On the boundedness of subsequences of Vilenkin-Fejér means on the martingale Hardy spaces

arXiv:2002.04403v1 [math.CA] 4 Feb 2020

ON THE BOUNDEDNESS OF SUBSEQUENCES OF VILENKIN-FEJÉR MEANS ON THE MARTINGALE

HARDY SPACES

L-E. PERSSON, G. TEPHNADZE AND G. TUTBERIDZE

Abstract. In this paper we characterize subsequences of Fejér means with respect to Vilenkin systems, which are bounded from the Hardy spaceHpto the Lebesgue spaceLp,for all0< p <1/2.The result is in a sense sharp.

2000 Mathematics Subject Classification. 42C10, 42B25.

Key words and phrases: Vilenkin system, Vilenkin group, Vilenkin- Fejér means, martingale Hardy space, maximal operator, Vilenkin-Fourier series.

1. Introduction

In the one-dimensional case the weak (1,1)-type inequality for the maximal operator of Fejér means

σ^∗f := sup

n∈N

|σ_nf|

can be found in Schipp [12] for Walsh series and in Pál, Simon [10] for bounded Vilenkin series. Here, as usual, the symbol σ_n denotes the Fejér mean with respect to the Vilenkin system (and thus also called the Vilenkin- Fejér means, see Section 2).

Fujji [6] and Simon [14] verified thatσ^∗ is bounded fromH1 toL1. Weisz [23] generalized this result and proved boundedness ofσ^∗from the martingale space H_p to the Lebesgue space L_p for p > 1/2. Simon [13] gave a coun- terexample, which shows that boundedness does not hold for 0< p < 1/2.

A counterexample for p = 1/2 was given by Goginava [8] (see also [2] and [3]). Weisz [24] proved that the maximal operator of the Fejér means σ^∗ is bounded from the Hardy spaceH_1/2 to the spaceweak−L_1/2. The bound- edness of weighted maximal operators are considered in [9], [16] and [17].

Weisz [22] (see also [21]) also proved that the following theorem is true:

Theorem W: (Weisz)Letp > 0.Then the maximal operator

(1) σ^∇,∗f = sup

n∈N

|σ_Mnf|

Second author was supported by grant of Shota Rustaveli National Science Foundation of Georgia, no. YS-18-043 and third author was supported by grant of Shota Rustaveli National Science Foundation of Georgia, no. PHDF-18-476.

1

where M₀ := 1, M_n+1 := m_nM_n (n ∈ N) and m := (m₀, m₁, . . .) be a sequences of the positive integers not less than 2, which generate Vilenkin systems, is bounded from the Hardy spaceHp to the space Lp.

In [11] the result of Weisz was generalized and it was found the maximal subspace S⊂Nof positive numbers, for which the restricted maximal oper- ator on this subspace sup

n∈S⊂N

|σ_nf|of Fejér means is bounded from the Hardy spaceH_pto the spaceL_p for all0< p≤1/2.The new theorem (Theorem 1) in this paper show in particular that this result is in a sense sharp. In partic- ular, for every natural number n=P∞

k=0n_kM_k,where n_k∈Z_mk (k∈N₊) we define numbers

hni:= min{j ∈N:n_j 6= 0}, |n|:= max{j ∈N:n_j 6= 0}, ρ(n) =|n|−hni and prove that

S ={n∈N:ρ(n)≤c <∞.}

Sinceρ(M_n) = 0 for alln∈Nwe obtain that{M_n:n∈N} ⊂S and that follows i.e. that result of Weisz [22] (see also [21]) that restricted maximal operator (1) is bounded from the Hardy spaceH_p to the space L_p.

The main aim of this paper is to generalize Theorem W and find the maximal subspace of positive numbers, for which the restricted maximal operator of Fejér means in this subspace is bounded from the Hardy space Hp to the space Lp for all 0 < p ≤ 1/2. As applications, both some well- known and new results are pointed out.

This paper is organized as follows: In order not to disturb our discussions later on some preliminaries (definitions, notations and lemmas) are presented in Section 2. The main result (Theorem 1) and some of its consequences can be found in Section 3. The detailed proof of Theorem 1 is given in Section 4.

2. Preliminaries

Denote byN₊ the set of the positive integers, N:= N₊∪ {0}. Let m :=

(m₀, m₁, . . .) be a sequence of the positive integers not less than 2. Denote by Z_mn := {0,1, . . . , m_n−1} the additive group of integers modulo m_n. Define the groupG_m as the complete direct product of the groupsZ_mn with the product of the discrete topologies of Zmn‘s. In this paper we discuss bounded Vilenkin groups, i.e. the case when sup_n∈Nm_n<∞.

The direct productµof the measures µ_n({j}) := 1/m_n, (j ∈Z_mn)is the Haar measure onG_m withµ(G_m) = 1.

The elements of G_m are represented by sequences x:= (x₀, x₁, . . . , x_n, . . .), (x_n∈Z_mn). It is easy to give a base for the neighbourhood of G_m :

I₀(x) :=G_m, I_n(x) :={y ∈G_m |y₀=x₀, . . . , y_n−1 =x_n−1} (x∈G_m, n∈N). SetIn:=In(0),for n∈N₊ and

en:= (0, . . . ,0, xn= 1,0, . . .)∈Gm (n∈N). Denote

I_N^k,l:=

I_N(0, . . . ,0, x_k6= 0,0, . . . ,0, x_l 6= 0, x_l+1,...,x_N−1 ), k < l < N, I_N(0, . . . ,0, x_k6= 0,0, . . . ,0), l=N.

It is easy to show that

(2) I_N =





N−2[

i=0 N−1[

j=i+1

I_N^i,j



[ ^N−1[

i=0

I_N^i,N

!

, n= 2,3, ...

If we define the so-called generalized number system based on m in the following way :

M₀ := 1, M_n+1 :=m_nM_n (n∈N), then every n ∈ N can be uniquely expressed as n = P∞

k=0n_kM_k, where n_k∈Z_mk (k∈N₊) and only a finite number ofn_k‘s differ from zero. Let

hni:= min{j∈N:nj 6= 0} and |n|:= max{j ∈N:nj 6= 0}, that is M_|n|≤n≤M_|n|+1.Setρ(n) =|n| − hni, for all n∈N.

Next, we introduce on Gm an orthonormal system, which is called the Vilenkin system. At first, we define the complex-valued function r_k(x) : G_m→C,the generalized Rademacher functions, by

r_k(x) := exp (2πix_k/m_k), i²=−1, x∈G_m, k∈N .

Now, define the Vilenkin system ψ:= (ψn:n∈N) on Gm as:

ψn(x) :=

Y∞ k=0

rⁿ_k^k(x) (n∈N).

Specifically, we call this system the Walsh-Paley system, whenm≡2.

The norms (or quasi-norms) of the spaces L_p(G_m) and weak−L_p(G_m) (0< p <∞) are respectively defined by

kfk^p_p :=

Z

Gm

|f|^pdµ, kfk^p_Lp,

∞ := sup

λ>0

λ^pµ(f > λ)<∞.

The Vilenkin system is orthonormal and complete inL₂(G_m) (see [20]).

If f ∈L₁(G_m) we can define Fourier coefficients, partial sums, Dirichlet kernels, Fejér means, Fejér kernels with respect to the Vilenkin system in the usual manner:

fb(k) :=

Z

Gm

f ψ_kdµ ( k∈N),

Snf : =

n−1X

k=0

fb(k)ψ_k, Dn:=

n−1X

k=0

ψ_k ( n∈N₊ ),

σ_nf : = 1 n

n−1X

k=0

S_kf, K_n:= 1 n

n−1X

k=0

D_k ( n∈N₊ ).

Recall that (see e.g. [1])

(3) D_Mn(x) =

M_n, if x∈I_n, 0, if x /∈I_n, and

(4) D_snMn =D_snMn

sXn−1 k=0

ψ_kMn =D_Mn

sXn−1 k=0

r_n^k, where n∈Nand 1≤s_n≤m_n−1.

Theσ-algebra generated by the intervals{I_n(x) :x∈G_m}will be denoted by ̥_n (n∈N). Denote by f = f⁽ⁿ⁾, n∈N

a martingale with respect to

̥_n(n∈N) (for details see e.g. [21]). The maximal function of a martingale f is defined by

f^∗= sup

n∈N

f⁽ⁿ⁾.

In the case f ∈L₁(G_m),the maximal functions are just also given by f^∗(x) = sup

n∈N

1

|I_n(x)|

Z

In(x)

f(u)µ(u) .

For0 < p <∞ the Hardy martingale spaces H_p(G_m) consist of all mar- tingales f, for which

kfk_Hp :=kf^∗k_p <∞.

Iff ∈L1(Gm),then it is easy to show that the sequence(SMn(f) :n∈N) is a martingale. If f = f⁽ⁿ⁾, n∈N

is a martingale, then the Vilenkin- Fourier coefficients must be defined in a slightly different manner:

fb(i) := lim

k→∞

Z

Gm

f^(k)(x)ψ_i(x)dµ(x).

The Vilenkin-Fourier coefficients off ∈L₁(G_m) are the same as those of the martingale(S_Mnf :n∈N)obtained from f.

A bounded measurable functionais said to be a p-atom if there exists an interval I, such that

Z

I

adµ= 0, kak_∞≤µ(I)^−1/p, supp(a)⊂I.

For the proof of the main result (Theorem 1) we need the following Lem- mas:

Lemma 1(see e.g. [22]). A martingalef = f⁽ⁿ⁾, n∈N

is inH_p(0< p≤1) if and only if there exist a sequence (a_k, k∈N) of p-atoms and a sequence (µ_k, k∈N) of real numbers such that for every n∈N:

(5)

X∞ k=0

µ_kSMna_k=f⁽ⁿ⁾

and

X∞ k=0

|µ_k|^p <∞.

Moreover, kfk_Hp ∽inf (P∞

k=0|µ_k|^p)^1/p, where the infimum is taken over all decomposition of f of the form (5).

Lemma 2 (see e.g. [22]). Suppose that an operator T is σ-linear and for some0< p≤1

Z

−

I

|T a|^pdµ≤c_p <∞,

for everyp-atoma, where I denotes the support of the atom. IfT is bounded fromL∞ to L∞, then

kT fk_p ≤cpkfk_Hp.

Lemma 3 (see [7]). Let n > t, t, n∈N, x∈I_t\ I_t+1. Then K_Mn(x) =

0, if x−x_te_t∈/ I_n,

Mt

1−rt(x), if x−x_te_t∈I_n.

Lemma 4 (see [17]). Letx∈I_N^i,j, i= 0, . . . , N−1, j=i+ 1, . . . , N. Then Z

IN

|K_n(x−t)|dµ(t)≤ cM_iM_j

M_N² , for n≥M_N. Lemma 5 (see [11]). Let n∈N. Then

(6) |Kn(x)| ≤ c

n X|n|

l=hni

M_l|KM_l| ≤c X|n|

l=hni

|KM_l|

and

(7) |nK_n| ≥ M_hni²

2πλ, x∈I_hni+1 e_hni−1+e_hni , where λ:= supmn.

3. The Main Result and applications Our main result reads:

Theorem 1. a) Let 0 < p < 1/2, f ∈ H_p. Then there exists an absolute constant c_p, depending only on p, such that

kσn_kfk_Hp≤ c_pM_|n^1/p−2

k|

M_hn^1/p−2

ki

kfk_Hp.

b) (sharpness) Let 0< p <1/2 and Φ (n) be any nondecreasing function, such that

(8) sup

k∈N

ρ(n_k) =∞, lim

k→∞

M_|n^1/p−2

k|

M_hn^1/p−2

ki Φ (n_k) =∞.

Then there exists a martingale f ∈H_p,such that

sup

k∈N

σ_nkf Φ (n_k)

Lp,∞

=∞.

Corollary 1. Let 0< p <1/2, and f ∈ H_p. Then there exists an absolute constant c_p, depending only on p, such that

kσ_nkfk_Hp ≤c_pkfk_Hp, k∈N if and only if

sup

k∈N

ρ(n_k)< c <∞.

As an application we also obtain the previous mentioned result by Weisz [21], [22] (Theorem W).

Corollary 2. Let 0 < p < 1/2, f ∈ Hp. Then there exists an absolute constant c_p, depending only on p, such that

kσ_Mnfk_Hp ≤c_pkfk_Hp, n∈N. On the other hand, the following unexpected result is true:

Corollary 3. a) Let 0 < p < 1/2, f ∈ H_p. Then there exists an absolute constant c_p, depending only on p, such that

kσMn+1fk_Hp ≤cpM_n^1/p−2kfk_Hp, n∈N.

b) Let 0< p <1/2 and Φ (n) be any nondecreasing function, such that

k→∞lim

M_k^1/p−2 Φ (k) =∞.

Then there exists a martingale f ∈H_p,such that sup

k∈N

σ_Mk+1f Φ (k)

Lp,∞

=∞.

Remark 1. From Corollary 2 we obtain that σ_Mn are bounded from H_p to H_p, but from Corollary 3 we conclude that σ_Mn+1 are not bounded from H_p to Hp. The main reason is that Fourier coefficients of martingales f ∈Hp

are not uniformly bounded (for details see e.g. [18]).

In the next corollary we state some estimates for the Walsh system only to clearly see the difference of divergence rates for the various subsequences:

Corollary 4. a)Let 0 < p < 1/2, f ∈ H_p. Then there exists an absolute constant c_p, depending only on p, such that

(9) kσ₂ⁿ₊₁fk_Hp ≤c_p2^(1/p−2)nkfk_Hp, n∈N and

(10) kσ₂ⁿ₊₁fk_Hp ≤c_p2^(1/p2−2)n kfk_Hp, n∈N.

b) The rates 2^(1/p−2)n and2^(1/p−22)n in inequalities (9) and (10) are sharp in the same sense as in Theorem 1.

4. Proof of Theorem 1 Proof. a) Since

(11) sup

n∈N

Z

Gm

|K_n(x)|dµ(x)≤c <∞,

we obtain that

M_hn^1/p−2

ki |σ_nka(x)|

M_|n^1/p−2

k|

is bounded from L∞ to L∞. According to Lemma 2 we find that the proof of Theorem 1 will be complete, if we show that

Z

IN

M_hn^1/p−2

ki σn_ka(x) M_|n^1/p−2

k|

p

< c <∞,

for every p-atom a, with support I andµ(I) =M_N⁻¹.We may assume that I =I_N.It is easy to see that σ_nk(a) = 0 whenn_k≤M_N.Therefore, we can suppose thatn_k > MN.

Sincekak_∞≤M_N^1/p we find that M_hn^1/p−2

ki |σn_ka(x)|

M_|n^1/p−2

k|

≤ M_hn^1/p−2

ki

M_|n^1/p−2

k|

Z

IN

|a(t)| |K_nk(x−t)|dµ(t) (12)

≤ M_hn^1/p−2

ki kak_∞ M_|n^1/p−2

k|

Z

IN

|K_nk(x−t)|dµ(t)

≤ M_hn^1/p−2

ki M_N^1/p M_|n^1/p−2

k|

Z

IN

|K_nk(x−t)|dµ(t)

≤ M_hn^1/p−2

ki M_|n²

k|

Z

I_N

|K_nk(x−t)|dµ(t).

Without loss the generality we may assume that i < j. Let x ∈I_N^i,j and j <hn_ki.Thenx−t∈I_N^i,j for t∈I_N and, according to Lemma 3, we obtain that

|K_Ml(x−t)|= 0, for all hn_ki ≤ l≤ |n_k|.

By applying (12) and (6) in Lemma 5, for x∈I_N^i,j, 0≤i < j < hn_ki we get that

M_hn^1/p−2

ki |σ_nka(x)|

M_|n^1/p−2

k|

≤M_hn^1/p−2

ki M_|n²

k|

|nk|

X

l=hn_ki

Z

I_N

|K_Ml(x−t)|dµ(t) = 0.

(13)

Letx∈I_N^i,j,where hn_ki ≤j≤N.Then, in the view of Lemma 4, we have that

Z

I_N

|K_nk(x−t)|dµ(t)≤ cM_iM_j M_N² .

By using again (12) we find that M_hn^1/p−2

ki |σn_ka(x)|

M_|n^1/p−2

k|

≤ M_hn^1/p−2

ki M_N^1/p M_|n^1/p−2

k|

Z

IN

|K_nk(x−t)|dµ(t) (14)

≤ M_hn^1/p−2

ki M_N^1/p M_|n^1/p−2

k|

MiMj

M_N² ≤M_hn^1/p−2

ki M_iM_j.

By combining (2) and (12)-(14) we get that Z

IN

M_hn^1/p−2

ki |σ_nka(x)|

M_|n^1/p−2

k|

p

dµ

=

N−2X

i=0 N−1X

j=i+1

Z

I_N^i,j

M_hn^1/p−2

ki |σ_nka(x)|

M_|n^1/p−2

k|

p

dµ

+

N−1X

i=0

Z

I_N^k,N

M_hn^1/p−2

ki |σ_nka(x)|

M_|n^1/p−2

k|

p

dµ

≤

hnXki−1 i=0

N−1X

j=hnki

Z

I_N^i,j

M_hn^1/p−2

ki |σ_nka(x)|

M_|n^1/p−2

k|

p

dµ

+

N−2X

i=hn_ki NX−1 j=i+1

Z

I_N^i,j

M_hn^1/p−2

ki |σ_nka(x)|

M_|n^1/p−2

k|

p

dµ

+

NX−1 i=0

Z

I_N^i,N

M_hn^1/p−2

ki |σ_nka(x)|

M_|n^1/p−2

k|

p

dµ

≤

hnXki−1 i=0

N−1X

j=hn_ki

Z

I_N^i,j

M_hn^1/p−2

ki M_iM_j^pdµ+

NX−2 i=hn_ki

N−1X

j=i+1

Z

I_N^i,j

M_hn^1/p−2

ki M_iM_j^pdµ +

NX−1 i=0

Z

I_N^i,N

M_hn^1/p−2

ki M_iM_N^pdµ

≤ cpM_hn^1−2p

ki hnXki−1

i=0 NX−1 j=hnki

(M_iM_j)^p

M_j +cpM_hn^1−2p

ki N−2X

i=hnki NX−1 j=i+1

(M_iM_j)^p M_j

+c_pM_hn^1−2p

ki

X

i=0

(M_iM_N)^p M_N

≤ c_pM_hn^1−2p

ki hnXki

i=0

M_i^p

N−1X

j=hn_ki+1

1

M_j^1−p +M_hn^1−2p

ki NX−2 i=hn_ki

M_i^p

N−1X

j=i+1

1 M_j^1−p +c_p

N−1X

i=0

M_i^p M_N^p

≤ c_pM_hn^1−2p

ki M_hn^p

ki

1 M_hn^1−p

ki

+c_pM_hn^1−2p

ki N−2X

i=hnki

1

M_i^1−2p +c_p ≤c_p <∞.

The proof of the a) part is complete.

b) Let{n_k :k≥0}be a sequence of positive numbers, satisfying condition (8). Then

(15) sup

k∈N

M_|nk| M_hnki =∞.

Under condition (15) there exists a sequence{α_k: k≥0} ⊂ {n_k: k≥0}

such that α₀ ≥3 and (16)

X∞ k=0

M_hα^(1−2p)/2

ki Φ^p/2(α_k) M_|α^(1−2p)/2

k|

< c <∞.

Let

f⁽ⁿ⁾ = X

{k;|α_k|<n}

λ_ka_k,

where

λ_k= λM_hα^(1/p−2)/2

ki Φ^1/2(α_k) M_|α^(1/p−2)/2

k|

and

a_k= M_|α^1/p−1

k|

λ

DM|_αk|+1−DM|_αk|

. B applying Lemma 1 we can conclude that f ∈H_p. It is evident that

(17) fb(j) =











 M_|α^1/2p

k|M_hα^(1/p−2)/2

ki Φ^1/2(α_k), if j∈

M_|αk|, ..., M_|αk|+1−1 , k = 0,1,2..., 0 ,

if j /∈ S^∞

k=0

M_|αk|, ..., M_|αk|+1−1 .

Moreover, σ_αkf

Φ (α_k) = 1 α_kΦ (α_k)

MX|_αk| j=1

S_jf + 1 α_kΦ (α_k)

αk

X

j=M|αk|+1

S_jf :=I+II.

LetM_|αk|< j ≤α_k.Then, by applying (17) we get that (18) S_jf =S_M

|αk|f +M_|α^1/2p

k|M_hα^(1/p−2)/2

ki Φ^1/2(α_k)

D_j−D_M

|αk|

.

By using (18) we can rewriteII as II = α_k−M_|αk|

α_kΦ (α_k) SM|_αk|f+ M_|α^1/2p

k|M_hα^(1/p−2)/2

ki

α_kΦ^1/2(α_k)

αk

X

j=M|αk|

Dj−DM|_αk|

:= II1+II2.

Since (for details see e.g. [5] and [19])

S_M

|_αk|f

weak−Lp

≤c_pkfk_Hp

we obtain that

kII₁k^p_weak−Lp ≤

α_k−M_|αk| α_kΦ (α_k)

pS_M

|αk|f^p_weak−L

p

≤ S_M

|αk|f^p

weak−Lp

≤c_pkfk^p_Hp <∞.

By using part a) of Theorem 1 we find that kIk^p_weak−Lp =

M_|αk| α_kΦ (α_k)

pσM|_αk|f^p_weak−L

p

≤cpkfk^p_Hp <∞.

Letx∈I^{hαki−1,hαki}

hαki⁺¹ .Under condition (8) we can conclude thathα_ki 6=|α_k| and

α_k−M_|αk|

=hα_ki.Since

(19) D_j+Mn =D_Mn+ψ_MnD_j =D_Mn+r_nD_j, when j < M_n if we apply estimate (7) in Lemma 5 forII₂ we obtain that

|II₂| = M_|α^1/2p

k| M_hα^(1/p−2)/2

ki

α_kΦ^1/2(α_k)

αk−MX|αk| j=1

D_j+M

|αk|−D_M

|αk|

= M_|α^1/2p

k|M_hα^(1/p−2)/2

ki

α_kΦ^1/2(α_k) ψM|_αk|

α_k−MX|αk| j=1

Dj

≥ c_pM_|α^1/2p−1

k| M_hα^(1/p−2)/2

ki

Φ^1/2(α_k) α_k−M_|αk| K_αk−M

|αk|

≥ c_pM_|α^1/2p−1

k| M_hα^(1/p+2)/2

ki

Φ^1/2(α_k) . It follows that

kII₂k^p_weak−Lp

≥ c_p



M_|α^(1/p−2)/2

k| M_hα^(1/p+2)/2

ki

Φ^1/2(α_k)





p

µn

x∈G_m:|IV₂| ≥c_pM_|α^(1/p−2)/2

k| M_hα^(1/p+2)/2

ki

o

≥ c_p

M_|α^1/2−p

k| M_hα^1/2+p

ki µn

I^{hαki−1,hαki}

hαki⁺¹

o

Φ^p/2(α_k) ≥ c_pM_|α^1/2−p

k|

M_hα^1/2−p

ki Φ^p/2(α_k).

Hence, for large k,

kσ_αkfk^p_weak−Lp

≥ kII2k^p_weak−Lp− kII1k^p_weak−Lp− kIk^p_weak−Lp

≥ 1

2kII₂k^p_weak−L

p ≥ c_pM_|α^1/2−p

k|

2M_hα^1/2−p

ki Φ^p/2(α_k) → ∞, ask→ ∞.

The proof is complete.

Acknowledgment: We thank the careful referee for some good sugges- tions which have improved the final version of this paper.

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L-E. Persson, Department of Computer Science and Computational En- gineering, UiT -The Arctic University of Norway, P.O. Box 385, N-8505, Narvik, Norway and Department of Mathematics and Computer Science, Karl- stad University, Sweden

E-mail address: [email protected] [email protected]

G. Tephnadze, The University of Georgia, School of Science and Tech- nology, 77a Merab Kostava St, Tbilisi, 0128, Georgia.

E-mail address: [email protected]

G.Tutberidze, The University of Georgia, School of science and technol- ogy, 77a Merab Kostava St, Tbilisi 0128, Georgia and Department of Com- puter Science and Computational Engineering, UiT -The Arctic University of Norway, P.O. Box 385, N-8505, Narvik, Norway.

E-mail address: [email protected]