also well-known for the Walsh-Paley-Dirichlet kernel functions the following lower pointwise estimates holds. Let ๐‘๐ด=22๐ด+โ‹ฏ+22+20 (๐ดโˆˆ๐). Then for any 2โˆ’2๐ดโˆ’1โ‰ค๐‘ฅ<1 and ๐ดโˆˆ๐, we have||๐ท๐‘ค๐‘๐ด||โ‰ฅ1(๐‘ฅ)4๐‘ฅ.(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, Gรกt, and Tkebuchava proved the inequality||๐น๐‘ค๐‘๐ด||(๐‘ฅ)โ‰ฅ๐‘log(1/๐‘ฅ)๐‘ฅlog๐‘๐ด(3.4) for all 1โ‰ค๐ดโˆˆ๐, and ๐‘ฅโˆˆ(2โˆ’2๐ดโˆ’1,1)โงต๐ฝ. We have the exceptional set ๐ฝ, such that it is โ€œrare around zero.โ€ For ๐‘ก=๐‘ก0,๐‘ก0+1,โ€ฆ,2๐ด,๐‘ก0=inf{๐‘กโˆถโŒŠ(๐‘™๐‘[๐‘ก/2]โˆ’1/16)โˆ’215โŒ‹>1} set ฬƒ๐‘กโˆถ=โŒŠ(๐‘™๐‘[๐‘ก/2]โˆ’1/16)โˆ’215โŒ‹ (where โŒŠ๐‘ขโŒ‹ denotes the lower integral part of ๐‘ข), and we take a โ€œsmall partโ€™โ€™ of the interval ๐ผ๐‘กโงต๐ผ๐‘ก+1=[2โˆ’๐‘กโˆ’1,2โˆ’๐‘ก). This way the intervals are defined as follows:ฮ”๐‘ก๎ƒฌ1โˆถ=2๐‘ก+1,12๐‘ก+1+12๐‘ก+ฬƒ๐‘ก๎ƒช.(3.5) The exceptional set ๐ฝ is given by๐ฝโˆถ=โˆž๎š๐‘ก=๐‘ก0ฮ”๐‘ก.(3.6)

For the logarithmic kernels ๐…๐œ…๐‘๐ด 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, Gรกt and Tkebuchava [22].

Lemma 3.1 (see [22]). Let ๐‘ ,๐‘ก=0,1,โ€ฆ,2๐ด, ๐‘ โ‰ค๐‘ก, and (๐‘ฅ,๐‘ฆ)โˆˆ(๐ฝ๐‘ ร—๐ฝ๐‘ก)โงต(๐‘„(๐‘ ,๐‘ก)๐‘ก+ฬƒ๐‘ โˆช(ฮ”๐‘ ร—ฮ”๐‘ก)). Then one has ||๐…๐‘ค๐‘๐ด||2(๐‘ฅ,๐‘ฆ)โ‰ฅ๐‘๐‘ก+๐‘ ๐‘ log๐‘๐ด,(3.7) where ๐ฝ๐‘กโˆถ=๐ผ๐‘กโงต๐ผ๐‘ก+1 and ๐‘„๐ต(๐‘ ,๐‘ก)๎šโˆถ=๐‘ฅ๐‘ +1+โ‹ฏ+๐‘ฅ๐‘กโˆ’11โ‰ค1๎š๐‘ฅ๐‘ก=0โ‹ฏ1๎š๐‘ฅ๐ตโˆ’1=0๐ผ๐ต๎€ท0,โ€ฆ,0,1,๐‘ฅ๐‘ +1,โ€ฆ,๐‘ฅ๐ตโˆ’1๎€ธร—๐ผ๐ต๎€ท0,โ€ฆ,0,1,๐‘ฆ๐‘ก+1=๐‘ฅ๐‘ก+1,โ€ฆ,๐‘ฆ๐ตโˆ’1=๐‘ฅ๐ตโˆ’1๎€ธ.(3.8)

Now, for the Walsh-Kaczmarz logarithmic kernels ๐…๐œ…๐‘๐ด, we prove the following.

Lemma 3.2. Let ๐‘ฅโˆˆ๐ผ2๐ด(1,๐‘ฅ1,โ€ฆ,๐‘ฅ๐‘กโˆ’2,๐‘ฅ๐‘กโˆ’1=1,๐‘ฅ๐‘ก,โ€ฆ,๐‘ฅ๐‘ โˆ’1,๐‘ฅ๐‘ =1,๐‘ฅ๐‘ +1=1,0,โ€ฆ,0)=โˆถ๐ผ๐‘ก,๐‘ 2๐ด,โ€‰โ€‰๐‘ฆโˆˆ๐ผ2๐ด(1,๐‘ฆ1,โ€ฆ,๐‘ฆ๐‘กโˆ’2,๐‘ฆ๐‘กโˆ’1=0,๐‘ฆ๐‘ก=1,๐‘ฆ๐‘ก+1๎‚๐ผ=1,0,โ€ฆ,0)=โˆถ๐‘ก2๐ด for ๐‘ ,๐‘ก=2,3,โ€ฆ,๐ด and ๐‘กโ‰ค๐‘ . Then ||๐…๐œ…๐‘๐ด||(๐‘ฅ,๐‘ฆ)โ‰ฅ๐‘24๐ดโˆ’๐‘กโˆ’๐‘ .(3.9)

Proof. Set (๐‘ฅ,๐‘ฆ)โˆˆ๐ผ๐‘ก,๐‘ 2๐ดร—๎‚๐ผ๐‘ก2๐ด. Let ๐†๐›ผ๐‘๐ด(๐‘ฅ,๐‘ฆ)โˆถ=๐‘™๐‘๐ด๐…๐›ผ๐‘๐ด(๐‘ฅ,๐‘ฆ),๐บ๐›ผ๐‘๐ด(๐‘ฅ)โˆถ=๐‘™๐‘๐ด๐น๐›ผ๐‘๐ด(๐‘ฅ)(3.10) for ๐›ผ=๐‘ค or ๐œ…. Now, we write that ๐†๐œ…๐‘๐ด(๐‘ฅ,๐‘ฆ)=22๐ด๎“๐‘—=1๐ท๐œ…๐‘—(๐‘ฅ)๐ท๐œ…๐‘—(๐‘ฆ)๐‘๐ด+โˆ’๐‘—๐‘๐ดโˆ’1๎“๐‘—=22๐ด+1๐ท๐œ…๐‘—(๐‘ฅ)๐ท๐œ…๐‘—(๐‘ฆ)๐‘๐ดโˆ’๐‘—โˆถ=๐ผ+๐ผ๐ผ.(3.11)
First, we discuss ๐ผ๐ผ. We use equalityโ€‰โ€‰(1.13)๐ผ๐ผ=๐‘๐ดโˆ’1โˆ’1๎“๐‘—=1๐ท๐œ…๐‘—+22๐ด(๐‘ฅ)๐ท๐œ…๐‘—+22๐ด(๐‘ฆ)๐‘๐ดโˆ’1โˆ’๐‘—=๐‘™๐‘๐ดโˆ’1๐ท22๐ด(๐‘ฅ)๐ท22๐ด(๐‘ฆ)+๐ท22๐ด(๐‘ฅ)๐‘Ÿ2๐ด(๐‘ฆ)๐บ๐‘ค๐‘๐ดโˆ’1๎€ท๐œ2๐ด(๎€ธ๐‘ฆ)+๐ท22๐ด(๐‘ฆ)๐‘Ÿ2๐ด(๐‘ฅ)๐บ๐‘ค๐‘๐ดโˆ’1๎€ท๐œ2๐ด๎€ธ(๐‘ฅ)+๐‘Ÿ2๐ด(๐‘ฅ)๐‘Ÿ2๐ด(๐‘ฆ)๐†๐‘ค๐‘๐ดโˆ’1๎€ท๐œ2๐ด(๐‘ฅ),๐œ2๐ด๎€ธ.(๐‘ฆ)(3.12)
If (๐‘ฅ,๐‘ฆ)โˆˆ๐ผ๐‘ก,๐‘ 2๐ดร—๎‚๐ผ๐‘ก2๐ด, then (see (1.14))๐ท22๐ด(๐‘ฅ)=๐ท22๐ด๐œ(๐‘ฆ)=0,2๐ด๎€ท(๐‘ฅ)=0,โ€ฆ,0,๐‘ฅ๐‘ +1=1,๐‘ฅ๐‘ =1,๐‘ฅ๐‘ โˆ’1,โ€ฆ,๐‘ฅ๐‘ก,๐‘ฅ๐‘กโˆ’1=1,๐‘ฅ๐‘กโˆ’2,โ€ฆ,๐‘ฅ1,๐‘ฅ0=1,๐‘ฅ2๐ด๎€ธ,๐œ,โ€ฆ2๐ด๎€ท(๐‘ฆ)=0,โ€ฆ,0,๐‘ฆ๐‘ก+1=1,๐‘ฆ๐‘ก=1,๐‘ฆ๐‘กโˆ’1=0,๐‘ฆ๐‘กโˆ’2,โ€ฆ,๐‘ฆ1,๐‘ฆ0=1,๐‘ฆ2๐ด๎€ธ.,โ€ฆ(3.13) Moreover, by Lemma 3.1, we have ||||=||๐†๐ผ๐ผ๐‘ค๐‘๐ดโˆ’1๎€ท๐œ2๐ด(๐‘ฅ),๐œ2๐ด๎€ธ||(๐‘ฆ)โ‰ฅ๐‘(2๐ดโˆ’๐‘ )24๐ดโˆ’๐‘กโˆ’๐‘ .(3.14)
Now, we discuss ๐ผ by the help of โ€‰(1.13)๐ผ=2๐ดโˆ’1๎“2๐‘™=0๐‘™+1โˆ’1๎“๐‘—=2๐‘™๐ท๐œ…๐‘—(๐‘ฅ)๐ท๐œ…๐‘—(๐‘ฆ)๐‘๐ด+๐ทโˆ’๐‘—22๐ด(๐‘ฅ)๐ท22๐ด(๐‘ฆ)๐‘๐ดโˆ’1=2๐ดโˆ’1๎“2๐‘™=0๐‘™โˆ’1๎“๐‘—=0๐ท๐œ…๐‘—+2๐‘™(๐‘ฅ)๐ท๐œ…๐‘—+2๐‘™(๐‘ฆ)๐‘๐ดโˆ’๐‘—โˆ’2๐‘™=2๐ดโˆ’1๎“2๐‘™=0๐‘™โˆ’1๎“๐‘—=0๐ท2๐‘™(๐‘ฅ)๐ท2๐‘™(๐‘ฆ)๐‘๐ดโˆ’๐‘—โˆ’2๐‘™+2๐ดโˆ’1๎“๐‘™=0๐ท2๐‘™(๐‘ฅ)2๐‘™โˆ’1๎“๐‘—=0๐‘Ÿ๐‘™(๐‘ฆ)๐ท๐‘ค๐‘—๎€ท๐œ๐‘™๎€ธ(๐‘ฆ)๐‘๐ดโˆ’๐‘—โˆ’2๐‘™+2๐ดโˆ’1๎“๐‘™=0๐ท2๐‘™(๐‘ฆ)2๐‘™โˆ’1๎“๐‘—=0๐‘Ÿ๐‘™(๐‘ฅ)๐ท๐‘ค๐‘—๎€ท๐œ๐‘™(๎€ธ๐‘ฅ)๐‘๐ดโˆ’๐‘—โˆ’2๐‘™+2๐ดโˆ’1๎“2๐‘™=0๐‘™โˆ’1๎“๐‘—=0๐‘Ÿ๐‘™(๐‘ฅ)๐‘Ÿ๐‘™(๐‘ฆ)๐ท๐‘ค๐‘—๎€ท๐œ๐‘™(๎€ธ๐ท๐‘ฅ)๐‘ค๐‘—๎€ท๐œ๐‘™(๎€ธ๐‘ฆ)๐‘๐ดโˆ’๐‘—โˆ’2๐‘™.(3.15) Since ๐‘ฅ0=๐‘ฆ0=1, ๐ท2๐‘™(๐‘ฅ)=๐ท2๐‘™(๐‘ฆ)=0 for all ๐‘™โ‰ฅ1, thus, 1๐ผ=๐‘๐ด+โˆ’12๐ดโˆ’1๎“๐‘™=1๐‘Ÿ๐‘™(๐‘ฅ)๐‘Ÿ๐‘™(๐‘ฆ)2๐‘™โˆ’1๎“๐‘—=1๐ท๐‘ค๐‘—๎€ท๐œ๐‘™๎€ธ๐ท(๐‘ฅ)๐‘ค๐‘—๎€ท๐œ๐‘™๎€ธ(๐‘ฆ)๐‘๐ดโˆ’๐‘—โˆ’2๐‘™1=โˆถ๐‘๐ด+โˆ’12๐ดโˆ’1๎“๐‘™=1๐ผ๐‘™.(3.16)
We use Abelโ€™s transformation for ๐ผ๐‘™ (๐‘™โ‰ฅ1)๐ผ๐‘™=๐‘Ÿ๐‘™(๐‘ฅ)๐‘Ÿ๐‘™(๐‘ฆ)2๐‘™โˆ’2๎“๐‘—=1๎‚ต1๐‘๐ดโˆ’๐‘—โˆ’2๐‘™โˆ’1๐‘๐ดโˆ’๐‘—โˆ’2๐‘™๎‚ถโˆ’1๐‘—๐’ฆ๐‘ค๐‘—๎€ท๐œ๐‘™(๐‘ฅ),๐œ๐‘™๎€ธ(๐‘ฆ)+๐‘Ÿ๐‘™(๐‘ฅ)๐‘Ÿ๐‘™๎€ท2(๐‘ฆ)๐‘™๎€ธ๐’ฆโˆ’1๐‘ค2๐‘™โˆ’1๎€ท๐œ๐‘™(๐‘ฅ),๐œ๐‘™(๎€ธ๐‘ฆ)๐‘๐ดโˆ’2๐‘™+1+1=โˆถ๐ผ1๐‘™+๐ผ2๐‘™.(3.17) We write ||๐ผ1๐‘™||โ‰ค๐‘224๐ด๐‘™โˆ’2๎“๐‘—=1๐‘—||๐’ฆ๐‘ค๐‘—๎€ท๐œ๐‘™(๐‘ฅ),๐œ๐‘™๎€ธ||,||๐ผ(๐‘ฆ)2๐‘™||โ‰ค๐‘22๐ด๎€ท2๐‘™๎€ธ||๐’ฆโˆ’1๐‘ค2๐‘™โˆ’1๎€ท๐œ๐‘™(๐‘ฅ),๐œ๐‘™(๎€ธ||.๐‘ฆ)(3.18) Since |๐ท๐‘–|โ‰ค๐‘–, we have ๐‘›||๐’ฆ๐‘ค๐‘›||โ‰ค|||||๐‘›๎“๐‘–=0๐ท๐‘ค๐‘–๐ท๐‘ค๐‘–|||||โ‰ค๐‘๐‘›3.(3.19) For ๐‘™โ‰ค๐‘ +2, we obtain that ||๐ผ1๐‘™||2โ‰ค๐‘4๐‘™24๐ด,||๐ผ2๐‘™||2โ‰ค๐‘3๐‘™22๐ด.(3.20) Moreover, ๐‘ +2๎“๐‘™=1||๐ผ1๐‘™||โ‰ค๐‘24๐ด๐‘ +2๎“๐‘™=124๐‘™2โ‰ค๐‘4๐‘ 24๐ด<๐‘,๐‘ +2๎“๐‘™=1||๐ผ2๐‘™||โ‰ค๐‘22๐ด๐‘ +2๎“๐‘™=123๐‘™โ‰ค๐‘23๐‘ 22๐ดโ‰ค๐‘2๐ด.(3.21) Let ๐‘ +2<๐‘™<2๐ด. Then we have ๐œ๐‘™๎€ท(๐‘ฅ)=0,โ€ฆ,0,1,1,๐‘ฅ๐‘ โˆ’1,โ€ฆ,๐‘ฅ๐‘กโˆ’1=1,โ€ฆ,๐‘ฅ1,1,0,โ€ฆ,0,๐‘ฅ2๐ด๎€ธ,๐œ,โ€ฆ๐‘™(๎€ท๐‘ฆ)=0,โ€ฆ,0,1,1,๐‘ฆ๐‘กโˆ’1=0,๐‘ฆ๐‘กโˆ’2,โ€ฆ,๐‘ฆ1,1,0,โ€ฆ,0,๐‘ฆ2๐ด๎€ธ,,โ€ฆ(3.22) that is, ๐œ๐‘™(๐‘ฅ)โˆˆ๐ผ๐‘™โˆ’๐‘ โˆ’3โงต๐ผ๐‘™โˆ’๐‘ โˆ’2, ๐œ๐‘™(๐‘ฆ)โˆˆ๐ผ๐‘™โˆ’๐‘กโˆ’3โงต๐ผ๐‘™โˆ’๐‘กโˆ’2.
Now, we introduce the following notation:๐’ฆ๐‘ค๐‘Ž,๐‘(๐‘ฅ,๐‘ฆ)โˆถ=๐‘Ž+๐‘โˆ’1๎“๐‘—=๐‘Ž๐ท๐‘ค๐‘—(๐‘ฅ)๐ท๐‘ค๐‘—(๐‘ฆ),(3.23) and ๐‘›(๐‘ )โˆ‘โˆถ=โˆž๐‘–=๐‘ ๐‘›๐‘–2๐‘– (๐‘›,๐‘ โˆˆ๐). A simple consideration gives ๐‘›๐’ฆ๐‘ค๐‘›(๐‘ฅ,๐‘ฆ)=|๐‘›|๎“๐‘ =0๐‘›๐‘ ๐’ฆ๐‘›(๐‘ +1),2๐‘ (๐‘ฅ,๐‘ฆ)+๐ท๐‘ค๐‘›(๐‘ฅ)๐ท๐‘ค๐‘›(๐‘ฆ).(3.24) It is known [23] that if ๐‘กโ‰ค๐‘˜ and (๐‘ฅ,๐‘ฆ)โˆˆ(๐ผ๐‘กโงต๐ผ๐‘ก+1)ร—(๐ผ๐‘˜โงต๐ผ๐‘˜+1), then ||๐’ฆ๐‘ค๐‘›(๐‘ +1),2๐‘ ||(๐‘ฅ,๐‘ฆ)โ‰ค๐‘2๐‘ +๐‘ก+๐‘˜.(3.25) Using this inequality, we have that ||๐ผ1๐‘™||โ‰ค๐‘24๐ด๐‘™โˆ’1๎“2๐‘˜=0๐‘˜+1โˆ’1๎“๐‘—=2๐‘˜๐‘—||๐’ฆ๐‘ค๐‘—๎€ท๐œ๐‘™(๐‘ฅ),๐œ๐‘™๎€ธ||โ‰ค๐‘(๐‘ฆ)24๐ด๐‘™โˆ’1๎“2๐‘˜=0๐‘˜+1โˆ’1๎“๐‘—=2๐‘˜๎ƒฉ๐‘˜๎“๐œˆ=0|||๐’ฆ๐‘ค๐‘—(๐œˆ+1),2๐œˆ๎€ท๐œ๐‘™(๐‘ฅ),๐œ๐‘™(๎€ธ|||+||๐ท๐‘ฆ)๐‘—๎€ท๐œ๐‘™(๎€ธ๐ท๐‘ฅ)๐‘—๎€ท๐œ๐‘™(๎€ธ||๎ƒชโ‰ค๐‘๐‘ฆ)24๐ด๐‘™โˆ’1๎“2๐‘˜=0๐‘˜+1โˆ’1๎“๐‘—=2๐‘˜๎ƒฉ๐‘˜๎“๐œˆ=02๐œˆ+2๐‘™โˆ’๐‘กโˆ’๐‘ +๐‘—2๎ƒช2โ‰ค๐‘4๐‘™โˆ’๐‘กโˆ’๐‘ 24๐ด2+๐‘3๐‘™24๐ด.(3.26) Moreover, 2๐ดโˆ’1๎“๐‘™=๐‘ +3||๐ผ1๐‘™||2โ‰ค๐‘8๐ดโˆ’๐‘กโˆ’๐‘ 24๐ด2+๐‘6๐ด24๐ดโ‰ค๐‘24๐ดโˆ’๐‘กโˆ’๐‘ .(3.27) Analogously, ||๐ผ2๐‘™||โ‰ค๐‘22๐ด๎ƒฉ๐‘™โˆ’1๎“๐œˆ=02๐œˆ+2๐‘™โˆ’๐‘กโˆ’๐‘ +22๐‘™๎ƒช2โ‰ค๐‘3๐‘™โˆ’๐‘กโˆ’๐‘ 22๐ด2+๐‘2๐‘™22๐ด,(3.28)2๐ดโˆ’1๎“๐‘™=๐‘ +3||๐ผ2๐‘™||โ‰ค๐‘24๐ดโˆ’๐‘กโˆ’๐‘ .(3.29) Taking into account (3.11), we get ||๐…๐œ…๐‘๐ด||โ‰ฅ||||โˆ’||๐ผ||(๐‘ฅ,๐‘ฆ)๐ผ๐ผ๐‘™๐‘๐ดโ‰ฅ๐‘1||||โˆ’||๐ผ||๐ผ๐ผ2๐ด.(3.30) Therefore by (3.14)โ€“(3.29), it follows that ||๐…๐œ…๐‘๐ด||(๐‘ฅ,๐‘ฆ)โ‰ฅ๐‘โ‹…๐‘1๎‚€๐‘ 1โˆ’๎‚22๐ด4๐ดโˆ’๐‘กโˆ’๐‘ โˆ’ฬƒ๐‘22๐ด4๐ดโˆ’๐‘กโˆ’๐‘ (3.31) with suitable constants ๐‘,๐‘1,ฬƒ๐‘>0. It is clear that for ๐ด large enough we have ||๐…๐œ…๐‘๐ด||(๐‘ฅ,๐‘ฆ)โ‰ฅ๐‘โˆ—24๐ดโˆ’๐‘กโˆ’๐‘ ,(3.32) where ๐‘โˆ—>0 is an absolute constant.

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

Theorem A. Let ๐ปโˆถ๐ฟ1(๐ผ2)โ†’๐ฟ0(๐ผ2) be a linear continuous operator, which commutes with family of translations โ„ฐ, that is, for all ๐ธโˆˆโ„ฐ for all ๐‘“โˆˆ๐ฟ1(๐ผ2)๐ป๐ธ๐‘“=๐ธ๐ป๐‘“. Let โ€–๐‘“โ€–๐ฟ1(๐ผ2)=1 and ๐œ†>1. Then for any 1โ‰ค๐‘Ÿโˆˆโ„• under condition, mes{(๐‘ฅ,๐‘ฆ)โˆˆ๐ผ2โˆถ|๐ป๐‘“|>๐œ†}โ‰ฅ1/๐‘Ÿ, there exist ๐ธ1,โ€ฆ,๐ธ๐‘Ÿ,๐ธ๎…ž1,โ€ฆ,๐ธ๎…ž๐‘Ÿโˆˆโ„ฐ and ๐œ€๐‘–=ยฑ1, ๐‘–=1,โ€ฆ,๐‘Ÿ, such that โŽงโŽชโŽจโŽชโŽฉmes(๐‘ฅ,๐‘ฆ)โˆˆ๐ผ2โˆถ|||||๐ป๎ƒฉ๐‘Ÿ๎“๐‘–=1๐œ€๐‘–๐‘“๎€ท๐ธ๐‘–๐‘ฅ,๐ธ๎…ž๐‘–๐‘ฆ๎€ธ๎ƒช|||||โŽซโŽชโŽฌโŽชโŽญโ‰ฅ1>๐œ†8.(3.33)

Theorem B. Let {๐ป๐‘š}โˆž๐‘š=1 be a sequence of linear continuous operators, acting from Orlicz space ๐ฟฮฆ(๐ผ2) into the space ๐ฟ0(๐ผ2). Suppose that there exist the sequence of functions {๐œ‰๐‘˜}โˆž๐‘˜=1 from unit bull ๐‘†ฮฆ(0,1) of space ๐ฟฮฆ(๐ผ2), and sequences of integers {๐‘š๐‘˜}โˆž๐‘˜=1 and {๐œˆ๐‘˜}โˆž๐‘˜=1 increasing to infinity such that ๐œ€0=inf๐‘˜๎€ฝmes(๐‘ฅ,๐‘ฆ)โˆˆ๐ผ2โˆถ||๐ป๐‘š๐‘˜๐œ‰๐‘˜||(๐‘ฅ,๐‘ฆ)>๐œˆ๐‘˜๎€พ>0.(3.34)
Then ๐ตโ€”the set of functions ๐‘“ from space ๐ฟฮฆ(๐ผ2), for which the sequence {๐ป๐‘š๐‘“} converges in measure to an a.e. finite functionโ€”is of first Baire category in space ๐ฟฮฆ(๐ผ2).

Theorem GGT. Let ๐ฟฮฆ be an Orlicz space, and let ๐œ‘โˆถ[0,โˆž)โ†’[0,โˆž) be measurable function with condition ๐œ‘(๐‘ฅ)=๐‘œ(ฮฆ(๐‘ฅ)) as ๐‘ฅโ†’โˆž. Then there exists Orlicz space ๐ฟ๐œ”, such that ๐œ”(๐‘ฅ)=๐‘œ(ฮฆ(๐‘ฅ))as ๐‘ฅโ†’โˆž, and ๐œ”(๐‘ฅ)โ‰ฅ๐œ‘(๐‘ฅ) for ๐‘ฅโ‰ฅ๐‘โ‰ฅ0.

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 {๐ด๐‘˜โˆถ๐‘˜โ‰ฅ1} and {๐œˆ๐‘˜โˆถ๐‘˜โ‰ฅ1} increasing to infinity, and a sequence of functions {๐œ‰๐‘˜โˆถ๐‘˜โ‰ฅ1} from the unit bull ๐‘†ฮฆ(0,1) of Orlicz space ๐ฟฮฆ(๐ผ2), such that for all ๐‘˜๎‚†mes(๐‘ฅ,๐‘ฆ)โˆˆ๐ผ2โˆถ|||๐ญ๐œ…๐‘๐ด๐‘˜๎€ท๐œ‰๐‘˜๎€ธ|||,๐‘ฅ,๐‘ฆ>๐œˆ๐‘˜๎‚‡โ‰ฅ18.(4.1)
First, we prove that there exists ๐‘>0 such that๎‚†mes(๐‘ฅ,๐‘ฆ)โˆˆ๐ผ2โˆถ|||๐ญ๐œ…๐‘๐ด๐‘˜๎€ท๐ท22๐ด+1โŠ—๐ท22๐ด+1๎€ธ|||;๐‘ฅ,๐‘ฆ>๐‘23๐ด๎‚‡๐ด>๐‘23๐ด.(4.2)
It is easy to show that๐ญ๐œ…๐‘๐ด๐‘˜๎€ท๐ท22๐ด+1โŠ—๐ท22๐ด+1๎€ธ;๐‘ฅ,๐‘ฆ=๐‘†22๐ด+1,22๐ด+1๎€ท๐…๐œ…๐‘๐ด๎€ธ;๐‘ฅ,๐‘ฆ=๐…๐œ…๐‘๐ด(๐‘ฅ,๐‘ฆ).(4.3) Hence, from Lemma 3.2, we can write |||๐ญ๐œ…๐‘๐ด๐‘˜๎€ท๐ท22๐ด+1โŠ—๐ท22๐ด+1๎€ธ|||=||๐…;๐‘ฅ,๐‘ฆ๐œ…๐‘๐ด||(๐‘ฅ,๐‘ฆ)โ‰ฅ๐‘24๐ดโˆ’๐‘กโˆ’๐‘ ,(4.4) for (๐‘ฅ,๐‘ฆ)โˆˆ๐ผ๐‘ก,๐‘ 2๐ดร—๎‚๐ผ๐‘ก2๐ด.
Since๐‘24๐ดโˆ’๐‘กโˆ’๐‘ >๐‘23๐ด(4.5) for ๐‘ก+๐‘ <๐ด and 0โ‰ค๐‘กโ‰ค๐‘ <๐ด, from (4.4) we conclude that ๎€ฝmes(๐‘ฅ,๐‘ฆ)โˆˆ๐ผ2โˆถ||๐ญ๐œ…๐‘๐ด๎€ท๐ท22๐ด+1โŠ—๐ท22๐ด+1๎€ธ||;๐‘ฅ,๐‘ฆ>๐‘23๐ด๎€พโ‰ฅ๐‘๐ด๎“[]๐‘ =๐ด/2๐ดโˆ’๐‘ ๎“๐‘ก=0๎‚€๐ผmes๐‘ก,๐‘ 2๐ดร—๎‚๐ผ๐‘ก2๐ด๎‚โ‰ฅ๐‘๐ด๎“[]๐‘ =๐ด/2๐ดโˆ’๐‘ ๎“๐‘ก=02๐‘ 22๐ด2๐‘ก22๐ดโ‰ฅ๐‘๐ด๎“[]๐‘ =๐ด/22๐‘ 22๐ด2๐ดโˆ’๐‘ 22๐ดโ‰ฅ๐‘๐ด23๐ด.(4.6) Hence, (4.2) is proved.
From the condition of the theorem, we write [1]liminf๐‘ขโ†’โˆžฮฆ(๐‘ข)๐‘ขlog๐‘ข=0.(4.7) Consequently, there exists a sequence of integers {๐ด๐‘˜}โˆž๐‘˜=1 increasing to infinity, such that lim๐‘˜โ†’โˆžฮฆ๎€ท24๐ด๐‘˜๎€ธ2โˆ’4๐ด๐‘˜๐ด๐‘˜โˆ’1ฮฆ๎€ท2=0,4๐ด๐‘˜๎€ธ24๐ด๐‘˜โ‰ฅ4,โˆ€๐‘˜.(4.8)
From (4.2) we write๎‚†mes(๐‘ฅ,๐‘ฆ)โˆˆ๐ผ2โˆถ|||๐ญ๐œ…๐‘๐ด๐‘˜๎€ท๐ท2๐‘˜2๐ด+1โŠ—๐ท2๐‘˜2๐ด+1๎€ธ|||;๐‘ฅ,๐‘ฆ>๐‘23๐ด๐‘˜๎‚‡๐ด>๐‘๐‘˜23๐ด๐‘˜.(4.9) Then by the virtue of Theorem A, there exists ๐‘’1,โ€ฆ,๐‘’๐‘Ÿ,๐‘’๎…ž1,โ€ฆ,๐‘’๎…ž๐‘Ÿโˆˆ[0,1] and ๐œ€1,โ€ฆ,๐œ€๐‘Ÿ=ยฑ1 such that โŽงโŽชโŽจโŽชโŽฉmes(๐‘ฅ,๐‘ฆ)โˆˆ๐ผ2โˆถ|||||๐‘Ÿ๎“๐‘–=1๐œ€๐‘–๐ญ๐œ…๐‘๐ด๐‘˜๎€ท๐ท2๐‘˜2๐ด+1โŠ—๐ท2๐‘˜2๐ด+1,๐‘’๐‘–โŠ•๐‘ฅ,๐‘’๎…ž๐‘–๎€ธ|||||โŠ•๐‘ฆ>23๐ด๐‘˜โŽซโŽชโŽฌโŽชโŽญ>18,(4.10) where ๐‘Ÿ=[23๐ด๐‘˜/(๐‘๐ด๐‘˜)]+1.
Set๐œ‰๐‘˜2(๐‘ฅ,๐‘ฆ)=4๐ด๐‘˜โˆ’1ฮฆ๎€ท24๐ด๐‘˜๎€ธ๐‘€๐‘˜(๐‘ฅ,๐‘ฆ),(4.11) where ๐‘€๐‘˜1(๐‘ฅ,๐‘ฆ)=๐‘Ÿ๐‘Ÿ๎“๐‘–=1๐œ€๐‘–๐ท2๐‘˜2๐ด+1๎€ท๐‘’๐‘–๎€ธ๐ทโŠ•๐‘ฅ2๐‘˜2๐ด+1๎€ท๐‘’๎…ž๐‘–๎€ธโŠ•๐‘ฆ.(4.12) Denote ๐œˆ๐‘˜=27๐ด๐‘˜โˆ’1๎€ท2๐‘Ÿฮฆ4๐ด๐‘˜๎€ธ.(4.13) It is easy to show that ๎‚†mes(๐‘ฅ,๐‘ฆ)โˆˆ๐ผ2โˆถ|||๐ญ๐œ…๐‘๐ด๐‘˜๎€ท๐œ‰๐‘˜๎€ธ|||,๐‘ฅ,๐‘ฆ>๐œˆ๐‘˜๎‚‡โŽงโŽชโŽจโŽชโŽฉ=mes(๐‘ฅ,๐‘ฆ)โˆˆ๐ผ2โˆถ|||||๐‘Ÿ๎“๐‘–=1๐œ€๐‘–๐ญ๐œ…๐‘๐ด๐‘˜๎€ท๐ท2๐‘˜2๐ด+1โŠ—๐ท2๐‘˜2๐ด+1,๐‘’๐‘–โŠ•๐‘ฅ,๐‘’๎…ž๐‘–๎€ธ|||||โŠ•๐‘ฆ>23๐ด๐‘˜โŽซโŽชโŽฌโŽชโŽญ>18.(4.14) Hence, (4.1) is proved.
Sinceโ€–โ€–๐‘€๐‘˜โ€–โ€–๐ฟโˆž(๐ผ2)โ‰ค24๐‘š๐‘˜+2,โ€–โ€–๐‘€๐‘˜โ€–โ€–๐ฟ1(๐ผ2)โ‰ค1.(4.15) Moreover, ||||๎€œ๐ผ2||||โ‰ค1๐‘“๐‘”2๎‚ธ๎€œ๐ผ2ฮฆ๎€ท2||๐‘“||๎€ธ๎‚น,+1ฮฆ(๐‘ข)๐‘ข<ฮฆ๎€ท๐‘ข๎…ž๎€ธ๐‘ข๎…ž๎€ท0<๐‘ข<๐‘ข๎…ž๎€ธ.(4.16) From (4.8) we can write โ€–โ€–๐œ‰๐‘˜โ€–โ€–๐ฟฮฆ(๐ผ2)โ‰ค12๎ƒฌ๎€œ1+๐ผ2ฮฆ๎ƒฉ24๐ด๐‘˜||๐‘€๐‘˜||(๐‘ฅ,๐‘ฆ)ฮฆ๎€ท24๐ด๐‘˜๎€ธ๎ƒช๎ƒญโ‰ค1๐‘‘๐‘ฅ๐‘‘๐‘ฆ2๎ƒฌ๎€œ1+๐ผ2ฮฆ๎€ท24๐ด๐‘˜24๐ด๐‘˜+2/๎€ทฮฆ๎€ท24๐ด๐‘˜๎€ธ๎€ธ๎€ธ24๐ด๐‘˜๎€ท24๐ด๐‘˜+2/๎€ทฮฆ๎€ท24๐ด๐‘˜2๎€ธ๎€ธ๎€ธ4๐ด๐‘˜||๐‘€๐‘˜(||๐‘ฅ,๐‘ฆ)ฮฆ๎€ท24๐ด๐‘˜๎€ธ๎ƒญโ‰ค1๐‘‘๐‘ฅ๐‘‘๐‘ฆ2๎ƒฌ๎€œ1+๐ผ2ฮฆ๎€ท24๐ด๐‘˜๎€ธ24๐ด๐‘˜24๐ด๐‘˜||๐‘€๐‘˜||(๐‘ฅ,๐‘ฆ)ฮฆ๎€ท24๐ด๐‘˜๎€ธ๎ƒญ๐‘‘๐‘ฅ๐‘‘๐‘ฆโ‰ค1.(4.17)
Hence, ๐œ‰๐‘˜โˆˆ๐‘†ฮฆ(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, โ€œOn Fourier series with respect to the Walsh-Kaczmarz system,โ€ Analysis Mathematica, vol. 7, no. 2, pp. 141โ€“150, 1981. View at: Publisher Site | Google Scholar
  5. A. A. ล neฤญder, โ€œOn series of Walsh functions with monotonic coefficients,โ€ Izvestiya Akademii Nauk SSSR, vol. 12, pp. 179โ€“192, 1948. View at: Google Scholar
  6. L. A. Balaลกov, โ€œSeries with respect to the Walsh system with monotone coefficients,โ€ Sibirskiฤญ Matematiฤeskiฤญ ลฝurnal, vol. 12, pp. 25โ€“39, 1971 (Russian). View at: Google Scholar
  7. F. Schipp, โ€œPointwise convergence of expansions with respect to certain product systems,โ€ Analysis Mathematica, vol. 2, no. 1, pp. 65โ€“76, 1976. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  8. W. S. Young, โ€œOn the a.e. convergence of Walsh-Kaczmarz-Fourier series,โ€ Proceedings of the American Mathematical Society, vol. 44, pp. 353โ€“358, 1974. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  9. G. Gรกt, โ€œOn (C, 1) summability of integrable functions with respect to the Walsh-Kaczmarz system,โ€ Studia Mathematica, vol. 130, no. 2, pp. 135โ€“148, 1998. View at: Google Scholar | Zentralblatt MATH
  10. P. Simon, โ€œOn the Cesaro summability with respect to the Walsh-Kaczmarz system,โ€ Journal of Approximation Theory, vol. 106, no. 2, pp. 249โ€“261, 2000. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  11. P. Simon, โ€œ(C,ฮฑ) summability of Walsh-Kaczmarz-Fourier series,โ€ Journal of Approximation Theory, vol. 127, no. 1, pp. 39โ€“60, 2004. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  12. K. Nagy, โ€œOn the two-dimensional Marcinkiewicz means with respect to Walsh-Kaczmarz system,โ€ Journal of Approximation Theory, vol. 142, no. 2, pp. 138โ€“165, 2006. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  13. L. V. Zhizhiashvili, โ€œSome problems of multidimensional harmonic analysis,โ€ TGU, Tbilisi, 1996. View at: Google Scholar
  14. R. Getsadze, โ€œOn the divergence in measure of multiple Fourier seties,โ€ Some Problems of Functions Theory, vol. 4, pp. 84โ€“117, 1988 (Russian). View at: Google Scholar
  15. S. V. Konyagin, โ€œOn a subsequence of Fourier-Walsh partial sums,โ€ Rossiฤญskaya Akademiya Nauk, vol. 54, no. 4, pp. 69โ€“75, 1993. View at: Publisher Site | Google Scholar
  16. G. Tkebuchava, โ€œSubsequences of partial sums of multiple Fourier and Fourier-Walsh series,โ€ Bulletin of the Georgian Academy of Sciences, vol. 169, no. 2, pp. 252โ€“253, 2004. View at: Google Scholar
  17. U. Goginava, โ€œConvergence in measure of partial sums of double Fourier series with respect to the Walsh-Kaczmarz system,โ€ Journal of Mathematical Analysis and Approximation Theory, vol. 2, no. 2, pp. 160โ€“167, 2007. View at: Google Scholar | Zentralblatt MATH
  18. U. Goginava and K. Nagy, โ€œWeak type inequality for logarithmic means of Walsh-Kaczmarz-Fourier series,โ€ Real Analysis Exchange, vol. 35, no. 2, pp. 445โ€“461, 2010. View at: Google Scholar | Zentralblatt MATH
  19. G. Gรกt and U. Goginava, โ€œOn the divergence of Nรถrlund logarithmic means of Walsh-Fourier series,โ€ Acta Mathematica Sinica, vol. 25, no. 6, pp. 903โ€“916, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  20. G. Gรกt, U. Goginava, and G. Tkebuchava, โ€œConvergence in measure of logarithmic means of double Walsh-Fourier series,โ€ Georgian Mathematical Journal, vol. 12, no. 4, pp. 607โ€“618, 2005. View at: Google Scholar | Zentralblatt MATH
  21. G. Gรกt, U. Goginava, and G. Tkebuchava, โ€œConvergence of logarithmic means of multiple Walsh-Fourier series,โ€ Analysis in Theory and Applications, vol. 21, no. 4, pp. 326โ€“338, 2005. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  22. G. Gรกt, U. Goginava, and G. Tkebuchava, โ€œConvergence in measure of logarithmic means of quadratical partial sums of double Walsh-Fourier series,โ€ Journal of Mathematical Analysis and Applications, vol. 323, no. 1, pp. 535โ€“549, 2006. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  23. K. Nagy, โ€œSome convergence properties of the Walsh-Kaczmarz system with respect to the Marcinkiewicz means,โ€ Rendiconti del Circolo Matematico di Palermo. Serie II, no. 76, pp. 503โ€“516, 2005. View at: Google Scholar | Zentralblatt MATH
  24. A. M. Garsia, Topics in Almost Everywhere Convergence, vol. 4, Markham Publishing Co., Chicago, Ill, USA, 1970.

Copyright © 2012 Ushangi Goginava and Károly 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.

Related articles

No related content is available yet for this article.
 PDF Download Citation Citation
 Download other formatsMore
 Order printed copiesOrder
Views598
Downloads652
Citations

Related articles

No related content is available yet for this article.

Article of the Year Award: Outstanding research contributions of 2021, as selected by our Chief Editors. Read the winning articles.