An Analogy of the Carleson–Hunt Theorem with Respect to Vilenkin Systems

In this paper we discuss and prove an analogy of the Carleson–Hunt theorem with respect to Vilenkin systems. In particular, we use the theory of martingales and give a new and shorter proof of the almost everywhere convergence of Vilenkin–Fourier series of f∈Lp(Gm)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f\in L_p(G_m)$$\end{document} for p>1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p>1$$\end{document} in case the Vilenkin system is bounded. Moreover, we also prove sharpness by stating an analogy of the Kolmogorov theorem for p=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p=1$$\end{document} and construct a function f∈L1(Gm)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f\in L_1(G_m)$$\end{document} such that the partial sums with respect to Vilenkin systems diverge everywhere.


Journal of Fourier Analysis and Applications
(2022) 28:48 (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. The classical theory of Hilbert spaces (for details see e.g the books [58,60]) says that if we consider the partial sums S n f := n−1 k=0 f (k) ψ k , with respect to Vilenkin systems, then In the same year Schipp [45], Simon [51] and Young [67] (see also the book [49]) generalized this inequality for 1 < p < ∞: there exists an absolute constant c p , depending only on p, such that S n f p ≤ c p f p , when f ∈ L p (G m ).
From this it follows that for every f ∈ L p (G m ) with 1 < p ≤ ∞, S n f − f p → 0, as n → ∞.
The boundedness does not hold for p = 1, but Watari [64] (see also Gosselin [23] and Young [67]) proved that there exists an absolute constant c such that, for n = 1, 2, . . . , the weak type estimate holds: The almost-everywhere convergence of Fourier series for L 2 functions was postulated by Luzin [39] in 1915 and the problem was known as Luzin's conjecture. Carleson's theorem is a fundamental result in mathematical analysis establishing the pointwise (Lebesgue) almost everywhere convergence of Fourier series of L 2 functions, proved by Carleson [10] in 1966. The name is also often used to refer to the extension of the result by Hunt [26] which was given in 1968 to L p functions for p ∈ (1, ∞) (also known as the Carleson-Hunt theorem).
Carleson's original proof is exceptionally hard to read, and although several authors have simplified the arguments there are still no easy proofs of his theorem. Expositions of the original Carleson's paper were published by Kahane [28], Mozzochi [40], Jorsboe and Mejlbro [27] and Arias de Reyna [43]. Moreover, Fefferman [16] published a new proof of Hunt's extension, which was done by bounding a maximal operator of partial sums This, in turn, inspired a much simplified proof of the L 2 result by Lacey and Thiele [35], explained in more detail in Lacey [33]. The books Fremlin [17] and Grafakos [24] also give proofs of Carleson's theorem. An interesting extension of Carleson-Hunt result much more closer to L 1 space then L p for any p > 1 was done by Carleson's student Sjölin [56] and later on, by Antonov [2]. Already in 1923, Kolmogorov [31] showed Recent proof of almost everywhere convergence of Walsh-Fourier series was given by Demeter [12] in 2015. By using some methods of martingale Hardy spaces, almost everywhere convergence of subsequences of Vilenkin-Fourier series was considered in [8]. Antonov [3] proved that for f ∈ L 1 (log + L)(log + log + log + L)(G m ) its Walsh-Fourier series converges a.e. Similar result for the bounded Vilenkin systems was proved by Oniani [42]. However, there exists a function from L 1 (log + log + L)(G m ) whose Vilenkin-Fourier series diverges everywhere, where in this result G m is a general (not necessary "bounded") Vilenkin group (see Tarkaev [59]).
Stein [57] constructed an integrable function whose Walsh-Fourier series diverges almost everywhere. Later Schipp [44,49] proved that there exists an integrable function whose Walsh-Fourier series diverges everywhere. Kheladze [29,30] proved that for any set of measure zero there exists a function in f ∈ L p (G m ) (1 < p < ∞) whose Vilenkin-Fourier series diverges on the set, while the result for continuous or bounded function was proved by Harris [25] or Bitsadze [6]. Moreover, Simon [53] constructed an integrable function such that its Vilenkin-Fourier series diverges everywhere. Bochkarev [9] considered rearrangements of Vilenkin-Fourier series of bounded type.
In this paper, we use the theory of martingales and give a new an shorter proof of the almost everywhere convergence of Vilenkin-Fourier series of f ∈ L p (G m ) for p > 1 in the case the Vilenkin system is bounded. The positive results of this paper are derived in Sect. 3. In Theorem 2 we prove the boundedness of the maximal operator on L p (1 < p < ∞) spaces. By using this result, we derive the L p norm convergence of the partial sums of Vilenkin-Fourier series (Theorem 3) as well as the analogue of the Carleson-Hunt theorem, i.e., the almost everywhere convergence of the partial sums of f ∈ L p (Theorem 4), when 1 < p < ∞. The proof is built up by proving some new lemmas of independent interest. The corresponding sharpness and almost everywhere divergence are stated and proved in Sect. 4, see Theorems 5 and 6. Especially Theorem 6 is the Kolmogorov type result and also here the proof is built up by proving some lemmas of independent interest. In order not to disturb our discussion later, some necessary preliminaries are presented in Sect. 2. Define the group G m as the complete direct product of the groups Z m i with the product of the discrete topologies of Z m j '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, i.e. the case when sup n m n < ∞.
The elements of G m are represented by sequences It is easy to give a base for the neighborhood of G m : where x ∈ G m , n ∈ N. Denote I n := I n (0) for n ∈ N + , and I n := G m \ I n .
If we define the so-called generalized number system based on m by then every n ∈ N can be uniquely expressed as n = ∞ j=0 n j M j , where n j ∈ Z m j ( j ∈ N + ) and only a finite number of n j 's differ from zero. For two natural numbers n = ∞ j=1 n j M j and k = ∞ j=1 k j M j , we define that Next, we introduce on G m an orthonormal system which is called the Vilenkin system. First, we define the complex-valued function r k (x) : G m → C, the generalized Rademacher functions, by Now, define the Vilenkin system ψ := (ψ n : n ∈ N) on G m as: Specifically, we call this system the Walsh-Paley system when m ≡ 2. The norms (or quasi-norms) of the spaces L p (G m ) (0 < p < ∞) is defined by The Vilenkin system is orthonormal and complete in L 2 (G m ) (for details see e.g. the books [1,49]). Now, 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 Dirichlet kernels with respect to the Vilenkin system in the usual manner: respectively. Recall that (see e.g. Simon [50,54] and Golubov et al. [22]) and It is known that (for the details see e.g. [1,7,37,38]) there exist absolute constants C 1 and C 2 such that A function P is called Vilenkin polynomial if P = n k=0 c k ψ k . The spectra of the Vilenkin polynomial P is defined by

Martingale Inequalities
We will also need some martingale inequalities. The σ -algebra generated by the intervals {I n (x) : x ∈ G m } will be denoted by F n (n ∈ N). If F denotes the set of Haar measurable subsets of G m , then obviously F n ⊂ F. By a Vilenkin interval we mean one of the form I n (x), n ∈ N, x ∈ G m . The conditional expectation operators relative to F n are denoted by E n . An integrable sequence f = ( f n ) n∈N is said to be a martingale if f n is F n -measurable for all n ∈ N and E n f m = f n in the case n ≤ m. We can see that if f ∈ L 1 (G m ), then (E n f ) n∈N is a martingale. Martingales with respect to (F n , n ∈ N) are called Vilenkin martingales. It is easy to show (see e.g. Weisz [65, p. 11]) that the sequence (F n , n ∈ N) is regular, i.e., for all non-negative Vilenkin martingales ( f n ), where R := max n∈N m n .
. By the well known martingale theorems, this implies that For a Vilenkin martingale f = ( f n ) n∈N , the maximal function is defined by For denote the martingale differences, where f −1 := 0. The square function and the conditional square function of f are defined by We have shown the following theorem in [65].
We will use the following convexity, concavity theorem proved in [65].

A.E. Convergence of Vilenkin-Fourier Series
We introduce some notations. For j, k ∈ N we define the following subsets of N : We introduce also the partial sums taken in these intervals: For simplicity, we suppose that f (0) = 0. The last author has proved in [66] that, for an arbitrary n ∈ I k j M k , For I = I k n(k) , let Since I k n(k) = I k n(k) implies n(k + 1) =ñ(k + 1), the operators T I (I ∈ I) are well defined. Note that there are n k summands in (8).
Lemma 1 For all n ∈ N, we have where I k n(k) is defined in (7). Proof We sketch the proof, only. It is proved in [66] that Moreover, n is contained in I k n(k) and I k n(k) ⊂ I k+1 n(k+1) . Since we get that This finishes the proof of Lemma 1.

Lemma 2
For all k, n ∈ N, we have Proof Equalities (9) and (6) imply which shows the lemma.
Proof First, ψ n T I k n(k) f is F k+1 measurable because of (9) and the fact that r k is F k+1 measurable. Since E k (r i k ) = 0 for i = 1, . . . , m n − 1, we can see that Before proving our main theorem, we need some further notations and lemmas. In what follows, I , J , K denote some elements of I. Let Since |s I + f | is F I + measurable, by the regularity condition (4), Then Proof Let us fix I J in I and t in G m . Set Therefore K = τ K + K . Consequently, if the set is empty then T ;I ,J f (t) = 0 or else let K 1 be its minimum element. Moreover, denote by K 0 one of the minimum elements of the set This means that if L K 0 , then τ L (t) = 0. Thus α K 0 (t) = 1 and By Lemma 2 and (11), On the other hand, Taking the supremum over all I J , we get (12).
where α I is defined in Lemma 4. Observe that α I can be rewritten as Denote by P p,q the set of functions f ∈ L 1 which satisfy f p,q < ∞. For q = ∞, define It is easy to see that and where α I is defined in Lemma 4.
Proof Equality (9) implies that for any F K measurable function ξ . By Lemma 3, for a suitable n ∈ I , ψ n T K f I ⊂K is a martingale difference sequence relative to (F K + ) I ⊂K . We have Using Burkholder-Gundy's inequality (see Theorem 1) together with (10), we obtain where p 0 > 1. Applying again Theorem 1, one can establish that For fixed I and t ∈ G m let us denote by K 0 (t) ∈ I (resp. K 1 (t) ∈ I) the smallest (resp. largest) interval K ⊃ I for which K 0 (t) = 1 (resp. K 1 (t) = 1). Then By (11) and by the definition of K , Hence By Tsebisev's inequality and the concavity theorem (see Proposition 1), for p 0 ≥ q > 1, one can see that Set p 0 := q 2 / p ≥ q > 1 and observe that which shows the lemma.

Lemma 6
Let max(1, p) < q < ∞ and f ∈ P p,q . Then Proof First we define a decomposition generated by the sequences k = ( k K , K ∈ I), where Notice that (10) and (14) imply Henceforth Let us apply Lemma 4 to k and x = 2 k to write Choosing j ∈ Z such that 2 j < y ≤ 2 j+1 , we get that By Lemma 5, for any k ∈ Z and z k > 0, we have Consequently, To use (15), observe that c β k≤ Thus, by (15), so the lemma is proved.

Lemma 7
Suppose that 1 < p, q < ∞ satisfy (1/ p, 1/q) ∈ . Then, for all f ∈ L p , Proof For an arbitrary x > 0, let us use the definition of α I given in (13). Then α I is F I − measurable and, obviously, For all I ∈ I, introduce the projections F I := α I s I and observe that s I • s J = 0 for every incomparable I and J . Therefore, we get for every g ∈ L 1 and I , J ∈ I that where δ I ,J is the Kronecker symbol. Thus the projections F I are orthogonal and Bessel's inequality implies for any g ∈ L 2 that Let us introduce the operators where (η I , I ∈ I) is a fixed sequence of functions satisfying η I ∞ ≤ 1 for each I ∈ I. Then Furthermore, by Doob's inequality, for any 1 < s ≤ ∞ and g ∈ L s . It follows by interpolation that where 1/ p = (1 − t)/2 + t/s and 1/q = (1 − t)/2 for any 0 ≤ t ≤ 1. Setting g := f and η t := sign s I f , we have Using the fact that we can see that which finishes the proof.
Now we are ready to formulate our first main result. Then Proof It is easy to see that Lemma 1 implies S * f ≤ T * f . It follows from Lemmas 6 and 7 that sup y>0 y p μ S * f > y ≤ C p f p for 1 < p < ∞. Now the proof of the theorem follows by the Marcinkiewicz interpolation theorem.
The next norm convergence result in L p spaces for 1 < p < ∞ follow from the density of the Vilenkin polynomials in L p (G m ) and from Theorem 2.
Our announced Carleson-Hunt type theorem reads: Then S n f → f , a.e., as n → ∞.
The proof follows directly by using Theorem 2 and the fact that the Vilinkin polynomials are dense in L p .

Almost Everywhere Divergence of Vilenkin-Fourier Series
A set E ⊂ G m is called a set of divergence for L p (G m ) if there exists a function f ∈ L p (G m ) whose Vilenkin-Fourier series diverges on E.
Proof We claim that given any g ∈ L 1 (G m ), there is an unbounded monotone increasing sequence λ = λ j , j ∈ N of positive real numbers and a function f ∈ L 1 (G m ) such thatf To prove this claim use (5) for p = 1 to choose a strictly increasing sequence of positive integer n 1 , n 2 , . . . such that Consider the function f defined by By (17), the series converges in the norm of L 1 (G m ).
In particular, f belongs to for j ∈ N. Therefore, the claim follows from orthogonality if we set To prove the theorem, suppose that g ∈ L 1 (G m ) is a function whose Vilenkin-Fourier series diverges on E. Use the claim to choose a monotone increasing, unbounded sequence λ which satisfies (16). By Abel's transformation, for any integers n, m ∈ N with n > m. Since λ is increasing, it follows that Since λ is unbounded, it follows that S n g converge at x when S * f (x) is finite. In particular, (S * f )(x) = ∞ for all x ∈ E.
Proof Suppose first that E is a set of divergence for L 1 (G m ). Let g ∈ L 1 (G m ) be a function whose Vilenkin-Fourier series diverges on E. By repeating the proof of Lemma 8, we can choose an unbounded, monotone increasing positive sequence (λ j , j ∈ N) and a function f ∈ L 1 (G m ) such that for all integers n, m ∈ N, m < n. Let (ω j , j ∈ N) be an unbounded sequence of positive, increasing numbers which satisfy For example, let then |S n g(x) − S m g(x)| → 0, as n, m → ∞ and we get that S n g(x) is a convergent series for any x ∈ E, which is contradiction. Consequently, the inequality holds for infinitely many integers n ∈ N. Use (5) for p = 1 to choose strictly increasing sequences of positive integers (n j , j ∈ N) and (α j , j ∈ N) which satisfy n j < α j + 1, and Consider the functions defined by Clearly, these functions are Vilenkin polynomials. We will show that they satisfy (18) and (19). Since S M n h 1 ≤ h 1 , for n ∈ N and h ∈ L 1 (G m ), (18) is a direct consequence of (21). To verify (19), fix x ∈ E and choose an n ∈ N satisfying (20) which is large enough so that α j < n ≤ α j+1 for some j ∈ N + . Since the definition of P j implies S n P j = S n f − S n (S M n j f ), we have by (20) and (22) that Hence (19) follows from the fact that ω n → ∞ as n → ∞. Conversely, suppose that is a sequence of Vilenkin polynomials which satisfies (18) and (19). Let n 1 := α 1 and for j > 1 set n j := 1 + max{n j−1 , α j }. Then (n j , j ∈ N) is a strictly increasing sequence of integers and it is easy to see that for any choice of integers k 0 and k 1 which satisfy 0 ψ M n j P j and observe by (18) that f ∈ L 1 (G m ). It is clear that the series defining f converges in L 1 (G m ) norm. Consequently, this series is the Vilenkin-Fourier series of f . Moreover, (23) can be used to see that Thus there exist integers α 1 < α 2 < . . . such that ∞ j=α n P (n) Let (Q j , j ∈ N + ) be any enumeration of the polynomials P (n) j : j ≥ α n , n = 1, 2, . . . . e.g., In particular, by Lemma 9 it suffices to show that But this follows from the construction and from (24) since every x ∈ E necessarily belongs to some E n . Proof We begin with a general remark. If A ⊆ G m is a finite union of intervals I 1 , I 2 , . . . , I n for some n ∈ N + and if N is any non-negative integer, then there exists a Vilenkin polynomial P such that, for some i ≥ N , Indeed, if i := max{M N , 1/μ(I j ) : 1 ≤ j ≤ n}, then, in view of (2), we find that P := χ(A)ψ i is such a polynomial.
To prove the theorem, suppose E ⊆ G m satisfies μ(E) = 0. Cover E with intervals (I k , k ∈ N) such that ∞ k=0 μ(I k ) < 1 and each x ∈ E belongs to infinitely many of the sets I k . Set n 0 := 0 and choose integers n 0 < n 1 < n 2 · · · such that Apply the general remark above successively to the sets to generate integers α 0 := 0 < α 1 < α 2 < · · · and Vilenkin polynomials P 0 , P 1 , . . . such that sp(P j ) ⊂ M α j , M α j +1 : and P j (x) = 1 x ∈ A j , for j ∈ N.
Setting f := ∞ j=1 P j , we observe by (25) that this series converges in L p (G m ) norm. Hence f ∈ L p (G m ) and this series is the Vilenkin-Fourier series of f . Moreover, since the spectra of the polynomials P j are pairwise disjoint, we have S M α j +1 f − S M α j f = P j ( j ∈ N + ).
Since every x ∈ E belongs to infinitely many of the sets A j , it follows from (26) that the Vilenkin-Fourier series of f diverges at every point x ∈ E.
This theorem cannot be improved for 1 < p < ∞ and measurable sets with nonzero measure. Indeed, in this case the Vilenkin-Fourier series of an f ∈ L p (G m ) converges a.e. (see Theorem 4). However, it can be improved considerably for p = 1.

Theorem 6
There is a function f ∈ L 1 (G m ) whose Vilenkin-Fourier series diverges everywhere.
Proof Fix α n ∈ M n−1 , M n ) , where n ∈ N + is odd. By using the lower estimate for the Lebesgue constant in (3), we can conclude that there exists an absolute constant C > 0, which does not depend on n, such that α n −1 k=0 ψ k 1 = D α n 1 > Cn.
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