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, Károly 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 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)×[0,1). mes(𝐴) is the Lebesgue measure of the set 𝐴⊂𝐼2. The constants appearing in this paper are denoted by 𝑐.

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

This space is endowed with the norm‖𝑓‖𝐿Φ(𝐼2)=inf𝑘>0∶𝐼2Φ||||𝑓(𝑥,𝑦)𝑘𝑑𝑥𝑑𝑦≤1.(1.2) In particular case, if Φ(𝑢)=𝑢ln(1+𝑢), 𝑢>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 𝑝∈𝐍, 0≤𝑝<2𝑛. Given 𝑛∈𝐍 and 𝑥∈[0,1), let 𝐼𝑛(𝑥) denote the dyadic interval of length 2−𝑛 which contains the point 𝑥.

Every point 𝑥∈𝐼 can be written in the following form:𝑥=âˆžî“ğ‘˜=0𝑥𝑘2𝑘+1𝑥=∶0,𝑥1,…,𝑥𝑛,…,𝑥𝑘∈{0,1}.(1.3) In that case when there are two different forms we choose the one for which limğ‘˜â†’âˆžğ‘¥ğ‘˜=0. We use the notation𝑒𝑗1∶=2𝑗+1=0,…,0,𝑥𝑗=1,0,….(1.4)

It is well-known that [2, 3]𝐼𝑛𝑥0,…,𝑥𝑛−1∶=𝐼𝑛𝑝(𝑥)=2𝑛,𝑝+12𝑛,(1.5) where ∑𝑝=𝑛−1𝑗=0𝑥𝑗2𝑛−1−𝑗.

Let 𝑟0(𝑥) be a function defined by𝑟0⎧⎪⎨⎪⎩1(𝑥)=1,if𝑥∈0,2,1−1,if𝑥∈2,𝑟,10(𝑥+1)=𝑟0(𝑥).(1.6) The Rademacher functions are defined by𝑟𝑛(𝑥)=𝑟0(2𝑛[𝑥),𝑛≥0,𝑥∈0,1).(1.7)

Let 𝑤0,𝑤1,… represent the Walsh functions, that is, 𝑤0(𝑥)=1, and if ∑𝑛=âˆžğ‘–=0𝑛𝑖2𝑖 is a positive integer with 𝑛𝑖∈{0,1}, then𝑤𝑛(𝑥)∶=âˆžî‘ğ‘–=0𝑟𝑖(𝑥)𝑛𝑖.(1.8) The order of the 𝑛 is denoted by |𝑛|∶=max{𝑗∈𝐍∶𝑛𝑗≠0}.

The Walsh-Kaczmarz functions are given by 𝜅0∶=1, and for 𝑛≥1𝜅𝑛(𝑥)∶=𝑟|𝑛|(𝑥)|𝑛|−1𝑘=0𝑟|𝑛|−1−𝑘(𝑥)𝑛𝑘.(1.9) For 𝐴∈𝐍 and 𝑥∈𝐼, Skvorcov [4] defined a transformation 𝜏𝐴∶𝐼→𝐼 by𝜏𝐴(𝑥)∶=𝐴−1𝑘=0𝑥𝐴−𝑘−12−(𝑘+1)+âˆžî“ğ‘—=𝐴𝑥𝑗2−(𝑗+1).(1.10) By the definition of 𝜏𝐴, we have (see [4])𝜅𝑛(𝑥)=𝑟|𝑛|(𝑥)𝑤𝑛−2|𝑛|𝜏|𝑛|(𝑥)(𝑛∈𝐍,𝑥∈𝐼).(1.11)

The Dirichlet kernels are defined by𝐷𝛼𝑛(𝑥)∶=𝑛−1𝑘=0𝛼𝑘(𝑥),(1.12) where 𝛼𝑘=𝑤𝑘 for all 𝑘∈𝐍 or 𝜅𝑘 for all 𝑘∈𝐍.

It is well-known that [2, 4]𝐷𝜅𝑛(𝑥)=𝐷2|𝑛|(𝑥)+𝑤2|𝑛|(𝑥)𝐷𝑤𝑛−2|𝑛|𝜏|𝑛|,𝐷(𝑥)(1.13)2𝑛(𝑥)∶=𝐷𝑤2𝑛(𝑥)=𝐷𝜅2𝑛(⎧⎪⎨⎪⎩2𝑥)=𝑛1,if𝑥∈0,2𝑛,10,if𝑥∈2𝑛.,1(1.14)

The 𝑛th Fejér means of the Walsh-(Kaczmarz-)Fourier series of function 𝑓 are given byğœŽğ›¼ğ‘›1(𝑓,𝑥)∶=𝑛𝑛𝑗=1𝑆𝛼𝑗(𝑓,𝑥),(1.15) where𝑆𝛼𝑛(𝑓,𝑥)=𝑛−1𝑘=0𝑓𝛼(𝑘)𝛼𝑘(𝑥)(1.16) is the 𝑛th partial sum of the Fourier series and 𝑓𝛼∫(𝑛)∶=𝐼𝑓𝛼𝑛(𝑛∈𝐍) is said to be the 𝑛th Walsh-(Kaczmarz-)Fourier coefficient of the function 𝑓.

The Nörlund logarithmic (simply we say logarithmic) means and kernels of one-dimensional Walsh-(Kaczmarz-)Fourier series are defined as follows:𝑡𝛼𝑛1(𝑓,𝑥)=𝑙𝑛𝑛−1𝑘=1𝑆𝛼𝑘(𝑓,𝑥)𝑛−𝑘,𝐹𝛼𝑛1(𝑡)=𝑙𝑛𝑛−1𝑘=1𝐷𝛼𝑘(𝑡)𝑛−𝑘,(1.17) where 𝑙𝑛=∑𝑛−1𝑘=11/𝑘.

The Kronecker product (𝛼𝑚,𝑛∶𝑛,𝑚∈𝐍) of two Walsh(-Kaczmarz) systems is said to be the two-dimensional Walsh(-Kaczmarz) system. That is,𝛼𝑚,𝑛(𝑥,𝑦)=𝛼𝑚(𝑥)𝛼𝑛(𝑦).(1.18)

If 𝑓∈𝐿(𝐼2), then the number 𝑓𝛼∫(𝑚,𝑛)∶=𝐼2𝑓𝛼𝑚,𝑛(𝑛,𝑚∈𝐍) is said to be the (𝑚,𝑛)th Walsh-(Kaczmarz-)Fourier coefficient of 𝑓. The rectangular partial sums of double Fourier series with respect to both system are defined by𝑆𝛼𝑚,𝑛(𝑓,𝑥,𝑦)=𝑚−1𝑖=0𝑛−1𝑗=0𝑓𝛼(𝑖,𝑗)𝛼𝑖(𝑥)𝛼𝑗(𝑦).(1.19)

The logarithmic means and kernels of quadratical partial sums of double Walsh-(Kaczmarz)-Fourier series are given by𝐭𝛼𝑛1(𝑓,𝑥,𝑦)=𝑙𝑛𝑛−1𝑘=1𝑆𝛼𝑘,𝑘(𝑓,𝑥,𝑦)𝑛−𝑘,𝐅𝛼𝑛1(𝑥,𝑦)=𝑙𝑛𝑛−1𝑘=1𝐷𝛼𝑘(𝑥)𝐷𝛼𝑘(𝑦)𝑛−𝑘.(1.20)

It is evident that𝐭𝛼𝑛(𝑓,𝑥,𝑦)−𝑓(𝑥,𝑦)=𝐼2[]𝐅𝑓(𝑥⊕𝑡,𝑦⊕𝑠)−𝑓(𝑥,𝑦)𝛼𝑛(𝑡,𝑠)𝑑𝑡𝑑𝑠,(1.21) where ⊕ denotes the dyadic addition [3]. The 𝑛th Marcinkiewicz kernels are given by𝒦𝛼𝑛1(𝑥,𝑦)∶=𝑛𝑛𝑘=1𝐷𝛼𝑘(𝑥)𝐷𝛼𝑘(𝑦).(1.22)

In 1948 Å neÄ­der [5] showed that the inequalitylimsupğ‘›â†’âˆžğ·ğœ…ğ‘›(𝑥)log𝑛≥𝐶>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 “spreadness” 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 Fejér means with respect to the Walsh-Kaczmarz system converge uniformly to 𝑓 for any continuous functions 𝑓. For any integrable functions, Gát [9] proved that the Fejér means with respect to the Walsh-Kaczmarz system converge a.e. to the function. Recently, Gát’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 𝑆𝑤𝑛(𝑓) of the Walsh-Fourier series of a function 𝑓∈𝐿(𝐼), 𝐼=[0,1) converge in measure on 𝐼 [2]. The condition 𝑓∈𝐿log+𝐿(𝐼2) provides convergence in measure on 𝐼2 of the rectangular partial sums 𝑆𝑤𝑛,𝑚(𝑓) of double Walsh-Fourier series [13]. The first example of a function from classes wider than 𝐿log+𝐿(𝐼2) with 𝑆𝑤𝑛,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 Nörlund logarithmic means 𝑡𝜅𝑛𝑓 of one-dimensional Walsh-Kaczmarz-Fourier series are of weak type (1,1), and this fact implies that 𝑡𝜅𝑛𝑓 converge in measure on 𝐼 for all functions 𝑓∈𝐿(𝐼) and 𝑡𝜅𝑛,𝑚𝑓 converge in measure on 𝐼2 for all functions 𝑓∈𝐿log+𝐿(𝐼2).

At last, we note that the Walsh-Nörlund logarithmic means are closer to the partial sums than to the classical logarithmic means or the Fejér 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-Nörlund 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 sums1𝑙𝑛𝑛−1𝑖=1𝑆𝑖,𝑖(𝑓,𝑥,𝑦)𝑛−𝑖(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 𝐭𝜅𝑛(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𝐼2̸⊆𝐿log+𝐿𝐼2.(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 𝜑∶[0,∞[→[0,∞[ be a nondecreasing function satisfying for 𝑥→+∞ the condition 𝜑(𝑥)=𝑜(𝑥log𝑥).(2.2)
Then there exists a function 𝑓∈𝐿(𝐼2) such that
(a)𝐼2𝜑||||𝑓(𝑥,𝑦)𝑑𝑥𝑑𝑦<∞,(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||𝐷𝑤𝑛||<1(𝑥)𝑥(3.1) for any 0<𝑥<1. Then for these 𝑥’s we also get||𝐹𝑤𝑛||<1(𝑥)𝑥,(3.2) where 𝑛∈𝐍 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+⋯+22+20 (𝐴∈𝐍). Then for any 2−2𝐴−1≤𝑥<1 and 𝐴∈𝐍, we have||𝐷𝑤𝑝𝐴||≥1(𝑥)4𝑥.(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, Gát, and Tkebuchava proved the inequality||𝐹𝑤𝑝𝐴||(𝑥)≥𝑐log(1/𝑥)𝑥log𝑝𝐴(3.4) for all 1≤𝐴∈𝐍, and 𝑥∈(2−2𝐴−1,1)⧵𝐽. We have the exceptional set 𝐽, such that it is “rare around zero.” For 𝑡=𝑡0,𝑡0+1,…,2𝐴,𝑡0=inf{𝑡∶⌊(𝑙𝑝[𝑡/2]−1/16)−215⌋>1} set ̃𝑡∶=⌊(𝑙𝑝[𝑡/2]−1/16)−215⌋ (where ⌊𝑢⌋ denotes the lower integral part of 𝑢), and we take a “small part’’ of the interval 𝐼𝑡⧵𝐼𝑡+1=[2−𝑡−1,2−𝑡). This way the intervals are defined as follows:Δ𝑡1∶=2𝑡+1,12𝑡+1+12𝑡+̃𝑡.(3.5) The exceptional set 𝐽 is given by𝐽∶=âˆžîšğ‘¡=𝑡0Δ𝑡.(3.6)

For the logarithmic kernels 𝐅𝜅𝑝𝐴 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, Gát and Tkebuchava [22].

Lemma 3.1 (see [22]). Let 𝑠,𝑡=0,1,…,2𝐴, 𝑠≤𝑡, and (𝑥,𝑦)∈(𝐽𝑠×𝐽𝑡)⧵(𝑄(𝑠,𝑡)𝑡+̃𝑠∪(Δ𝑠×Δ𝑡)). Then one has ||𝐅𝑤𝑝𝐴||2(𝑥,𝑦)≥𝑐𝑡+𝑠𝑠log𝑝𝐴,(3.7) where 𝐽𝑡∶=𝐼𝑡⧵𝐼𝑡+1 and 𝑄𝐵(𝑠,𝑡)∶=𝑥𝑠+1+⋯+𝑥𝑡−11≤1𝑥𝑡=0⋯1𝑥𝐵−1=0𝐼𝐵0,…,0,1,𝑥𝑠+1,…,𝑥𝐵−1×𝐼𝐵0,…,0,1,𝑦𝑡+1=𝑥𝑡+1,…,𝑦𝐵−1=𝑥𝐵−1.(3.8)

Now, for the Walsh-Kaczmarz logarithmic kernels 𝐅𝜅𝑝𝐴, we prove the following.

Lemma 3.2. Let 𝑥∈𝐼2𝐴(1,𝑥1,…,𝑥𝑡−2,𝑥𝑡−1=1,𝑥𝑡,…,𝑥𝑠−1,𝑥𝑠=1,𝑥𝑠+1=1,0,…,0)=∶𝐼𝑡,𝑠2𝐴,  𝑦∈𝐼2𝐴(1,𝑦1,…,𝑦𝑡−2,𝑦𝑡−1=0,𝑦𝑡=1,𝑦𝑡+1𝐼=1,0,…,0)=∶𝑡2𝐴 for 𝑠,𝑡=2,3,…,𝐴 and 𝑡≤𝑠. Then ||𝐅𝜅𝑝𝐴||(𝑥,𝑦)≥𝑐24𝐴−𝑡−𝑠.(3.9)

Proof. Set (𝑥,𝑦)∈𝐼𝑡,𝑠2𝐴×𝐼𝑡2𝐴. Let 𝐆𝛼𝑝𝐴(𝑥,𝑦)∶=𝑙𝑝𝐴𝐅𝛼𝑝𝐴(𝑥,𝑦),𝐺𝛼𝑝𝐴(𝑥)∶=𝑙𝑝𝐴𝐹𝛼𝑝𝐴(𝑥)(3.10) for 𝛼=𝑤 or 𝜅. Now, we write that 𝐆𝜅𝑝𝐴(𝑥,𝑦)=22𝐴𝑗=1𝐷𝜅𝑗(𝑥)𝐷𝜅𝑗(𝑦)𝑝𝐴+−𝑗𝑝𝐴−1𝑗=22𝐴+1𝐷𝜅𝑗(𝑥)𝐷𝜅𝑗(𝑦)𝑝𝐴−𝑗∶=𝐼+𝐼𝐼.(3.11)
First, we discuss 𝐼𝐼. We use equality  (1.13)𝐼𝐼=𝑝𝐴−1−1𝑗=1𝐷𝜅𝑗+22𝐴(𝑥)𝐷𝜅𝑗+22𝐴(𝑦)𝑝𝐴−1−𝑗=𝑙𝑝𝐴−1𝐷22𝐴(𝑥)𝐷22𝐴(𝑦)+𝐷22𝐴(𝑥)𝑟2𝐴(𝑦)𝐺𝑤𝑝𝐴−1𝜏2𝐴(𝑦)+𝐷22𝐴(𝑦)𝑟2𝐴(𝑥)𝐺𝑤𝑝𝐴−1𝜏2𝐴(𝑥)+𝑟2𝐴(𝑥)𝑟2𝐴(𝑦)𝐆𝑤𝑝𝐴−1𝜏2𝐴(𝑥),𝜏2𝐴.(𝑦)(3.12)
If (𝑥,𝑦)∈𝐼𝑡,𝑠2𝐴×𝐼𝑡2𝐴, then (see (1.14))𝐷22𝐴(𝑥)=𝐷22𝐴𝜏(𝑦)=0,2𝐴(𝑥)=0,…,0,𝑥𝑠+1=1,𝑥𝑠=1,𝑥𝑠−1,…,𝑥𝑡,𝑥𝑡−1=1,𝑥𝑡−2,…,𝑥1,𝑥0=1,𝑥2𝐴,𝜏,…2𝐴(𝑦)=0,…,0,𝑦𝑡+1=1,𝑦𝑡=1,𝑦𝑡−1=0,𝑦𝑡−2,…,𝑦1,𝑦0=1,𝑦2𝐴.,…(3.13) Moreover, by Lemma 3.1, we have ||||=||𝐆𝐼𝐼𝑤𝑝𝐴−1𝜏2𝐴(𝑥),𝜏2𝐴||(𝑦)≥𝑐(2𝐴−𝑠)24𝐴−𝑡−𝑠.(3.14)
Now, we discuss 𝐼 by the help of  (1.13)𝐼=2𝐴−12𝑙=0𝑙+1−1𝑗=2𝑙𝐷𝜅𝑗(𝑥)𝐷𝜅𝑗(𝑦)𝑝𝐴+𝐷−𝑗22𝐴(𝑥)𝐷22𝐴(𝑦)𝑝𝐴−1=2𝐴−12𝑙=0𝑙−1𝑗=0𝐷𝜅𝑗+2𝑙(𝑥)𝐷𝜅𝑗+2𝑙(𝑦)𝑝𝐴−𝑗−2𝑙=2𝐴−12𝑙=0𝑙−1𝑗=0𝐷2𝑙(𝑥)𝐷2𝑙(𝑦)𝑝𝐴−𝑗−2𝑙+2𝐴−1𝑙=0𝐷2𝑙(𝑥)2𝑙−1𝑗=0𝑟𝑙(𝑦)𝐷𝑤𝑗𝜏𝑙(𝑦)𝑝𝐴−𝑗−2𝑙+2𝐴−1𝑙=0𝐷2𝑙(𝑦)2𝑙−1𝑗=0𝑟𝑙(𝑥)𝐷𝑤𝑗𝜏𝑙(𝑥)𝑝𝐴−𝑗−2𝑙+2𝐴−12𝑙=0𝑙−1𝑗=0𝑟𝑙(𝑥)𝑟𝑙(𝑦)𝐷𝑤𝑗𝜏𝑙(𝐷𝑥)𝑤𝑗𝜏𝑙(𝑦)𝑝𝐴−𝑗−2𝑙.(3.15) Since 𝑥0=𝑦0=1, 𝐷2𝑙(𝑥)=𝐷2𝑙(𝑦)=0 for all 𝑙≥1, thus, 1𝐼=𝑝𝐴+−12𝐴−1𝑙=1𝑟𝑙(𝑥)𝑟𝑙(𝑦)2𝑙−1𝑗=1𝐷𝑤𝑗𝜏𝑙𝐷(𝑥)𝑤𝑗𝜏𝑙(𝑦)𝑝𝐴−𝑗−2𝑙1=∶𝑝𝐴+−12𝐴−1𝑙=1𝐼𝑙.(3.16)
We use Abel’s transformation for 𝐼𝑙 (𝑙≥1)𝐼𝑙=𝑟𝑙(𝑥)𝑟𝑙(𝑦)2𝑙−2𝑗=11𝑝𝐴−𝑗−2𝑙−1𝑝𝐴−𝑗−2𝑙−1𝑗𝒦𝑤𝑗𝜏𝑙(𝑥),𝜏𝑙(𝑦)+𝑟𝑙(𝑥)𝑟𝑙2(𝑦)𝑙𝒦−1𝑤2𝑙−1𝜏𝑙(𝑥),𝜏𝑙(𝑦)𝑝𝐴−2𝑙+1+1=∶𝐼1𝑙+𝐼2𝑙.(3.17) We write ||𝐼1𝑙||≤𝑐224𝐴𝑙−2𝑗=1𝑗||𝒦𝑤𝑗𝜏𝑙(𝑥),𝜏𝑙||,||𝐼(𝑦)2𝑙||≤𝑐22𝐴2𝑙||𝒦−1𝑤2𝑙−1𝜏𝑙(𝑥),𝜏𝑙(||.𝑦)(3.18) Since |𝐷𝑖|≤𝑖, we have 𝑛||𝒦𝑤𝑛||≤|||||𝑛𝑖=0𝐷𝑤𝑖𝐷𝑤𝑖|||||≤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 𝜏𝑙(𝑥)=0,…,0,1,1,𝑥𝑠−1,…,𝑥𝑡−1=1,…,𝑥1,1,0,…,0,𝑥2𝐴,𝜏,…𝑙(𝑦)=0,…,0,1,1,𝑦𝑡−1=0,𝑦𝑡−2,…,𝑦1,1,0,…,0,𝑦2𝐴,,…(3.22) that is, 𝜏𝑙(𝑥)∈𝐼𝑙−𝑠−3⧵𝐼𝑙−𝑠−2, 𝜏𝑙(𝑦)∈𝐼𝑙−𝑡−3⧵𝐼𝑙−𝑡−2.
Now, we introduce the following notation:ğ’¦ğ‘¤ğ‘Ž,𝑏(𝑥,𝑦)∶=ğ‘Ž+𝑏−1𝑗=ğ‘Žğ·ğ‘¤ğ‘—(𝑥)𝐷𝑤𝑗(𝑦),(3.23) and 𝑛(𝑠)∑∶=âˆžğ‘–=𝑠𝑛𝑖2𝑖 (𝑛,𝑠∈𝐍). A simple consideration gives 𝑛𝒦𝑤𝑛(𝑥,𝑦)=|𝑛|𝑠=0𝑛𝑠𝒦𝑛(𝑠+1),2𝑠(𝑥,𝑦)+𝐷𝑤𝑛(𝑥)𝐷𝑤𝑛(𝑦).(3.24) It is known [23] that if 𝑡≤𝑘 and (𝑥,𝑦)∈(𝐼𝑡⧵𝐼𝑡+1)×(𝐼𝑘⧵𝐼𝑘+1), then ||𝒦𝑤𝑛(𝑠+1),2𝑠||(𝑥,𝑦)≤𝑐2𝑠+𝑡+𝑘.(3.25) Using this inequality, we have that ||𝐼1𝑙||≤𝑐24𝐴𝑙−12𝑘=0𝑘+1−1𝑗=2𝑘𝑗||𝒦𝑤𝑗𝜏𝑙(𝑥),𝜏𝑙||≤𝑐(𝑦)24𝐴𝑙−12𝑘=0𝑘+1−1𝑗=2𝑘𝑘𝜈=0|||𝒦𝑤𝑗(𝜈+1),2𝜈𝜏𝑙(𝑥),𝜏𝑙(|||+||𝐷𝑦)𝑗𝜏𝑙(𝐷𝑥)𝑗𝜏𝑙(||≤𝑐𝑦)24𝐴𝑙−12𝑘=0𝑘+1−1𝑗=2𝑘𝑘𝜈=02𝜈+2𝑙−𝑡−𝑠+𝑗22≤𝑐4𝑙−𝑡−𝑠24𝐴2+𝑐3𝑙24𝐴.(3.26) Moreover, 2𝐴−1𝑙=𝑠+3||𝐼1𝑙||2≤𝑐8𝐴−𝑡−𝑠24𝐴2+𝑐6𝐴24𝐴≤𝑐24𝐴−𝑡−𝑠.(3.27) Analogously, ||𝐼2𝑙||≤𝑐22𝐴𝑙−1𝜈=02𝜈+2𝑙−𝑡−𝑠+22𝑙2≤𝑐3𝑙−𝑡−𝑠22𝐴2+𝑐2𝑙22𝐴,(3.28)2𝐴−1𝑙=𝑠+3||𝐼2𝑙||≤𝑐24𝐴−𝑡−𝑠.(3.29) Taking into account (3.11), we get ||𝐅𝜅𝑝𝐴||≥||||−||𝐼||(𝑥,𝑦)𝐼𝐼𝑙𝑝𝐴≥𝑐1||||−||𝐼||𝐼𝐼2𝐴.(3.30) Therefore by (3.14)–(3.29), it follows that ||𝐅𝜅𝑝𝐴||(𝑥,𝑦)≥𝑐⋅𝑐1𝑠1−22𝐴4𝐴−𝑡−𝑠−̃𝑐22𝐴4𝐴−𝑡−𝑠(3.31) with suitable constants 𝑐,𝑐1,̃𝑐>0. It is clear that for 𝐴 large enough we have ||𝐅𝜅𝑝𝐴||(𝑥,𝑦)≥𝑐∗24𝐴−𝑡−𝑠,(3.32) where 𝑐∗>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 ℰ, that is, for all 𝐸∈ℰ for all 𝑓∈𝐿1(𝐼2)𝐻𝐸𝑓=𝐸𝐻𝑓. Let ‖𝑓‖𝐿1(𝐼2)=1 and 𝜆>1. Then for any 1≤𝑟∈ℕ under condition, mes{(𝑥,𝑦)∈𝐼2∶|𝐻𝑓|>𝜆}≥1/𝑟, there exist 𝐸1,…,𝐸𝑟,ğ¸î…ž1,…,ğ¸î…žğ‘Ÿâˆˆâ„° and 𝜀𝑖=±1, 𝑖=1,…,𝑟, such that ⎧⎪⎨⎪⎩mes(𝑥,𝑦)∈𝐼2∶|||||𝐻𝑟𝑖=1𝜀𝑖𝑓𝐸𝑖𝑥,ğ¸î…žğ‘–ğ‘¦î€¸îƒª|||||⎫⎪⎬⎪⎭≥1>𝜆8.(3.33)

Theorem B. Let {𝐻𝑚}âˆžğ‘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 {𝜉𝑘}âˆžğ‘˜=1 from unit bull 𝑆Φ(0,1) of space 𝐿Φ(𝐼2), and sequences of integers {𝑚𝑘}âˆžğ‘˜=1 and {𝜈𝑘}âˆžğ‘˜=1 increasing to infinity such that 𝜀0=inf𝑘mes(𝑥,𝑦)∈𝐼2∶||𝐻𝑚𝑘𝜉𝑘||(𝑥,𝑦)>𝜈𝑘>0.(3.34)
Then 𝐵—the set of functions 𝑓 from space 𝐿Φ(𝐼2), for which the sequence {𝐻𝑚𝑓} converges in measure to an a.e. finite function—is of first Baire category in space 𝐿Φ(𝐼2).

Theorem GGT. Let 𝐿Φ be an Orlicz space, and let 𝜑∶[0,∞)→[0,∞) be measurable function with condition 𝜑(𝑥)=𝑜(Φ(𝑥)) as ğ‘¥â†’âˆž. Then there exists Orlicz space 𝐿𝜔, such that 𝜔(𝑥)=𝑜(Φ(𝑥))as ğ‘¥â†’âˆž, and 𝜔(𝑥)≥𝜑(𝑥) for 𝑥≥𝑐≥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 {𝐴𝑘∶𝑘≥1} and {𝜈𝑘∶𝑘≥1} increasing to infinity, and a sequence of functions {𝜉𝑘∶𝑘≥1} from the unit bull 𝑆Φ(0,1) of Orlicz space 𝐿Φ(𝐼2), such that for all 𝑘mes(𝑥,𝑦)∈𝐼2∶|||𝐭𝜅𝑝𝐴𝑘𝜉𝑘|||,𝑥,𝑦>𝜈𝑘≥18.(4.1)
First, we prove that there exists 𝑐>0 such thatmes(𝑥,𝑦)∈𝐼2∶|||𝐭𝜅𝑝𝐴𝑘𝐷22𝐴+1⊗𝐷22𝐴+1|||;𝑥,𝑦>𝑐23𝐴𝐴>𝑐23𝐴.(4.2)
It is easy to show that𝐭𝜅𝑝𝐴𝑘𝐷22𝐴+1⊗𝐷22𝐴+1;𝑥,𝑦=𝑆22𝐴+1,22𝐴+1𝐅𝜅𝑝𝐴;𝑥,𝑦=𝐅𝜅𝑝𝐴(𝑥,𝑦).(4.3) Hence, from Lemma 3.2, we can write |||𝐭𝜅𝑝𝐴𝑘𝐷22𝐴+1⊗𝐷22𝐴+1|||=||𝐅;𝑥,𝑦𝜅𝑝𝐴||(𝑥,𝑦)≥𝑐24𝐴−𝑡−𝑠,(4.4) for (𝑥,𝑦)∈𝐼𝑡,𝑠2𝐴×𝐼𝑡2𝐴.
Since𝑐24𝐴−𝑡−𝑠>𝑐23𝐴(4.5) for 𝑡+𝑠<𝐴 and 0≤𝑡≤𝑠<𝐴, from (4.4) we conclude that mes(𝑥,𝑦)∈𝐼2∶||𝐭𝜅𝑝𝐴𝐷22𝐴+1⊗𝐷22𝐴+1||;𝑥,𝑦>𝑐23𝐴≥𝑐𝐴[]𝑠=𝐴/2𝐴−𝑠𝑡=0𝐼mes𝑡,𝑠2𝐴×𝐼𝑡2𝐴≥𝑐𝐴[]𝑠=𝐴/2𝐴−𝑠𝑡=02𝑠22𝐴2𝑡22𝐴≥𝑐𝐴[]𝑠=𝐴/22𝑠22𝐴2𝐴−𝑠22𝐴≥𝑐𝐴23𝐴.(4.6) Hence, (4.2) is proved.
From the condition of the theorem, we write [1]liminfğ‘¢â†’âˆžÎ¦(𝑢)𝑢log𝑢=0.(4.7) Consequently, there exists a sequence of integers {𝐴𝑘}âˆžğ‘˜=1 increasing to infinity, such that limğ‘˜â†’âˆžÎ¦î€·24𝐴𝑘2−4𝐴𝑘𝐴𝑘−1Φ2=0,4𝐴𝑘24𝐴𝑘≥4,∀𝑘.(4.8)
From (4.2) we writemes(𝑥,𝑦)∈𝐼2∶|||𝐭𝜅𝑝𝐴𝑘𝐷2𝑘2𝐴+1⊗𝐷2𝑘2𝐴+1|||;𝑥,𝑦>𝑐23𝐴𝑘𝐴>𝑐𝑘23𝐴𝑘.(4.9) Then by the virtue of Theorem A, there exists 𝑒1,…,𝑒𝑟,ğ‘’î…ž1,…,ğ‘’î…žğ‘Ÿâˆˆ[0,1] and 𝜀1,…,𝜀𝑟=±1 such that ⎧⎪⎨⎪⎩mes(𝑥,𝑦)∈𝐼2∶|||||𝑟𝑖=1𝜀𝑖𝐭𝜅𝑝𝐴𝑘𝐷2𝑘2𝐴+1⊗𝐷2𝑘2𝐴+1,𝑒𝑖⊕𝑥,ğ‘’î…žğ‘–î€¸|||||⊕𝑦>23ğ´ğ‘˜âŽ«âŽªâŽ¬âŽªâŽ­>18,(4.10) where 𝑟=[23𝐴𝑘/(𝑐𝐴𝑘)]+1.
Set𝜉𝑘2(𝑥,𝑦)=4𝐴𝑘−1Φ24𝐴𝑘𝑀𝑘(𝑥,𝑦),(4.11) where 𝑀𝑘1(𝑥,𝑦)=𝑟𝑟𝑖=1𝜀𝑖𝐷2𝑘2𝐴+1𝑒𝑖𝐷⊕𝑥2𝑘2𝐴+1î€·ğ‘’î…žğ‘–î€¸âŠ•ğ‘¦.(4.12) Denote 𝜈𝑘=27𝐴𝑘−12𝑟Φ4𝐴𝑘.(4.13) It is easy to show that mes(𝑥,𝑦)∈𝐼2∶|||𝐭𝜅𝑝𝐴𝑘𝜉𝑘|||,𝑥,𝑦>ğœˆğ‘˜î‚‡âŽ§âŽªâŽ¨âŽªâŽ©=mes(𝑥,𝑦)∈𝐼2∶|||||𝑟𝑖=1𝜀𝑖𝐭𝜅𝑝𝐴𝑘𝐷2𝑘2𝐴+1⊗𝐷2𝑘2𝐴+1,𝑒𝑖⊕𝑥,ğ‘’î…žğ‘–î€¸|||||⊕𝑦>23ğ´ğ‘˜âŽ«âŽªâŽ¬âŽªâŽ­>18.(4.14) Hence, (4.1) is proved.
Sinceâ€–â€–ğ‘€ğ‘˜â€–â€–ğ¿âˆž(𝐼2)≤24𝑚𝑘+2,‖‖𝑀𝑘‖‖𝐿1(𝐼2)≤1.(4.15) Moreover, ||||𝐼2||||≤1𝑓𝑔2𝐼2Φ2||𝑓||,+1Φ(𝑢)𝑢<Î¦î€·ğ‘¢î…žî€¸ğ‘¢î…žî€·0<𝑢<ğ‘¢î…žî€¸.(4.16) From (4.8) we can write ‖‖𝜉𝑘‖‖𝐿Φ(𝐼2)≤121+𝐼2Φ24𝐴𝑘||𝑀𝑘||(𝑥,𝑦)Φ24𝐴𝑘≤1𝑑𝑥𝑑𝑦21+𝐼2Φ24𝐴𝑘24𝐴𝑘+2/Φ24𝐴𝑘24𝐴𝑘24𝐴𝑘+2/Φ24𝐴𝑘24𝐴𝑘||𝑀𝑘(||𝑥,𝑦)Φ24𝐴𝑘≤1𝑑𝑥𝑑𝑦21+𝐼2Φ24𝐴𝑘24𝐴𝑘24𝐴𝑘||𝑀𝑘||(𝑥,𝑦)Φ24𝐴𝑘𝑑𝑥𝑑𝑦≤1.(4.17)
Hence, 𝜉𝑘∈𝑆Φ(0,1), and Theorem 2.1 is proved.

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

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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.


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