Abstract
In this paper we introduce some new weighted maximal operators of the partial sums of the Walsh–Fourier series. We prove that for some “optimal” weights these new operators indeed are bounded from the martingale Hardy space \(H_{p}(G)\) to the Lebesgue space \(L_{p}(G),\) for \(0<p<1.\) Moreover, we also prove sharpness of this result. As a consequence we obtain some new and well-known results.
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1 Introduction
It is well-known that the Walsh system does not form a basis in the space \(L_{1}\) (see e.g. [2, 27]). Moreover, there exists a function f in the dyadic Hardy space \(H_{1},\) such that the partial sums of f are not bounded in \(L_{1}\)-norm, but the partial sums \(S_{n}\) of the Walsh–Fourier series of a function \(f\in L_{1}\) convergence in measure (see [9, 12]). Uniform and pointwise convergence and some approximation properties of partial sums in \(L_{1}\) norm were investigated by Onneweer [16], Goginava [10], Goginava and Tkebuchava [11], Nagy [15], Avdispahić and Memić [1], Persson et al. [18]. Fine [6] obtained sufficient conditions for the uniform convergence, which are complete analogies with the Dini-Lipschits conditions. Guličev [13] estimated the rate of uniform convergence of a Walsh–Fourier series by using Lebesgue constants
and modulus of continuity. These problems for Vilenkin groups were considered by Blahota [3], Fridli [7] and Gát [8].
Above, and in the sequel, all used notations can be found in Sect. 2. For example, the notations \(D_n\) and \(S_n\) are given in (6).
To study convergence of subsequences of partial sums in the martingale Hardy spaces \(H_p(G)\) for \(0<p\le 1,\) the central role plays uniquely expression of any natural number \(n\in \mathbb {N}\)
where only a finite numbers of \(n_{j}\) differ from zero and their important characters \(\left[ n\right] ,\) \(\left| n\right| ,\) \(\rho \left( n\right) \) and V(n) are defined by
and
In particular, (see [5, 14, 19])
from which it follows that, for any \(F\in L_1(G),\) there exists an absolute constant c such that
Moreover, for any \(f\in H_1,\)
In [23] and [24] it was proved that if \(0<p<1\) and \(F\in H_{p},\) then there exists an absolute constant \(c_{p},\) depending only on p, such that
Moreover, if \(0<p<1,\) \(\left\{ n_{k}:\text { }k\ge 0\right\} \) is any increasing sequence of positive integers such that
and \(\Phi :\mathbb {N}_{+}\rightarrow [1,\infty )\) is any nondecreasing function, satisfying the condition
then there exists a martingale \(F\in H_{p},\) such that
For \(0<p<1\) in [21, 22] the weighted maximal operator \(\overset{\sim }{S }^{*,p},\) defined by
was investigated and it was proved that the following estimate holds:
Moreover, it was also proved that the rate of the sequence \(\left( n+1\right) ^{ 1/p-1}\) given in denominator of (3) can not be improved, but it was proved only for the special subsequences.
For \(p=1\) analogical results for the maximal operator \(\overset{\sim }{S}^{*},\) defined by
was proved in [22].
One main aim of this paper is to generalize the estimate (4) for \(f\in H_p(G),\) \(0<p<1.\) Our main idea is to investigate much more general maximal operators by replacing the weights \(\left( n+1\right) ^{1/p-1} \) in (3) by more general “optimal” weights
where \(\varphi :\mathbb {R}_+\rightarrow \mathbb {R}_+\) is any nonnegative and nondecreasing function satisfying the condition
and prove that it is bounded from the martingale Hardy space \(H_p(G)\) to the Lebesgue space \(L_p(G),\) for \(0<p<1.\) As a consequence we obtain some new and well-known results. In particular, we prove that the maximal operator \({\widetilde{S}}^{*,\nabla },\) defined by
is bounded from the Hardy space \({{H}_{p}}(G)\) to the Lebesgue space \({{L}_{p}}(G)\) for any \(\varepsilon >0\) and is not bounded from the Hardy space \({{H}_{p}}(G)\) to the Lebesgue space \({{L}_{p}}(G)\) when \(\varepsilon =0.\)
This paper is organized as follows: In order not to disturb our discussions later on some definitions and notations are presented in Sect. 2. The main results and some of its consequences can be found in Sect. 3. The detailed proofs are given in Sect. 4.
2 Preliminaries
Let \(\mathbb {N}_{+}\) denote the set of the positive integers, \( \mathbb {N}:=\mathbb {N}_{+}\cup \{0\}.\) Denote by \(Z_{2}\) the discrete cyclic group of order 2, that is \(Z_{2}:=\{0,1\},\) where the group operation is the modulo 2 addition and every subset is open. The Haar measure on \(Z_{2}\) is given so that the measure of a singleton is 1/2.
Define the group G as the complete direct product of the group \(Z_{2},\) with the product of the discrete topologies of \(Z_{2}\)‘s. The elements of G are represented by sequences \(x:=(x_{0},x_{1},...,x_{j},...),\) where \( x_{k}=0\vee 1.\)
It is easy to give a base for the neighborhood of \(x\in G{:}\)
Denote \(I_{n}:=I_{n}\left( 0\right) ,\) \(\overline{I_{n}}:=G\) \({\backslash } \) \(I_{n}\) and
Then it is easy to show that
The norms (or quasi-norm) of the spaces \(L_{p}(G)\) and \(\text {weak}-L_{p}\left( G\right) ,\) \(\left( 0<p<\infty \right) \) are, respectively, defined by
The k-th Rademacher function \(r_{k}\left( x\right) \) is defined by
Now, define the Walsh system \(w:=(w_{n}:n\in \mathbb {N})\) on G by
The Walsh system is orthonormal and complete in \(L_{2}\left( G\right) \) (see e.g. [19]).
If \(f\in L_{1}\left( G\right) \) we can establish the Fourier coefficients, the partial sums of the Fourier series, the Dirichlet kernels with respect to the Walsh system in the usual manner:
and
Moreover, we have the following lower estimate (see [17]):
Lemma 1
Let \(n\in \mathbb {N}\) and \(\left[ n\right] \ne \left| n\right| .\) Then
The \(\sigma \)-algebra generated by the intervals \(\left\{ I_{n}\left( x\right) :x\in G\right\} \) will be denoted by \(\zeta _{n}\left( n\in \mathbb { N}\right) .\) It is easy to see that
It follows that
Denote by \(F=\left( F_{n},n\in \mathbb {N}\right) \) the martingale with respect to \(\digamma _{n}\) \(\left( n\in \mathbb {N}\right) \) (for details see e.g. [25]).
The maximal function \(F^{*}\) of a martingale F is defined by
In the case \(f\in L_{1}\left( G\right) ,\) the maximal function \(f^{*}\) is given by
For \(0<p<\infty \) the Hardy martingale spaces \(H_{p}\left( G\right) \) consists of all martingales for which
It is easy to check that for every martingale \(F=\left( F_{n},n\in \mathbb {N} \right) \) and every \(k\in \mathbb {N}\) the limit
exists and it is called the k-th Walsh–Fourier coefficients of F.
If \(F:=\) \(\left( S_{2^n}f:n\in \mathbb {N}\right) \) is a regular martingale, generated by \(f\in L_{1}\left( G\right) ,\) then (for details see e.g. [17, 20] and [25])
A bounded measurable function a is called p-atom, if there exists a dyadic interval I, such that
The dyadic Hardy martingale spaces \(H_{p}\) for \(0<p\le 1\) have an atomic characterization. Namely, the following theorem holds (see [17, 25, 26]):
Lemma 2
A martingale \(F=\left( F_{n},n\in \mathbb {N}\right) \) belongs to \(H_{p}\left( 0<p\le 1\right) \) if and only if there exists a sequence \( \left( a_{k},\text { }k\in \mathbb {N}\right) \) of p-atoms and a sequence \( \left( \mu _{k},k\in \mathbb {N}\right) \) of real numbers such that for every \(n\in \mathbb {N}\)
where
Moreover, \( \left\| F\right\| _{H_{p}}\backsim \inf \left( \sum _{k=0}^{\infty }\left| \mu _{k}\right| ^{p}\right) ^{1/p}, \) where the infimum is taken over all decomposition of F of the form (9).
3 The main results
Our first main result reads:
Theorem 1
Let \(0<p<1\), \(f\in {{H}_{p}}\left( G \right) \) and \(\varphi :\mathbb {N}_+\rightarrow \mathbb {R}_+\) be any nonnegative and nondecreasing function satisfying the condition
Then the weighted maximal operator \({\widetilde{S}}^{*,\nabla },\) defined by
is bounded from the Hardy space \({{H}_{p}}(G)\) to the Lebesgue space \({{L}_{p}}(G)\).
Theorem 1 can be of special interest even if we restrict to subsequences.
Corollary 1
Let \(0<p<1\), \(f\in {{H}_{p}}\left( G \right) \), \(\varphi :\mathbb {N}_+\rightarrow \mathbb {R}_+\) be any nonnegative and nondecreasing function satisfying the condition (10) and \(\left\{ n_{k}:k\ge 0\right\} \) be any sequence of positive numbers. Then the weighted maximal operator \({\widetilde{S}}^{*,\nabla },\) defined by
is bounded from the Hardy space \({{H}_{p}}(G)\) to the Lebesgue space \({{L}_{p}}(G)\).
We also prove sharpness of Theorem 1:
Theorem 2
Let \(0<p<1,\) \(\left\{ n_{k}:k\ge 0\right\} \) be a sequence of positive numbers and \(\varphi :\mathbb {N}_+\rightarrow \mathbb {R}_+\) be any nonnegative and nondecreasing function satisfying the condition
Then there exists p-atoms \(a_{k},\) such that
If we take
we get that condition (10) is fulfilled, on the other hand, if we take
then condition (12) holds true. Hence, Theorems 1 and 2 imply the following sharp result:
Corollary 2
a) Let \(0<p<1\) and \(f\in {{H}_{p}}\left( G \right) \). Then the weighted maximal operator \({\widetilde{S}}^{*,\nabla , \varepsilon },\) defined by
is bounded from the Hardy space \({{H}_{p}}(G)\) to the Lebesgue space \({{L}_{p}}(G)\).
b) The weighted maximal operator \({\widetilde{S}}^{*,\nabla , 0 },\) defined by
is not bounded from the Hardy space \({{H}_{p}}(G)\) to the Lebesgue space \({{L}_{p}}(G)\).
Remark 1
Suppose that \(\left\{ n_{k}:k\ge 0\right\} \) is a sequence of positive numbers, such that
Then
and
Let
Then, the maximal operator (11) can not be estimated by
Hence, Theorem 1 and Remark 1 and Theorem proved in [21, 22] follows that if \(0<p<1\), \(f\in {{H}_{p}}\left( G \right) \) and \(\varphi :\mathbb {N}_+\rightarrow \mathbb {R}_+\) be any nonnegative and nondecreasing function satisfying the condition (10), then the weighted maximal operator \({\widetilde{S}}^{*,\nabla },\) defined by
is bounded from the Hardy space \({{H}_{p}}(G)\) to the Lebesgue space \({{L}_{p}}(G)\).
Now, we formulate a result proved in [22], which follows from Theorems 1 and 2:
Corollary 3
a) Let \(0<p\le 1\) and \(\left\{ \alpha _{k},k\in \mathbb {N}\right\} \) be a subsequence of positive numbers such that \( \sup _{k\in \mathbb {N}}\rho \left( \alpha _{k}\right) <\infty . \) Then the maximal operator \({\widetilde{S}}^{*,\vartriangle }\) defined by
is bounded from the Hardy space \(H_{p}(G)\) to the Lebesgue space \(L_{p}(G).\)
b) Let \(0<p<1\) and \(\left( \alpha _{k},k\in \mathbb {N}\right) \) be a subsequence of positive numbers satisfying the condition \( \sup _{k\in \mathbb {N}}\rho \left( \alpha _{k}\right) =\infty . \) Then the maximal operator \({\widetilde{S}}^{*,\vartriangle }\) defined by (13) is not bounded from the Hardy space \(H_{p}(G)\) to the Lebesgue space \(L_{p}(G).\)
4 Proofs of the Theorems
Proof of Theorem 1
By using Lemma 2 the proof of Theorem 1 will be complete, if we prove that
for every p-atom a, with support I and \(\mu \left( I\right) =2^{-M}\). We may assume that this arbitrary p-atom a has support \(I=I_{M}.\) It is easy to see that \(S_{n}a\left( x\right) =0,\) when \(n\le 2^{M}\). Therefore, we can suppose that \(n>2^{M}\). Since \(\left\| a\right\| _{\infty }\le 2^{M/p}\) we find that
Let \(x\in I_{M}\). Since \(V\left( n\right) \le \rho \left( n\right) +2,\) by applying (2) we get that
so that
Let \(t\in I_{M}\) and \(x\in I_{s}\backslash I_{s+1}, \ 0\le s\le M-1<[ n] \ \text { or } \ 0\le s<[ n] \le M-1.\) Then \(x+t\) \(\in I_{s}\backslash I_{s+1}\). By using (8) we get that \(D_{n}\left( x+t\right) =0\) and
Let \(x\in I_{s}\backslash I_{s+1}, \ [ n] \le s\le M-1.\) Then \(x+t\in I_{s}\backslash I_{s+1},\) for \(t\in I_{M}\). By using (8) we find that
Hence, by applying (15) we get that
By now using (18) for \(0<[n]<s/2\) we can conclude that
Moreover, according to (18) for \(s/2\le [n]\le s\) we have that
By combining (17), (19) and (20), for all \( x\in I_{s}\backslash I_{s+1}, \ 0 \le s\le M-1\) we get that
By now combining (5) and (21) we obtain that
By combining (16) and (22) we obtain that (14) holds and the proof is complete. \(\square \)
Proof of Theorem 2
In view of the condition (12) we have that
Set
It is evident that
Then we easily can derive that
Since
from (7) it follows that
Let \(q_{n_k}^{s}\in \mathbb {N}\) be such that
By applying (24) we can conclude that
Let \(x\in I_{s}\backslash I_{s+1}\). By using Lemma 1 we obtain that
and
Hence,
Finally, by combining (23) and (25) we find that
The proof is complete. \(\square \)
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Acknowledgements
The research was supported by Shota Rustaveli National Science Foundation grant no. PHDF-21-1702. We thank the referee for some good suggestions, which have improved this paper.
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Open access funding provided by UiT The Arctic University of Norway (incl University Hospital of North Norway) The publication charges for this article have been funded by a grant from the publication fund of UiT The Arctic University of Norway.
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DB and GT gave the idea and initiated the writing of this paper. LEP followed up on this with some complementary ideas. All authors read and approved the final manuscript.
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Baramidze, D., Persson, LE. & Tephnadze, G. Some new \((H_p-L_p)\) type inequalities for weighted maximal operators of partial sums of Walsh–Fourier series. Positivity 27, 38 (2023). https://doi.org/10.1007/s11117-023-00989-3
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DOI: https://doi.org/10.1007/s11117-023-00989-3