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

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 𝐿_Φ=𝐿_Φ(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 andlim_𝑢→+∞Φ(𝑢)𝑢=+∞,lim^𝑢→0Φ(𝑢)𝑢=0.(1.1)

This space is endowed with the norm‖𝑓‖_𝐿Φ(𝐼²₎=inf𝑘>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/2^𝑛,(𝑝+1)/2^𝑛) for some 𝑝∈𝐍, 0≤𝑝<2^𝑛. Given 𝑛∈𝐍 and 𝑥∈[0,1), let 𝐼_𝑛(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=^∞_𝑘=0𝑥_𝑘2^𝑘+1=∶𝑥₀,𝑥₁,…,𝑥_𝑛,…,𝑥_𝑘∈{0,1}.(1.3) In that case when there are two different forms we choose the one for which lim_𝑘→∞𝑥_𝑘=0. We use the notation𝑒_𝑗∶=12^𝑗+1=0,…,0,𝑥_𝑗=1,0,….(1.4)

It is well-known that [2, 3]𝐼_𝑛𝑥₀,…,𝑥_𝑛−1∶=𝐼_𝑛(𝑥)=𝑝2^𝑛,𝑝+12^𝑛,(1.5) where 𝑝=∑^{𝑛−1𝑗=0}𝑥_𝑗2^{𝑛−1−𝑗}.

Let 𝑟₀(𝑥) be a function defined by𝑟₀(𝑥)=⎧⎪⎨⎪⎩1,if𝑥∈0,1−1,if𝑥∈12,2,1,𝑟⁰(𝑥+1)=𝑟₀(𝑥).(1.6) The Rademacher functions are defined by𝑟_𝑛(𝑥)=𝑟₀(2^𝑛𝑥),𝑛≥0,𝑥∈[0,1).(1.7)

Let 𝑤₀,𝑤₁,… represent the Walsh functions, that is, 𝑤₀(𝑥)=1, and if 𝑛=∑^∞𝑖=0𝑛_𝑖2^𝑖 is a positive integer with 𝑛_𝑖∈{0,1}, then𝑤_𝑛(𝑥)∶=^∞_𝑖=0𝑟_𝑖(𝑥)^𝑛𝑖.(1.8) The order of the 𝑛 is denoted by |𝑛|∶=max{𝑗∈𝐍∶𝑛_𝑗≠0}.

The Walsh-Kaczmarz functions are given by 𝜅₀∶=1, and for 𝑛≥1𝜅_𝑛(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 𝜏_𝐴∶𝐼→𝐼 by𝜏_𝐴(𝑥)∶=^𝐴−1_𝑘=0𝑥_{𝐴−𝑘−1}2^−(𝑘+1)+^∞_𝑗=𝐴𝑥_𝑗2^−(𝑗+1).(1.10) By the definition of 𝜏_𝐴, we have (see [4])𝜅_𝑛(𝑥)=𝑟_|𝑛|(𝑥)𝑤_𝑛−2^|𝑛|𝜏_|𝑛|(𝑥)(𝑛∈𝐍,𝑥∈𝐼).(1.11)

The Dirichlet kernels are defined by𝐷^𝛼𝑛(𝑥)∶=^𝑛−1_𝑘=0𝛼_𝑘(𝑥),(1.12) where 𝛼_𝑘=𝑤_𝑘 for all 𝑘∈𝐍 or 𝜅_𝑘 for all 𝑘∈𝐍.

It is well-known that [2, 4]𝐷𝐷^𝜅𝑛(𝑥)=𝐷₂^|𝑛|(𝑥)+𝑤₂^|𝑛|(𝑥)𝐷^𝑤𝑛−2_|𝑛|𝜏_|𝑛|(𝑥),(1.13)_2𝑛(𝑥)∶=𝐷^𝑤2_𝑛(𝑥)=𝐷^𝜅2_𝑛(𝑥)=⎧⎪⎨⎪⎩2^𝑛,if𝑥∈0,10,if𝑥∈12^𝑛,2^𝑛,1.(1.14)

The 𝑛th Fejér means of the Walsh-(Kaczmarz-)Fourier series of function 𝑓 are given by𝜎^𝛼𝑛(𝑓,𝑥)∶=1𝑛^𝑛_𝑗=1𝑆^𝛼𝑗(𝑓,𝑥),(1.15) where𝑆^𝛼𝑛(𝑓,𝑥)=^𝑛−1_𝑘=0𝑓^𝛼(𝑘)𝛼_𝑘(𝑥)(1.16) is the 𝑛th partial sum of the Fourier series and 𝑓^𝛼(𝑛)∶=∫_𝐼𝑓𝛼_𝑛(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:𝑡^𝛼𝑛(𝑓,𝑥)=1𝑙_{𝑛𝑛−1𝑘=1}𝑆^𝛼𝑘(𝑓,𝑥)𝑛−𝑘,𝐹^𝛼𝑛(𝑡)=1𝑙_{𝑛𝑛−1𝑘=1}𝐷^𝛼𝑘(𝑡)𝑛−𝑘,(1.17) where 𝑙_𝑛=∑^{𝑛−1𝑘=1}1/𝑘.

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

If 𝑓∈𝐿(𝐼²), then the number 𝑓^𝛼(𝑚,𝑛)∶=∫_𝐼2𝑓𝛼_𝑚,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 by𝑆^{𝛼𝑚,𝑛}(𝑓,𝑥,𝑦)=^𝑚−1_{𝑖=0𝑛−1𝑗=0}𝑓^𝛼(𝑖,𝑗)𝛼_𝑖(𝑥)𝛼_𝑗(𝑦).(1.19)

The logarithmic means and kernels of quadratical partial sums of double Walsh-(Kaczmarz)-Fourier series are given by𝐭^𝛼𝑛(𝑓,𝑥,𝑦)=1𝑙_{𝑛𝑛−1𝑘=1}𝑆^{𝛼𝑘,𝑘}(𝑓,𝑥,𝑦)𝑛−𝑘,𝐅^𝛼𝑛(𝑥,𝑦)=1𝑙_{𝑛𝑛−1𝑘=1}𝐷^𝛼𝑘(𝑥)𝐷^𝛼𝑘(𝑦)𝑛−𝑘.(1.20)

It is evident that𝐭^𝛼𝑛(𝑓,𝑥,𝑦)−𝑓(𝑥,𝑦)=_𝐼2[𝑓(𝑥⊕𝑡,𝑦⊕𝑠)−𝑓(𝑥,𝑦)]𝐅^𝛼𝑛(𝑡,𝑠)𝑑𝑡𝑑𝑠,(1.21) where ⊕ denotes the dyadic addition [3]. The 𝑛th Marcinkiewicz kernels are given by𝒦^𝛼𝑛(𝑥,𝑦)∶=1𝑛^𝑛_𝑘=1𝐷^𝛼𝑘(𝑥)𝐷^𝛼𝑘(𝑦).(1.22)

In 1948 Šneĭder [5] showed that the inequalitylimsup_𝑛→∞𝐷^𝜅𝑛(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 𝑆^𝑤𝑛(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 𝑆^{𝑤𝑛,𝑚}(𝑓) of double Walsh-Fourier series [13]. The first example of a function from classes wider than 𝐿log⁺𝐿(𝐼²) with 𝑆^{𝑤𝑛,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 𝑡^𝜅𝑛𝑓 of one-dimensional Walsh-Kaczmarz-Fourier series are of weak type (1,1), and this fact implies that 𝑡^𝜅𝑛𝑓 converge in measure on 𝐼 for all functions 𝑓∈𝐿(𝐼) and 𝑡^{𝜅𝑛,𝑚}𝑓 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 sums1𝑙_{𝑛𝑛−1𝑖=1}𝑆_𝑖,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 𝐭^𝜅𝑛(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 𝐿_Φ(𝐼²) be an Orlicz space, such that 𝐿_Φ𝐼²⊈𝐿log⁺𝐿𝐼².(2.1)
Then the set of the functions in the Orlicz space 𝐿_Φ(𝐼²) with logarithmic means of quadratical partial sums of double Walsh-Kaczmarz series convergent in measure on 𝐼² is of first Baire category in 𝐿_Φ(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||𝐷^𝑤𝑛(𝑥)||<1𝑥(3.1) for any 0<𝑥<1. Then for these 𝑥’s we also get||𝐹^𝑤𝑛(𝑥)||<1𝑥,(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 𝑝_𝐴=2^2𝐴+⋯+2²+2⁰ (𝐴∈𝐍). Then for any 2^−2𝐴−1≤𝑥<1 and 𝐴∈𝐍, we have||𝐷^𝑤𝑝_𝐴(𝑥)||≥14𝑥.(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||𝐹^𝑤𝑝_𝐴(𝑥)||≥𝑐log(1/𝑥)𝑥log𝑝_𝐴(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∶⌊(𝑙_{𝑝[𝑡/2]−1}/16)−2¹⁵⌋>1} set 𝑡̃∶=⌊(𝑙_{𝑝[𝑡/2]−1}/16)−2¹⁵⌋ (where ⌊𝑢⌋ denotes the lower integral part of 𝑢), and we take a “small part’’ of the interval 𝐼_𝑡⧵𝐼_𝑡+1=[2^−𝑡−1,2^−𝑡). This way the intervals are defined as follows:Δ_𝑡∶=12^𝑡+1,12^𝑡+1+12^𝑡+𝑡̃.(3.5) The exceptional set 𝐽 is given by𝐽∶=^∞_𝑡=𝑡0Δ_𝑡.(3.6)

For the logarithmic kernels 𝐅^𝜅𝑝_𝐴 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)∈(𝐽_𝑠×𝐽_𝑡)⧵(𝑄^{(𝑠,𝑡)𝑡+}𝑠̃∪(Δ^𝑠×Δ_𝑡)). Then one has ||𝐅^𝑤𝑝_𝐴(𝑥,𝑦)||≥𝑐2^𝑡+𝑠𝑠log𝑝_𝐴,(3.7) where 𝐽_𝑡∶=𝐼_𝑡⧵𝐼_𝑡+1 and 𝑄^{𝐵(𝑠,𝑡)}∶=_{𝑥𝑠+1+⋯+𝑥𝑡−11≤1}_𝑥𝑡=0⋯¹_𝑥_𝐵−1=0×𝐼_𝐵𝐼_𝐵0,…,0,1,𝑥_𝑠+1,…,𝑥_𝐵−10,…,0,1,𝑦_𝑡+1=𝑥_𝑡+1,…,𝑦_𝐵−1=𝑥_𝐵−1.(3.8)

Now, for the Walsh-Kaczmarz logarithmic kernels 𝐅^𝜅𝑝_𝐴, we prove the following.

Lemma 3.2. Let 𝑥∈𝐼_2𝐴(1,𝑥₁,…,𝑥_𝑡−2,𝑥_𝑡−1=1,𝑥_𝑡,…,𝑥_𝑠−1,𝑥_𝑠=1,𝑥_𝑠+1=1,0,…,0)=∶𝐼^{𝑡,𝑠2𝐴},  𝑦∈𝐼_2𝐴(1,𝑦₁,…,𝑦_𝑡−2,𝑦_𝑡−1=0,𝑦_𝑡=1,𝑦_𝑡+1=1,0,…,0)=∶𝐼^𝑡2𝐴 for 𝑠,𝑡=2,3,…,𝐴 and 𝑡≤𝑠. Then ||𝐅^𝜅𝑝_𝐴(𝑥,𝑦)||≥𝑐2^{4𝐴−𝑡−𝑠}.(3.9)

Proof. Set (𝑥,𝑦)∈𝐼^{𝑡,𝑠2𝐴}×𝐼^𝑡2𝐴. Let 𝐆^𝛼𝑝_𝐴(𝑥,𝑦)∶=𝑙_𝑝𝐴𝐅^𝛼𝑝_𝐴(𝑥,𝑦),𝐺^𝛼𝑝_𝐴(𝑥)∶=𝑙_𝑝𝐴𝐹^𝛼𝑝_𝐴(𝑥)(3.10) for 𝛼=𝑤 or 𝜅. Now, we write that 𝐆^𝜅𝑝_𝐴(𝑥,𝑦)=^22𝐴_𝑗=1𝐷^𝜅𝑗(𝑥)𝐷^𝜅𝑗(𝑦)𝑝_𝐴−𝑗+^𝑝𝐴−1_{𝑗=22𝐴+1}𝐷^𝜅𝑗(𝑥)𝐷^𝜅𝑗(𝑦)𝑝_𝐴−𝑗∶=𝐼+𝐼𝐼.(3.11)
First, we discuss 𝐼𝐼. We use equality  (1.13)𝐼𝐼=^{𝑝𝐴−1−1}_𝑗=1𝐷^𝜅𝑗+2_2𝐴(𝑥)𝐷^𝜅𝑗+2_2𝐴(𝑦)=𝑙_𝑝𝐴−1𝑝_𝐴−1−𝑗+𝐷𝐷₂^2𝐴_22𝐴(𝑥)𝐷₂^2𝐴(𝑦)+𝐷₂^2𝐴(𝑥)𝑟_2𝐴(𝑦)𝐺^𝑤𝑝_𝐴−1𝜏_2𝐴(𝑦)(𝑦)𝑟_2𝐴(𝑥)𝐺^𝑤𝑝_𝐴−1𝜏_2𝐴(𝑥)+𝑟_2𝐴(𝑥)𝑟_2𝐴(𝑦)𝐆^𝑤𝑝_𝐴−1𝜏_2𝐴(𝑥),𝜏_2𝐴(𝑦).(3.12)
If (𝑥,𝑦)∈𝐼^{𝑡,𝑠2𝐴}×𝐼^𝑡2𝐴, then (see (1.14))𝜏𝐷₂^2𝐴(𝑥)=𝐷₂^2𝐴(𝑦)=0,_2𝐴(𝑥)𝜏=0,…,0,𝑥_𝑠+1=1,𝑥_𝑠=1,𝑥_𝑠−1,…,𝑥_𝑡,𝑥_𝑡−1=1,𝑥_𝑡−2,…,𝑥₁,𝑥₀=1,𝑥_2𝐴,…_2𝐴,(𝑦)=0,…,0,𝑦_𝑡+1=1,𝑦_𝑡=1,𝑦_𝑡−1=0,𝑦_𝑡−2,…,𝑦₁,𝑦₀=1,𝑦_2𝐴,….(3.13) Moreover, by Lemma 3.1, we have ||𝐼𝐼||=||𝐆^𝑤𝑝_𝐴−1𝜏_2𝐴(𝑥),𝜏_2𝐴(𝑦)||≥𝑐(2𝐴−𝑠)2^{4𝐴−𝑡−𝑠}.(3.14)
Now, we discuss 𝐼 by the help of  (1.13)𝐼=^2𝐴−1_2𝑙=0^𝑙+1−1_𝑗=2𝑙𝐷^𝜅𝑗(𝑥)𝐷^𝜅𝑗(𝑦)𝑝_𝐴−𝑗+𝐷^22𝐴(𝑥)𝐷₂^2𝐴(𝑦)𝑝_𝐴−1=^2𝐴−1_2𝑙=0^𝑙−1_𝑗=0𝐷^𝜅𝑗+2_𝑙(𝑥)𝐷^𝜅𝑗+2_𝑙(𝑦)𝑝_𝐴−𝑗−2^𝑙=^2𝐴−1_2𝑙=0^𝑙−1_𝑗=0𝐷₂^𝑙(𝑥)𝐷₂^𝑙(𝑦)𝑝_𝐴−𝑗−2^𝑙+^2𝐴−1_𝑙=0𝐷₂^𝑙(𝑥)^2𝑙−1_𝑗=0𝑟_𝑙(𝑦)𝐷^𝑤𝑗𝜏_𝑙(𝑦)𝑝_𝐴−𝑗−2^𝑙+^2𝐴−1_𝑙=0𝐷₂^𝑙(𝑦)^2𝑙−1_𝑗=0𝑟_𝑙(𝑥)𝐷^𝑤𝑗𝜏_𝑙(𝑥)𝑝_𝐴−𝑗−2^𝑙+^2𝐴−1_2𝑙=0^𝑙−1_𝑗=0𝑟_𝑙(𝑥)𝑟_𝑙(𝑦)𝐷^𝑤𝑗𝜏_𝑙(𝑥)𝐷^𝑤𝑗𝜏_𝑙(𝑦)𝑝_𝐴−𝑗−2^𝑙.(3.15) Since 𝑥₀=𝑦₀=1, 𝐷₂^𝑙(𝑥)=𝐷₂^𝑙(𝑦)=0 for all 𝑙≥1, thus, 𝐼=1𝑝_𝐴−1+^2𝐴−1_𝑙=1𝑟_𝑙(𝑥)𝑟_𝑙(𝑦)^2𝑙−1_𝑗=1𝐷^𝑤𝑗𝜏_𝑙(𝑥)𝐷^𝑤𝑗𝜏_𝑙(𝑦)𝑝_𝐴−𝑗−2^𝑙=∶1𝑝_𝐴−1+^2𝐴−1_𝑙=1𝐼_𝑙.(3.16)
We use Abel’s transformation for 𝐼_𝑙 (𝑙≥1)𝐼_𝑙=𝑟_𝑙(𝑥)𝑟_𝑙(𝑦)^2𝑙−2_𝑗=1𝑝_𝐴−1𝑗−2^𝑙−1𝑝_𝐴−𝑗−2^𝑙−1𝑗𝒦^𝑤𝑗+𝑟_𝑙𝜏_𝑙(𝑥),𝜏_𝑙(𝑦)(𝑥)𝑟_𝑙(𝑦)2^𝑙−1𝒦^𝑤2_𝑙−1𝜏_𝑙(𝑥),𝜏_𝑙(𝑦)𝑝_𝐴−2^𝑙+1+1=∶𝐼^1𝑙+𝐼^2𝑙.(3.17) We write ||𝐼^1𝑙||≤𝑐2^{24𝐴𝑙−2}_𝑗=1||𝐼𝑗||𝒦^𝑤𝑗𝜏_𝑙(𝑥),𝜏_𝑙(𝑦)_2𝑙||,||≤𝑐2^2𝐴2^𝑙−1||𝒦^𝑤2_𝑙−1𝜏_𝑙(𝑥),𝜏_𝑙(𝑦)||.(3.18) Since |𝐷_𝑖|≤𝑖, we have 𝑛||𝒦^𝑤𝑛||≤|||||^𝑛_𝑖=0𝐷^𝑤𝑖𝐷^𝑤𝑖|||||≤𝑐𝑛₃.(3.19) For 𝑙≤𝑠+2, we obtain that ||𝐼^1𝑙||≤𝑐2^4𝑙2^4𝐴,||𝐼^2𝑙||≤𝑐2^3𝑙2^2𝐴.(3.20) Moreover, ^𝑠+2_𝑙=1||𝐼^1𝑙||≤𝑐2^4𝐴𝑠+2_𝑙=12^4𝑙≤𝑐2^4𝑠2^4𝐴<𝑐,_𝑠+2_𝑙=1||𝐼^2𝑙||≤𝑐2^2𝐴𝑠+2_𝑙=12^3𝑙≤𝑐2^3𝑠2^2𝐴≤𝑐2^𝐴.(3.21) Let 𝑠+2<𝑙<2𝐴. Then we have 𝜏_𝑙(𝜏𝑥)=0,…,0,1,1,𝑥_𝑠−1,…,𝑥_𝑡−1=1,…,𝑥₁,1,0,…,0,𝑥_2𝐴,…_𝑙,(𝑦)=0,…,0,1,1,𝑦_𝑡−1=0,𝑦_𝑡−2,…,𝑦₁,1,0,…,0,𝑦_2𝐴,…,(3.22) that is, 𝜏_𝑙(𝑥)∈𝐼_{𝑙−𝑠−3}⧵𝐼_{𝑙−𝑠−2}, 𝜏_𝑙(𝑦)∈𝐼_{𝑙−𝑡−3}⧵𝐼_{𝑙−𝑡−2}.
Now, we introduce the following notation:𝒦^{𝑤𝑎,𝑏}(𝑥,𝑦)∶=^{𝑎+𝑏−1}_𝑗=𝑎𝐷^𝑤𝑗(𝑥)𝐷^𝑤𝑗(𝑦),(3.23) and 𝑛^(𝑠)∶=∑^∞𝑖=𝑠𝑛_𝑖2^𝑖 (𝑛,𝑠∈𝐍). A simple consideration gives 𝑛𝒦^𝑤𝑛(𝑥,𝑦)=^|𝑛|_𝑠=0𝑛_𝑠𝒦_{𝑛(𝑠+1),2}^𝑠(𝑥,𝑦)+𝐷^𝑤𝑛(𝑥)𝐷^𝑤𝑛(𝑦).(3.24) It is known [23] that if 𝑡≤𝑘 and (𝑥,𝑦)∈(𝐼_𝑡⧵𝐼_𝑡+1)×(𝐼_𝑘⧵𝐼_𝑘+1), then ||𝒦^𝑤𝑛_{(𝑠+1),2𝑠}(𝑥,𝑦)||≤𝑐2^{𝑠+𝑡+𝑘}.(3.25) Using this inequality, we have that ||𝐼^1𝑙||≤𝑐2^{4𝐴𝑙−1}_2𝑘=0^𝑘+1−1_𝑗=2𝑘≤𝑐𝑗||𝒦^𝑤𝑗𝜏_𝑙(𝑥),𝜏_𝑙(𝑦)||2^{4𝐴𝑙−1}_2𝑘=0^𝑘+1−1_𝑗=2𝑘^𝑘_𝜈=0|||𝒦_{𝑤𝑗(𝜈+1),2𝜈}≤𝑐𝜏_𝑙(𝑥),𝜏_𝑙(𝑦)|||+||𝐷_𝑗𝜏_𝑙(𝑥)𝐷_𝑗𝜏_𝑙(𝑦)||2^{4𝐴𝑙−1}_2𝑘=0^𝑘+1−1_𝑗=2𝑘^𝑘_𝜈=02^{𝜈+2𝑙−𝑡−𝑠}+𝑗²≤𝑐2_{4𝑙−𝑡−𝑠}2^4𝐴+𝑐2^3𝑙2^4𝐴.(3.26) Moreover, ^2𝐴−¹_𝑙=𝑠+3||𝐼^1𝑙||≤𝑐2^{8𝐴−𝑡−𝑠}2^4𝐴+𝑐2^6𝐴2^4𝐴≤𝑐2^{4𝐴−𝑡−𝑠}.(3.27) Analogously, ||𝐼^2𝑙||≤𝑐2^2𝐴^𝑙−1_𝜈=02^{𝜈+2𝑙−𝑡−𝑠}+2^2𝑙≤𝑐22^{3𝑙−𝑡−𝑠2𝐴}+𝑐2^2𝑙2^2𝐴,(3.28)_{2𝐴−1𝑙=}_𝑠+3||𝐼^2𝑙||≤𝑐2^{4𝐴−𝑡−𝑠}.(3.29) Taking into account (3.11), we get ||𝐅^𝜅𝑝_𝐴(𝑥,𝑦)||≥||𝐼𝐼||−||𝐼||𝑙_𝑝𝐴≥𝑐₁||𝐼𝐼||−||𝐼||2𝐴.(3.30) Therefore by (3.14)–(3.29), it follows that ||𝐅^𝜅𝑝_𝐴(𝑥,𝑦)||≥𝑐⋅𝑐₁1−2𝑠𝐴2^{4𝐴−𝑡−𝑠}−2𝑐𝐴̃2^{4𝐴−𝑡−𝑠}(3.31) with suitable constants 𝑐,𝑐₁,𝑐̃>0. It is clear that for 𝐴 large enough we have ||𝐅^𝜅𝑝_𝐴(𝑥,𝑦)||≥𝑐_∗2^{4𝐴−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,…,𝐸_𝑟,𝐸^1,…,𝐸^𝑟∈ℰ and 𝜀_𝑖=±1, 𝑖=1,…,𝑟, such that mes⎧⎪⎨⎪⎩(𝑥,𝑦)∈𝐼²∶|||||𝐻^𝑟_𝑖=1𝜀_𝑖𝑓𝐸_𝑖𝑥,𝐸^𝑖𝑦|||||>𝜆⎫⎪⎬⎪⎭≥18.(3.33)

Theorem B. Let {𝐻_𝑚}^∞𝑚=1 be a sequence of linear continuous operators, acting from Orlicz space 𝐿_Φ(𝐼²) into the space 𝐿⁰(𝐼²). Suppose that there exist the sequence of functions {𝜉_𝑘}^∞𝑘=1 from unit bull 𝑆_Φ(0,1) of space 𝐿_Φ(𝐼²), and sequences of integers {𝑚_𝑘}^∞𝑘=1 and {𝜈_𝑘}^∞𝑘=1 increasing to infinity such that 𝜀₀=inf_𝑘mes(𝑥,𝑦)∈𝐼²∶||𝐻_𝑚𝑘𝜉_𝑘(𝑥,𝑦)||>𝜈_𝑘>0.(3.34)
Then 𝐵—the set of functions 𝑓 from space 𝐿_Φ(𝐼²), for which the sequence {𝐻_𝑚𝑓} converges in measure to an a.e. finite function—is of first Baire category in space 𝐿_Φ(𝐼²).

Theorem GGT. Let 𝐿_Φ be an Orlicz space, and let 𝜑∶[0,∞)→[0,∞) be measurable function with condition 𝜑(𝑥)=𝑜(Φ(𝑥)) as 𝑥→∞. Then there exists Orlicz space 𝐿_𝜔, such that 𝜔(𝑥)=𝑜(Φ(𝑥))as 𝑥→∞, 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 {𝐴_𝑘∶𝑘≥1} and {𝜈_𝑘∶𝑘≥1} increasing to infinity, and a sequence of functions {𝜉_𝑘∶𝑘≥1} from the unit bull 𝑆_Φ(0,1) of Orlicz space 𝐿_Φ(𝐼²), such that for all 𝑘mes(𝑥,𝑦)∈𝐼²∶|||𝐭^𝜅𝑝_𝐴𝑘𝜉_𝑘,𝑥,𝑦|||>𝜈_𝑘≥18.(4.1)
First, we prove that there exists 𝑐>0 such thatmes(𝑥,𝑦)∈𝐼²∶|||𝐭^𝜅𝑝_𝐴𝑘𝐷₂^2𝐴+1⊗𝐷₂^2𝐴+1;𝑥,𝑦|||>𝑐2^3𝐴>𝑐𝐴2^3𝐴.(4.2)
It is easy to show that𝐭^𝜅𝑝_𝐴𝑘𝐷₂^2𝐴+1⊗𝐷₂^2𝐴+1;𝑥,𝑦=𝑆₂^2𝐴+1_,2^2𝐴+1𝐅^𝜅𝑝_𝐴;𝑥,𝑦=𝐅^𝜅𝑝_𝐴(𝑥,𝑦).(4.3) Hence, from Lemma 3.2, we can write |||𝐭_{𝜅𝑝𝐴𝑘}𝐷₂^2𝐴+1⊗𝐷₂^2𝐴+1;𝑥,𝑦|||=||𝐅^𝜅𝑝_𝐴(𝑥,𝑦)||≥𝑐2^{4𝐴−𝑡−𝑠},(4.4) for (𝑥,𝑦)∈𝐼^{𝑡,𝑠2𝐴}×𝐼^𝑡2𝐴.
Since𝑐2^{4𝐴−𝑡−𝑠}>𝑐2^3𝐴(4.5) for 𝑡+𝑠<𝐴 and 0≤𝑡≤𝑠<𝐴, from (4.4) we conclude that mes(𝑥,𝑦)∈≥𝐼𝑐²∶_𝐴||𝐭^𝜅𝑝_𝐴𝐷₂^2𝐴+1⊗𝐷₂^2𝐴+1;𝑥,𝑦||>𝑐2^3𝐴_{[]𝑠=𝐴/2𝐴−𝑠𝑡=0}≥𝑐_𝐴mes𝐼^{𝑡,𝑠2𝐴}×𝐼^𝑡2𝐴_{[]𝑠=𝐴/2𝐴−𝑠𝑡=0}2^𝑠2^2𝐴2^𝑡2^2𝐴≥𝑐^𝐴_[]_𝑠=𝐴/22^𝑠2^2𝐴2^𝐴−𝑠2^2𝐴≥𝑐𝐴2^3𝐴.(4.6) Hence, (4.2) is proved.
From the condition of the theorem, we write [1]liminf_𝑢→∞Φ(𝑢)𝑢log𝑢=0.(4.7) Consequently, there exists a sequence of integers {𝐴_𝑘}^∞𝑘=1 increasing to infinity, such that lim_𝑘→∞Φ2^4𝐴𝑘2^−4𝐴𝑘𝐴^𝑘−1=0,Φ2^4𝐴𝑘2^4𝐴𝑘≥4,∀𝑘.(4.8)
From (4.2) we writemes(𝑥,𝑦)∈𝐼²∶|||𝐭^𝜅𝑝_𝐴𝑘𝐷₂^𝑘2𝐴+1⊗𝐷₂^𝑘2𝐴+1;𝑥,𝑦|||>𝑐2^3𝐴𝑘>𝑐𝐴^𝑘2^3𝐴𝑘.(4.9) Then by the virtue of Theorem A, there exists 𝑒₁,…,𝑒_𝑟,𝑒^1,…,𝑒^𝑟∈[0,1] and 𝜀₁,…,𝜀_𝑟=±1 such that mes⎧⎪⎨⎪⎩(𝑥,𝑦)∈𝐼²∶|||||^𝑟_𝑖=1𝜀_𝑖𝐭^𝜅𝑝_𝐴𝑘𝐷₂^𝑘2𝐴+1⊗𝐷₂^𝑘2𝐴+1,𝑒_𝑖⊕𝑥,𝑒^𝑖⊕𝑦|||||>2^3𝐴𝑘⎫⎪⎬⎪⎭>18,(4.10) where 𝑟=[2^3𝐴𝑘/(𝑐𝐴_𝑘)]+1.
Set𝜉_𝑘(𝑥,𝑦)=2^{4𝐴𝑘−1}Φ2^4𝐴𝑘𝑀^𝑘(𝑥,𝑦),(4.11) where 𝑀_𝑘(𝑥,𝑦)=1𝑟^𝑟_𝑖=1𝜀_𝑖𝐷₂^𝑘2𝐴+1𝑒_𝑖⊕𝑥𝐷₂^𝑘2𝐴+1𝑒^𝑖⊕𝑦.(4.12) Denote 𝜈_𝑘=2^{7𝐴𝑘−1}𝑟Φ2^4𝐴𝑘.(4.13) It is easy to show that mes=(𝑥m,e𝑦s)⎧⎪⎨⎪⎩∈(𝑥𝐼,²𝑦)∶∈|||𝐭𝐼^𝜅𝑝_𝐴𝑘2𝜉_𝑘,𝑥,𝑦|||>𝜈_𝑘∶|||||^𝑟_𝑖=1𝜀_𝑖𝐭^𝜅𝑝_𝐴𝑘𝐷₂^𝑘2𝐴+1⊗𝐷₂^𝑘2𝐴+1,𝑒_𝑖⊕𝑥,𝑒^𝑖⊕𝑦|||||>2^3𝐴𝑘>1⎫⎪⎬⎪⎭8.(4.14) Hence, (4.1) is proved.
Since‖‖𝑀_𝑘‖‖_{𝐿∞(𝐼2)}≤2^4𝑚𝑘+2,‖‖𝑀_𝑘‖‖_𝐿1(𝐼2)≤1.(4.15) Moreover, ||||_𝐼2𝑓𝑔||||≤12_𝐼²Φ(𝑢)Φ2||𝑓||+1,𝑢<Φ𝑢^𝑢^0<𝑢<𝑢^.(4.16) From (4.8) we can write ‖‖𝜉_𝑘‖‖_{𝐿Φ(𝐼2)}≤121+_𝐼²Φ2^4𝐴𝑘||𝑀_𝑘(𝑥,𝑦)||≤1Φ2^4𝐴𝑘𝑑𝑥𝑑𝑦21+_𝐼²Φ2^4𝐴𝑘2^4𝐴𝑘+2/Φ2^4𝐴𝑘2^4𝐴𝑘2^4𝐴𝑘+2/Φ2^4𝐴𝑘2^4𝐴𝑘||𝑀_𝑘(𝑥,𝑦)||≤1Φ2^4𝐴𝑘𝑑𝑥𝑑𝑦21+_𝐼²Φ2^4𝐴𝑘2^4𝐴𝑘2^4𝐴𝑘||𝑀_𝑘(𝑥,𝑦)||≤1.Φ2^4𝐴𝑘𝑑𝑥𝑑𝑦(4.17)
Hence, 𝜉_𝑘∈𝑆_Φ(0,1), and Theorem 2.1 is proved.

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