Abstract
We prove and discuss some new weak type \((1,1 ) \) inequalities of maximal operators of Vilenkin–Nörlund means generated by monotone coefficients. Moreover, we use these results to prove a.e. convergence of such Vilenkin–Nörlund means. As applications, both some well-known and new inequalities are pointed out.
Similar content being viewed by others
1 Introduction
In 1947 Vilenkin [31] actually introduced a large class of compact groups (now called Vilenkin groups) and the corresponding characters. In particular, Vilenkin investigated the group \(G_{m}\), which is a direct product of the additive groups \(Z_{m_{k}}:=\{0,1,\ldots ,m_{k}-1\}\) of integers modulo \(m_{k}\), where \(m:=(m_{0},m_{1},\ldots )\) are positive integers not less than 2, and introduced the Vilenkin systems \(\{{\psi}_{j}\}_{j=0}^{\infty}\) as follows:
where \(\mathbb{N}_{+}\) denotes the set of positive integers and \(\mathbb{N}:=\mathbb{N}_{+}\cup \{0\}\). In this paper we discuss bounded Vilenkin groups only, that is, \(\sup_{n\in \mathbb{N}}m_{n}<\infty \). The Vilenkin system is orthonormal and complete in \(L^{2} ( G_{m} ) \) (see [31]). Specifically, we call this system the Walsh–Paley system when \(m\equiv 2\).
It is well known (see e.g. the books [1] and [27]) that if \(f\in L^{1}(G_{m})\) and the Vilenkin series \(T (x )=\sum_{j=0}^{\infty}c_{j}\psi _{j} (x ) \) converges to f in \(L^{1}\)-norm, then
where \(c_{j}\) is called the jth Vilenkin–Fourier coefficient and μ is the Haar measure on the locally compact abelian groups \(G_{m}\), which coincide with the direct product of measures \(\mu _{k} ( \{j\} ) :=1/m_{k}\) (\(j\in Z_{m_{k}}\)).
The classical theory of Hilbert spaces (for details, see e.g. the books [1, 27]) implies that if we consider the partial sums \(S_{n}\), defined by
with respect to any orthonormal systems and among them to Vilenkin systems, then the inequality \(\Vert S_{n} f \Vert _{2}\leq \Vert f \Vert _{2} \) holds. It follows that for every \(f\in L^{2}\),
Since
and the Dirichlet kernels
are not uniformly bounded in \(L^{1}(G_{m})\), the boundedness of partial sums does not hold from \(L^{1}(G_{m})\) to \(L^{1}(G_{m})\).
The analogue of Carleson’s theorem for the Walsh system was proved by Billard [4] for \(p=2\) and by Sjölin [29] for \(1 < p<\infty \), while for bounded Vilenkin systems it was proved by Gosselin [13]. In each proof, they show that the maximal operator of the partial sums is bounded on \(L^{p}(G_{m})\), i.e., there exists an absolute constant \(c_{p}\) such that
A recent proof of almost everywhere convergence of subsequences of Walsh–Fourier series was given by Demeter [7] in 2015. Hence, if \(f\in L^{p}(G_{m})\) for \(p>1\), then
Persson, Schipp, Tephnadze, and Weisz [22] (see also [25]) gave a new and shorter proof of almost everywhere convergence of Vilenkin–Fourier series of \(f\in L^{p}(G_{m})\), which was based on the theory of martingales.
The nth Nörlund mean \(L_{n}\) is defined by
In [9] Gát and Goginava proved some properties of the Nörlund logarithmic means of integrable functions in \(L^{1}\) norm. Moreover, in [10] they proved that weak type \((1,1)\) inequality does not hold for the maximal operator of Nörlund logarithmic means \(L^{\ast}\), defined by
but there exists an absolute constant \(c_{p}\) such that the inequality
holds.
If we define the so-called generalized number system based on m in the following way:
then every \(n\in \mathbb{N}\) can be uniquely expressed as \(n=\sum_{j=0}^{\infty }n_{j}M_{j}\), where \(n_{j}\in Z_{m_{j}}\) (\(j\in \mathbb{N}\)) and only a finite number of \(n_{j}\)s differ from zero. Moreover, if we consider the following restricted maximal operator \(\widetilde{L}_{\#}^{\ast}\), defined by
then
Hence, if \(f\in L^{1}(G_{m})\), then \(L_{M_{n}}f\to f\) a.e. on \(G_{m}\).
If we consider the Fejér means \(\sigma _{n}\) and Fejér kernels \(K_{n}\), defined by
it is obvious that
Since \(\Vert K_{n} \Vert _{1} \leq c<\infty \), we obtain that the Fejér means are bounded from the space \(L^{p}\) to the space \(L^{p}\) for \(1\leq p\leq \infty \). The a.e. convergence of Fejér means is due to Schipp [26] for Walsh series and Pál, Simon [21] (see also Simon, Weisz [28] and Weisz [28, 32–34]) for bounded Vilenkin series proved that the maximal operator of Fejér means \(\sigma ^{\ast }\), defined by
is of weak type \((1,1)\), from which the a.e. convergence follows by standard argument (see [14]). Another well-known summability method is the so-called \((C,\alpha )\)-means (denoted by \(\sigma _{n}^{\alpha}\)), which are defined by
It is well known that for \(\alpha =1\) this summability method coincides with the Fejér summation and for \(\alpha =0\) we just have the partial sums of the Vilenkin–Fourier series. Moreover, if we consider the maximal operator of the Cesáro means \(\sigma ^{\alpha ,\ast}\), defined by
then the following weak type inequality holds (for details, see [23]):
The boundedness of the maximal operator of the Cesáro means does not hold from \(L^{1}(G_{m})\) to the space \(L^{1}(G_{m})\). However,
The nth Nörlund mean \(t_{n}\) for the Fourier series of f is defined by
where \(\{q_{k}:k\in \mathbb{N}\}\) is a sequence of nonnegative numbers and \(Q_{n}:=\sum_{k=0}^{n-1}q_{k} \).
If we assume that \(q_{0}>0\) and \(\lim_{n\rightarrow \infty }Q_{n}=\infty \), then it is well known (see [15]) that the summability method generated by \(\{q_{k}:k\geq 0\}\) is regular if and only if \(\lim_{n\rightarrow \infty }\frac{q_{n-1}}{Q_{n}}=0 \). The representation
plays a central role in the sequel. The Nörlund means are generalizations of the Fejér, Cesàro, and Nörlund logarithmic means.
Móricz and Siddiqi [16] investigated the approximation properties of some special Nörlund means of Walsh–Fourier series of \(L^{p}\) functions in norm. Similar problems for the two-dimensional case can be found in papers by Nagy [17–20] (see also [5]).
Let us define the maximal operator \(t^{\ast}\) of Nörlund means by
and if \(\{q_{k}:k\in \mathbb{N}\}\) is nonincreasing and satisfying the condition
then in [23] it was proved that the weak type inequality
holds. When the sequence \(\{q_{k}:k\in \mathbb{N}\}\) is nonincreasing, then the weak type \((1,1)\) inequality (2) holds for every maximal operator of Nörlund means. The boundedness of the maximal operator of the Nörlund means does not hold from \(L^{1}(G_{m})\) to the space \(L^{1}(G_{m})\). However,
Moreover, if \(\{q_{k}:k\in \mathbb{N}\}\) is nondecreasing and satisfying the condition
or \(\{q_{k}:k\in \mathbb{N}\}\) is nonincreasing, then for any \(f\in L^{1}(G_{m})\) we have that
for all Vilenkin–Lebesgue points of f.
In this paper we investigate a wider class of Nörlund means and prove that if \(\{q_{k}:k\in \mathbb{N}\}\) is nondecreasing and satisfying the conditions
then the weak type inequality (2) holds. In particular, from this result follows almost everywhere convergence of such Nörlund means.
The paper is organized as follows: In Sect. 3 we present and prove the main results. Moreover, in order not to disturb our discussions in this section, some preliminaries are given in Sect. 2. Also some of these results are new and of independent interest.
2 Preliminaries
Lemma 1
Let \(n\in \mathbb{N}\). Then
Moreover, if \(n\in \mathbb{N}\) and \(1\leq s_{n}\leq m_{n}-1\), then
and
where \(n=\sum_{i=0}^{\infty}n_{i}M_{i}\). We note that \(\sum_{k=m_{j}-n_{j}}^{m_{j}-1}r_{j}^{k}\equiv 0\) for all \(n_{j}=0\).
Lemma 2
(see [8])
Let \(n>t\), \(t,n\in \mathbb{N}\). Then
Lemma 3
If \(n\geq M_{N}\) and \(\{q_{k}:k\in \mathbb{N}\}\) is a sequence of nondecreasing numbers, then there exists an absolute constant c such that
If the sequence \(\{q_{k}:k\in \mathbb{N}\}\) is either nondecreasing and satisfying condition (3) or nonincreasing and satisfying condition (1), then the inequality
holds. On the other hand, if \(\{q_{k}:k\in \mathbb{N}\}\) is a sequence of nonincreasing numbers satisfying (4) for \(0<\alpha <1\), then there exists a constant \(c_{\alpha}\), depending only on α, such that the following inequality holds:
Lemma 4
Let \(\{q_{k}:k\in \mathbb{N}\}\) be either a sequence of nondecreasing numbers or nonincreasing numbers satisfying condition (1) or nonincreasing numbers satisfying the conditions in (4). Then, for any \(n, N\in \mathbb{N_{+}}\),
where
for any \(x\in G_{m}\), \(n\in \mathbb{N}\).
The next lemma is very important to study problems concerning almost everywhere convergence.
Lemma 5
(see [32])
Suppose that the σ-sublinear operator V is bounded from \(L^{p_{1}}\) to \(L^{p_{1}}\) for some \(1< p_{1}\leq \infty \) and
for \(f\in L^{1}\) and Vilenkin interval I, which satisfy
Then the operator V is of weak type \(( 1,1 )\), i.e., the following inequality holds:
Lemma 6
(see [14])
Let
be sublinear operators for some \(1\leq p<\infty \) with T bounded and
for each \(f\in X_{0}\), where \(X_{0}\) is dense in \(L^{p} ( G_{m} )\). Set
If there is a constant \(C>0\), independent of f and n, such that the weak type inequalities
and
hold for all \(y>0\) and \(f\in L^{p} ( G_{m} )\), then
for every \(f\in L^{p} ( G_{m} )\).
Next we prove a new lemma of independent interest, which is very important to prove almost everywhere convergence of Nörlund means generated by nondecreasing sequences \(\{q_{k}:k\in \mathbb{N}\}\).
Lemma 7
Let \(n\in \mathbb{N}\) and \(\{q_{k}:k\in \mathbb{N}\}\) be a sequence of nondecreasing numbers. Then
where c is an absolute constant.
Proof
If we define
then we can decompose \(\overline{I_{N}}:=G_{m} \backslash I_{N}\) as
Let \(n>M_{N}\) and
By using Lemma 3, we get that
so that
Let \(n>M_{N}\) and \(x\in I_{N}^{k,N}\). By using Lemma 1, we can conclude that
and
so that
Hence, if we apply estimates (8) and (9), then we get that
The proof is complete. □
We also need the following new lemmas.
Lemma 8
Let \(\{q_{k}:k\in \mathbb{N}\}\) be a sequence of nonincreasing numbers satisfying condition (1). Then there exists an absolute constant c such that
Proof
The proof is analogous to that of Lemma 7. Hence, we leave out the details. □
Lemma 9
Let \(\{q_{k}:k\in \mathbb{N}\}\) be a sequence of nondecreasing numbers satisfying condition (3). Then there exists an absolute constant c such that
Proof
Also in this case the proof is analogous to that of Lemma 7, so we leave out the details. □
Finally, we prove the following new estimate of independent interest.
Lemma 10
Let \(n\in \mathbb{N}\) and \(\{q_{k}:k\in \mathbb{N}\}\) be a sequence of nonincreasing numbers satisfying the conditions in (4). Then there exists an absolute constant c such that
Proof
Let \(n>M_{N}\) and \(x\in I_{N}^{k,l}\), \(k=0,\dots ,N-2\), \(l=k+1,\dots ,N-1\). By combining Lemma 2 and (5) in Lemma 3, we get that
so that
Let \(n>M_{N}\) and \(x\in I_{N}^{k,N}\). By using Lemma 1, we can conclude that
so that
By combining (7), (11), and (12), we can conclude that
so (10) holds and the proof is complete. □
3 The main results
Our first main result reads as follows.
Theorem 1
Let \(t_{n}\) be the Nörlund means and \(F_{n}\) be the corresponding Nörlund kernels such that
If the maximal operator \(t^{*}\) of Nörlund means is bounded from \(L^{p_{1}}\) to \(L^{p_{1}}\) for some \(1< p_{1}\leq \infty \), then the operator \(t^{*}\) is of weak type \(( 1,1 ) \), i.e., for all \(f\in L^{1}(G_{m})\), the following weak type inequality holds:
Proof
In view of Lemma 5 we obtain that the proof is complete if we prove that
for every function f, which satisfies the conditions in (6), where I denotes the support of the function f.
Without loss of generality we may assume that f is a function with support I and \(\mu ( I ) =M_{N}\). We may also assume that \(I=I_{N}\). It is easy to see that
Therefore, we can suppose that \(n>M_{N}\). Moreover,
so that
which implies that
Hence,
Let \(t\in I_{N}\) and \(x\in \overline{I_{N}}\). Then \(x-t\in \overline{I_{N}}\) and (14) implies that
Thus (13) holds, so the proof is complete. □
By using the same technique of proof, we obtain in a similar way the following result.
Theorem 2
Let \(t_{n}\) be Nörlund means and \(F_{n}\) be the corresponding Nörlund kernels such that
If the maximal operator \(t^{*}\) of the Nörlund means is bounded from \(L^{p_{1}}\) to \(L^{p_{1}}\) for some \(1< p_{1}\leq \infty \), then the operator \(t^{*}\) is of weak type \(( 1,1 ) \), i.e., the following weak type inequality
holds for all \(f\in L^{1}(G_{m})\).
Next, we present a new related result concerning almost everywhere convergence of some summability methods. The study of almost everywhere convergence is one of the most difficult topics in Fourier analysis.
Theorem 3
Let \(f\in L^{1}(G_{m})\) and \(t_{n}\) be the regular Nörlund means with nondecreasing sequences \(\{q_{k}:k\in \mathbb{N}\}\). Then
Proof
Since
according to the regularity of Nörlund means with nondecreasing sequence \(\{q_{k}:k\in \mathbb{N}\}\), we obtain that
where \(P\in \mathcal{P}\) is dense in the space \(L^{1}\).
On the other hand, by combining Lemma 4, Lemma 7, and Theorem 1, we obtain that the maximal operator \(t^{\ast}\) of the Nörlund means with nondecreasing sequence \(\{q_{k}:k\in \mathbb{N}\}\) is bounded from the space \(L^{1}\) to the space \(weak-L^{1}\), that is, the following weak type inequality holds:
Hence, according to Lemma 6, we obtain the claimed almost everywhere convergence of Nörlund means with nondecreasing sequence \(\{q_{k}:k\in \mathbb{N}\}\):
The proof is complete. □
Theorem 4
Let \(f\in L^{1}\) and \(t_{n}\) be the Nörlund means with nondecreasing sequence \(\{q_{k}:k\geq 0\}\) satisfying the conditions in (3). Then
Proof
The proof is similar to the proof of Theorem 3 if we instead apply Lemma 4, Lemma 9, and Theorem 1, so we omit the details. □
Next we consider almost everywhere convergence of Nörlund means with nonincreasing sequence \(\{q_{k}:k\in \mathbb{N}\}\).
Theorem 5
Let \(f\in L^{1}\) and \(t_{n}\) be the Nörlund means with nonincreasing sequence \(\{q_{k}:k\in \mathbb{N}\}\) satisfying condition (1). Then
Proof
The proof is quite analogous to that of Theorem 3 if we apply Lemma 4, Lemma 8, and Theorem 1, so we omit the details. □
Theorem 6
Let \(f\in L^{1}\) and \(t_{n}\) be Nörlund means with nonincreasing sequence \(\{q_{k}:k\in \mathbb{N}\}\) satisfying the conditions in (4). Then
Proof
The proof is similar to the proof of Theorem 3 if we instead apply Lemma 4, Lemma 10, and Theorem 1, so we omit the details. □
Theorem 7
Let \(f\in L^{1}\) and \(t_{n}\) be Nörlund means with nonincreasing sequence \(\{q_{k}:k\in \mathbb{N}\}\). Then
Proof
If we apply the fact that (see [8–10], and [25])
we can prove that if \(\{q_{k}:k\in \mathbb{N}\}\) is a sequence of nonincreasing numbers, then, for any \(N\in \mathbb{N_{+}}\),
and
and also in this case the proof is absolutely analogous to that of Theorem 3, so we can omit the details. □
A number of special cases of our results are of particular interest and give both well-known and new information. We just give the following examples of such corollaries.
In particular, since \(\sigma _{n}\) and \(\sigma _{n}^{\alpha }\) are regular Nörlund means with nondecreasing sequence \(\{q_{k}:k\in \mathbb{N}\}\), we have the following consequences of our Theorems:
Corollary 1
Let \(f\in L^{1}\). Then
and
Corollary 2
Let \(f\in L^{1}\). Then
We also give the following examples of new consequences.
Corollary 3
Let \(f\in L^{1}\) and the summability method \(V_{n}^{\alpha}\) be defined by
Then
Proof
Since \(V_{n}^{\alpha }\) are Nörlund means with nonincreasing sequences \(\{q_{k}:k\in \mathbb{N}\}\) satisfying the conditions in (4). Hence, the proof is complete by just using Theorem 6. □
Corollary 4
Let \(f\in L^{1}\) and the summability method \(\beta _{n}^{\alpha}\) be defined by
Then
Proof
We note that \(\beta _{n}^{\alpha}\) are Nörlund means with nondecreasing sequences \(\{q_{k}:k\in \mathbb{N}\}\). Hence, the proof is complete by just using Theorem 3. □
Corollary 5
Let \(f\in L^{1}\) and \(B_{n}\) be the Nörlund means with monotone and bounded sequence \(\{ q_{k}:k\in \mathbb{N} \} \). Then
Proof
The proof follows from Theorems 4 and 5. □
Corollary 6
Let \(f\in L^{1}\) and the summability method \(U_{n}^{\alpha}\) be defined by
Then
Proof
Obviously, \(U^{\alpha}_{n}\) are regular Nörlund means with nonincreasing sequences \(\{q_{k}:k\in \mathbb{N}\}\), the proof follows from Theorem 7. □
Availability of data and materials
Not applicable.
References
Agaev, G., Vilenkin, N., Dzahafarly, G., Rubinstein, A.: Multiplicative Systems of Functions and Harmonic Analysis on Zero-Dimensional Groups. Ehim, Baku (1981)
Baramidze, D., Persson, L.-E., Singh, H., Tephnadze, G.: Some new results and inequalities for subsequences of Nörlund logarithmic means of Walsh-Fourier series. J. Inequal. Appl. (2022). https://doi.org/10.1186/s13660-022-02765-5
Baramidze, L., Persson, L.E., Tephnadze, G., Wall, P.: Sharp \(H_{p}-L_{p}\) type inequalities of weighted maximal operators of Vilenkin-Nörlund means and its applications. J. Inequal. Appl. (2016). https://doi.org/10.1186/s13660-016-1182-1
Billard, P.: Sur la convergence presque partout des séries de Fourier-Walsh des fonctions de l’espace \(L^{2} (0, 1)\). Stud. Math. 28, 363–388 (1967)
Blahota, I., Nagy, K., Tephnadze, G.: Approximation by Marcinkiewicz Θ-means of double Walsh-Fourier series. Math. Inequal. Appl. 22(3), 837–853 (2019)
Blahota, I., Persson, L.E., Tephnadze, G.: On the Nörlund means of Vilenkin-Fourier series. Czechoslov. Math. J. 65(4), 983–1002 (2015)
Demeter, C.: A guide to Carleson’s theorem. Rocky Mt. J. Math. 45(1), 169–212 (2015)
Gát, G.: Investigations of certain operators with respect to the Vilenkin system. Acta Math. Hung. 61, 131–149 (1993)
Gát, G., Goginava, U.: Uniform and L-convergence of logarithmic means of Walsh-Fourier series. Acta Math. Sin. Engl. Ser. 22(2), 497–506 (2006)
Gát, G., Goginava, U.: On the divergence of Nörlund logarithmic means of Walsh-Fourier series. Acta Math. Sin. Engl. Ser. 25(6), 903–916 (2009). (English summary)
Goginava, U.: Almost everywhere convergence of subsequences of logarithmic means of Walsh-Fourier series. Acta Math. Acad. Paedagog. Nyházi. 21, 169–175 (2005)
Golubov, B.I., Efimov, A.V., Skvortsov, V.A.: Walsh Series and Transforms. Mathematics and Its Applications, vol. 64. Nauka, Moscow (1987). (Russian), English transl, Kluwer Academic, Dordrecht, 1991
Gosselin, J.: Almost everywhere convergence of Vilenkin-Fourier series. Trans. Am. Math. Soc. 185, 345–370 (1973)
Marcinkiewicz, I., Zygmund, A.: On the summability of double Fourier series. Fundam. Math. 32, 112–132 (1939)
Moore, C.N.: Summable Series and Convergence Factors, Summable Series and Convergence Factors. Dover, New York (1966)
Móricz, F., Siddiqi, A.: Approximation by Nörlund means of Walsh-Fourier series. J. Approx. Theory 70(3), 375–389 (1992). (English summary)
Nagy, K.: Approximation by Nörlund means of quadratical partial sums of double Walsh-Fourier series. Anal. Math. 36(4), 299–319 (2010)
Nagy, K.: Approximation by Cesáro means of negative order of Walsh-Kaczmarz-Fourier series. East J. Approx. 16(3), 297–311 (2010)
Nagy, K.: Approximation by Nörlund means of Walsh-Kaczmarz-Fourier series. Georgian Math. J. 18(1), 147–162 (2011)
Nagy, K.: Approximation by Nörlund means of double Walsh-Fourier series for Lipschitz functions. Math. Inequal. Appl. 15(2), 301–322 (2012)
Pál, J., Simon, P.: On a generalization of the concept of derivative. Acta Math. Acad. Sci. Hung. 29(1–2), 155–164 (1977)
Persson, L.-E., Schipp, F., Tephnadze, G., Weisz, F.: An analogy of the Carleson-Hunt theorem with respect to Vilenkin systems. J. Fourier Anal. Appl. (2022). https://doi.org/10.1007/s00041-022-09938-2
Persson, L.E., Tephnadze, G., Wall, P.: On the maximal operators of Vilenkin-Nörlund means. J. Fourier Anal. Appl. 21(1), 76–94 (2015)
Persson, L.E., Tephnadze, G., Wall, P.: Some new \((H_{p},L_{p})\) type inequalities of maximal operators of Vilenkin-Nörlund means with non-decreasing coefficients. J. Math. Inequal. 9(4), 1055–1069 (2015)
Persson, L.E., Tephnadze, G., Weisz, F.: Martingale Hardy Spaces and Summability of One-Dimensional Vilenkin-Fourier Series. Springer, Basel (2022)
Schipp, F., Certain, F.: Rearrangements of series in the Walsh system. Mat. Zametki 18(2), 193–201 (1975). (Russian)
Schipp, F., Wade, W.R., Simon, P., Pál, J.: Walsh Series. An Introduction to Dyadic Harmonic Analysis. Hilger, Bristol (1990)
Simon, P., Weisz, F.: Weak inequalities for Cesáro and Riesz summability of Walsh-Fourier series. J. Approx. Theory 151(1), 1–19 (2008)
Sjölin, P.: An inequality of Paley and convergence a.e. of Walsh-Fourier series. Ark. Mat. 7, 551–570 (1969)
Tephnadze, G.: Martingale Hardy Spaces and Summability of the One Dimensional Vilenkin-Fourier Series. PhD thesis, Department of Engineering Sciences and Mathematics, LuleåUniversity of Technology (2015). ISSN 1402-1544
Vilenkin, N.Y.: On a class of complete orthonormal systems. Am. Math. Soc. Transl. 28(2), 1–35 (1963)
Weisz, F.: Martingale Hardy Spaces and Their Applications in Fourier Analysis. Lecture Notes in Mathematics, vol. 1568. Springer, Berlin (1994)
Weisz, F.: Cesáro summability of one- and two-dimensional Walsh-Fourier series. Anal. Math. 22(3), 229–242 (1996)
Weisz, F.: \(( C,\alpha ) \) summability of Walsh-Fourier series, summability of Walsh-Fourier series. Anal. Math. 27(2), 141–155 (2001)
Acknowledgements
The work of Davit Baramidze was supported by Shota Rustaveli National Science Foundation grant PHDF-21-1702. The publication charges for this article have been funded by a grant from the publication fund of UiT The Arctic University of Norway. We thank both careful referees for some generous advice, which has improved the final version of this paper.
Funding
The publication charges for this manuscript are funded by UiT The Arctic University of Norway. Open Access funding provided by UiT The Arctic University of Norway (incl University Hospital of North Norway).
Author information
Authors and Affiliations
Contributions
DB and NN gave the idea and initiated the writing of this paper. LEP and GT followed up this with some complementary ideas. All authors read and approved the final manuscript.
Corresponding author
Ethics declarations
Competing interests
The authors declare no competing interests.
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Baramidze, D., Nadirashvili, N., Persson, LE. et al. Some weak type inequalities and almost everywhere convergence of Vilenkin–Nörlund means. J Inequal Appl 2023, 66 (2023). https://doi.org/10.1186/s13660-023-02970-w
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13660-023-02970-w