Journal of Function Spaces

1. Definitions and Notations

We denote by the Lebesgue space of functions that are measurable and finite almost everywhere on . is the Lebesgue measure of the set . The constants appearing in this paper are denoted by .

Let be the Orlicz space [1] generated by Young’s function ; that is, is convex, continuous, even function such that and

This space is endowed with the norm In particular case, if , , then the corresponding space is denoted by .

We denote the set of nonnegative integers by . By a dyadic interval in we mean one of the form for some , . Given and , let denote the dyadic interval of length which contains the point .

Every point can be written in the following form: In that case when there are two different forms we choose the one for which . We use the notation

It is well-known that [2, 3] where .

Let be a function defined by The Rademacher functions are defined by

Let represent the Walsh functions, that is, , and if is a positive integer with , then The order of the is denoted by .

The Walsh-Kaczmarz functions are given by , and for For and , Skvorcov [4] defined a transformation by By the definition of , we have (see [4])

The Dirichlet kernels are defined by where for all or for all .

It is well-known that [2, 4]

The th Fejér means of the Walsh-(Kaczmarz-)Fourier series of function are given by where is the th partial sum of the Fourier series and is said to be the th Walsh-(Kaczmarz-)Fourier coefficient of the function .

The Nörlund logarithmic (simply we say logarithmic) means and kernels of one-dimensional Walsh-(Kaczmarz-)Fourier series are defined as follows: where .

The Kronecker product () of two Walsh(-Kaczmarz) systems is said to be the two-dimensional Walsh(-Kaczmarz) system. That is,

If , then the number is said to be the th Walsh-(Kaczmarz-)Fourier coefficient of . The rectangular partial sums of double Fourier series with respect to both system are defined by

The logarithmic means and kernels of quadratical partial sums of double Walsh-(Kaczmarz)-Fourier series are given by

It is evident that where denotes the dyadic addition [3]. The th Marcinkiewicz kernels are given by

In 1948 Šneĭder [5] showed that the inequality 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 for any continuous functions . 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 of the Walsh-Fourier series of a function , converge in measure on [2]. The condition provides convergence in measure on of the rectangular partial sums of double Walsh-Fourier series [13]. The first example of a function from classes wider than with divergent in measure on was obtained in [14, 15]. Moreover, in [16] Tkebuchava proved that in each Orlicz space wider than the set of functions where quadratic Walsh-Fourier sums converge in measure on 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 of one-dimensional Walsh-Kaczmarz-Fourier series are of weak type (1,1), and this fact implies that converge in measure on for all functions and converge in measure on for all functions .

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 sums of double Walsh-Kaczmarz series and prove Theorem 2.1 that is, for any Orlicz space, which is not a subspace of , 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 there exists functions for which logarithmic means 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 be an Orlicz space, such that
Then the set of the functions in the Orlicz space with logarithmic means of quadratical partial sums of double Walsh-Kaczmarz series convergent in measure on is of first Baire category in .

Corollary 2.2. Let be a nondecreasing function satisfying for the condition
Then there exists a function such that
(a)(b)logarithmic means of quadratical partial sums of double Walsh-Kaczmarz series of diverge in measure on .

3. Auxiliary Results

It is well-known [2, 3] for the Dirichlet kernel function that for any . Then for these ’s we also get where is a nonnegative integer. It is also well-known for the Walsh-Paley-Dirichlet kernel functions the following lower pointwise estimates holds. Let (). Then for any and , we have

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 for all , and . We have the exceptional set , such that it is “rare around zero.” For set (where denotes the lower integral part of ), and we take a “small part’’ of the interval . This way the intervals are defined as follows: The exceptional set is given by

For the logarithmic kernels 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 , , and . Then one has where and

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

Lemma 3.2. Let ,   for and . Then

Proof. Set . Let for or . Now, we write that
First, we discuss . We use equality  (1.13)
If , then (see (1.14)) Moreover, by Lemma 3.1, we have
Now, we discuss by the help of  (1.13) Since , for all , thus,
We use Abel’s transformation for () We write Since , we have For , we obtain that Moreover, Let . Then we have that is, , .
Now, we introduce the following notation: and (). A simple consideration gives It is known [23] that if and , then Using this inequality, we have that Moreover, Analogously, Taking into account (3.11), we get Therefore by (3.14)–(3.29), it follows that with suitable constants . It is clear that for large enough we have where is an absolute constant.

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

Theorem A. Let be a linear continuous operator, which commutes with family of translations , that is, for all for all . Let and . Then for any under condition, , there exist and , , such that

Theorem B. Let be a sequence of linear continuous operators, acting from Orlicz space into the space . Suppose that there exist the sequence of functions from unit bull of space , and sequences of integers and increasing to infinity such that
Then —the set of functions from space , for which the sequence converges in measure to an a.e. finite function—is of first Baire category in space .

Theorem GGT. Let be an Orlicz space, and let be measurable function with condition as . Then there exists Orlicz space , such that as , and for .

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 and increasing to infinity, and a sequence of functions from the unit bull of Orlicz space , such that for all
First, we prove that there exists such that
It is easy to show that Hence, from Lemma 3.2, we can write for .
Since for and , from (4.4) we conclude that Hence, (4.2) is proved.
From the condition of the theorem, we write [1] Consequently, there exists a sequence of integers increasing to infinity, such that
From (4.2) we write Then by the virtue of Theorem A, there exists and such that where .
Set where Denote It is easy to show that Hence, (4.1) is proved.
Since Moreover, From (4.8) we can write
Hence, , and Theorem 2.1 is proved.