Vilenkin–Lebesgue Points and Almost Everywhere Convergence for Some Classical Summability Methods

The concept of Vilenkin–Lebesgue points was introduced in [12], where the almost everywhere convergence of Fejer means of Vilenkin–Fourier series was proved. In this paper, we present a different (and simpler) approach to prove a similar result, which can be used to prove that the corresponding result holds also in a more general context, namely for regular Norlund and T-means.


Introduction
Concerning some definitions and notations used in this introduction, we refer to Sect. 2.
The fact that the Walsh system is the group of characters of a compact abelian group connects Walsh analysis with abstract harmonic analysis was discovered independently by Fine [7] and Vilenkin [40]. Later on, in 1947 Vilenkin [40][41][42] actually introduced a large class of compact groups (now called Vilenkin groups) and the corresponding characters which includes the dyadic group and the Walsh system as a special case. For general references to the haar measure and harmonic analysis on groups see Pontryagin [33], Rudin [34], and Hewitt and Ross [14]. In particular, Vilenkin investigated the group G m , which is a direct product of the additive groups Z m k =: {0, 1, . . . , m k −1} of integers modulo m k , where m =: (m 0 , m 1 , . . .) are positive integers not less than 2, and introduced the Vilenkin systems {ψ j } ∞ j=0 . These systems include as a special case the Walsh system and many of the proofs presented for the Walsh system can be generalized readily to the Vilenkin case. Fejer's theorem shows that (see, e.g., [1,5,6,37]) if one replaces ordinary summation by Fejer means σ n defined by σ n f := 1 n n k=1 S k f, then, for any 1 ≤ p ≤ ∞, there exists an absolute constant C p , depending only on p such that σ n f p ≤ C p f p . If we define the maximal operator σ * of Fejer means by holds for any integrable function. For example, this result can be found in Zygmund [47] (see also [9,19]) for trigonometric series, in Schipp [35] for Walsh series and in Pál, Simon [28] (see also [30,[44][45][46]) for bounded Vilenkin series. It follows that the Fejer means with respect to trigonometric and Vilenkin systems of any integrable function converges a.e to this function. It is known that almost every point x is a Lebesgue point of a function f ∈ L 1 and the Fejer means σ T n f of the trigonometric Fourier series of f ∈ L 1 converge to f at each Lebesgue point.
Weisz [43] introduced the Walsh-Lebesgue points and proved the analogue of the preceding result: almost every point is a Walsh-Lebesgue point of an integrable function f ∈ L 1 and the Walsh-Fejer means of f converge to f at each Walsh-Lebesgue point. Later, Goginava and Gogoladze [12] introduced the Vilenkin-Lebesgue points and proved similar result. They used methods of martingale Hardy spaces.
In this paper, we consider some more general summability methods, which are called Nörlund and T -means. In particular, the n-th Nörlund mean t n and T -mean T n of the Fourier series of f are, respectively, defined by and where Here, {q k : k ≥ 0} is a sequence of nonnegative numbers, where q 0 > 0 and lim n→∞ Q n = ∞. Then, the summability method (1) generated by {q k : k ≥ 0} is regular if and only if (see [17]) Moreover, the summability method (2) is regular if and only if It is well known (for details see, e.g., [32]) that every Nörlund summability method generated by non-increasing sequence (q k , k ∈ N) is regular, but Nörlund means generated by non-decreasing sequence (q k , k ∈ N) is not always regular. On the other hand, every T -mean generated by non-decreasing sequence (q k , k ∈ N) is regular, but T -means generated by non-increasing sequence (q k , k ∈ N) is not always regular. In this paper, we investigate only regular Nörlund and T -means. Almost everywhere convergence and summability of Nörlund and Tmeans were studied by several authors. We mentioned Bhahota, Persson and Tephnadze [3] (see also [2,4,16,31]), Tutberidze [38,39], Fridli, Manchanda, Siddiqi [8], Móricz and Siddiqi [18] Nagy [20][21][22][23] (see also [24][25][26][27]).
We also define the maximal operator t * of Nörlund means by If {q k : k ∈ N} is non-increasing and satisfying the condition then the weak-type inequality was proved in [30]. When the sequence {q k : k ∈ N} is non-decreasing, then the weak-(1,1) type inequality (4) holds for every maximal operator of Nörlund means. It follows that for such Nörlund means of f ∈ L 1 (G m ), we have that lim Define the maximal operator T * of T -means by It was proved in [38] that if {q k : k ∈ N} is non-increasing or if {q k : k ∈ N} is non-decreasing and satisfying the condition then the following weak-type inequality holds: It follows that for such T -means and for f ∈ L 1 (G m ), we have that The main aim of this paper is to find a different and simpler approach, with the help of which we can generalize the results in [12] and prove them for a more large class of regular Norlund and T -means.
The paper is organized as follows: the main results are presented, proved and discussed in Sect. 3. In particular, Theorems 1 and 2 are parts of this new approach. The announced results for Norlund and T -means can be found in Theorems 3 and 4, respectively. In order not to disturb the presentations in Sect. 3, we use Sect. 2 for some necessary preliminaries (e.g., definitions, notations, lemmas). In particular, Lemma 2 is new and of independent interest. Define the groups G m as the complete direct product of the group Z mj with the product of the discrete topologies of Z mj , s. The direct product μ of the measures

Preliminaries
is the Haar measure on G m with μ (G m ) = 1. In this paper, we discuss bounded Vilenkin groups only, that is The elements of G m are represented by the sequences It is easy to give a base for the neighborhood of G m , namely The intervals I n (x) (n ∈ N, x ∈ G m ) are called Vilenkin intervals. Denote I n := I n (0) for n ∈ N and I n := G m \I n . Let Defining I n := G m \I n and we have Next, we introduce on G m an orthonormal system, which is called the Vilenkin system. First, define the complex valued function r k (x) : G m → C, the generalized Rademacher functions, as We define the Vilenkin system ψ := (ψ n : n ∈ N) on G m as Especially, we call this system the Walsh-Paley one if m ≡ 2 (for details see [13,36]). The Vilenkin system is orthonormal and complete in L 2 (G m ) (for details see, e.g., [1,36,40]). Next, we introduce analogues of the usual definitions in Fourier analysis. If f ∈ L 1 (G m ), we can define the Fourier coefficients, the partial sums of the Fourier series, the Fejer means, the Dirichlet and Fejer kernels with respect to the Vilenkin system ψ in the usual manner: Recall that (for details see, e.g., [1,10,11]), and Moreover, if n > t, t, n ∈ N, then A point x is called a Lebesgue point of an integrable function f if Weisz [43] introduced the concept of Walsh-Lebesgue points for the dyadic group with the help of the operator Similarly to [12], now we generalize this by We also define the operator V A by It is evident that where It is obvious that Next, we state the following Lemma, which is very important to study almost everywhere convergence of Vilenkin-Fejer means (see, e.g., [44]).
Then, the operator V is of weak type (1, 1), i.e., We also need the following new Lemma of independent interest: where c is an absolute constant.
Proof. Let A > N and x ∈ I k,l N , k = 0, . . . , N − 2 and l = k + 1, . . . , N − 1. Then it is easy to prove that x − re s ∈ G m \I N for all r = 1, . . . , m s − 1. Using (7), we get that Let A > N and x ∈ I k,N N . Using again (7), we can conclude that D MA (x − re s ) = 0 if s = k and D MA (x − re k ) = 0 if r = x k . Moreover, By combining (6), (12) and (13), we find that The proof is complete.

The Main Results with Applications
In our first main result, we consider the maximal operator V * defined by Theorem 1. Let f ∈ L 1 (G m ). Then, the operator V * is of weak type (1, 1), i.e., sup

Proof.
Since we obtain that V * is bounded from L ∞ (G m ) to L ∞ (G m ). According to Lemma 1, the proof will be complete if we prove that for every function f satisfying the conditions in (11), where I denotes the support of the function f. Without loss the generality, we may assume that f is a function with support I and μ (I) = M N . We may assume that I = I N . It is easy to see that V n f = 0 when n ≤ M N . Therefore, we can suppose that n > M N . Hence, Let t ∈ I N and x ∈ I N . Then x − t ∈ I N and by applying Lemma 2, we get that which means that (14) holds so the proof is complete.
Next, we state the following convergence result for the operator W A : Proof. It is easy to see that lim A→∞ W A f (x) = 0 for every Vilenkin polynomial. Hence, since the Vilenkin polynomials are dense in L 1 (G m ), the usual density argument (see Marcinkiewicz and Zygmund [15]) and Theorem 1 imply the proof.
Our convergence result for the Fejer means reads: for all Vilenkin-Lebesgue points of f .