Two-sided estimates of the Lebesgue constants with respect to Vilenkin systems and applications

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Two-sided estimates of the Lebesgue constants with respect to Vilenkin systems and applications

Journal: Glasgow Mathematical Journal Manuscript ID GMJ-15-0240

Manuscript Type: Original Research Article Date Submitted by the Author: 28-Nov-2015

Complete List of Authors: Blahota, István; College of Nyíregyháza

Persson, Lars-Erik; Luleå University of Technology; Narvik University College

Tephnadze, Giorgi; Tbilisi State University; Luleå University of Technology Speciality Areas: Vilenkin system, partial sums, Lebesgue constant

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TWO-SIDED ESTIMATES OF THE LEBESGUE CONSTANTS WITH RESPECT TO VILENKIN SYSTEMS

AND APPLICATIONS

I. BLAHOTA, L.E. PERSSON, G. TEPHNADZE

Abstract. In this paper we derive two-sided estimates of the Lebesgue constants for bounded Vilenkin systems, we also present some applica- tions of importance e.g. we obtain a characterization for the bounded- ness of a subsequence of partial sums with respect to Vilenkin-Fourier series ofH₁martingales in terms ofn^,s variation. The conditions given in this paper are in a sense necessary and sufficient.

2000 Mathematics Subject Classification. 42C10.

Key words and phrases: Vilenkin system, partial sums, Lebesgue con- stant, two-sided estimates, n^,s variation, modulus of continuity, martingale Hardy space.

1. Introduction It is known that for every Vilenkin systems

L_n:=kD_nk₁ ≤clogn

holds. For the definitions of Dn, the Vilenkin systems and other objects in this Section (e.g. v(n) and v^∗(n)) we refer to our Section 2.

For some concrete systems it is possible to write two-sided estimations of Lebesgue constants L_nk. In particular, for every bounded Vilenkin sys- tems Lukyanenko [4] proved two-sided estimates for the Lebesgue constants Lnk for some concrete indices n_k ∈N.Lukomskii [3] generalized this result and proved two-sided estimates for the Lebesgue constants L_n without the conditions on the indexes. He showed that for n = P∞

j=0njMj and ev- ery bounded Vilenkin systems we have the following two-sided estimates of Lebesgue constants:

(1) 1

4λv(n) + 1

λv^∗(n) + 1

2λ≤L_n≤ 3

2v(n) + 4v^∗(n)−1.

It is well-known that (see e.g. [1] and [2]) Vilenkin systems do not form bases in the spaceL₁.Moreover, there exists a function in the dyadic Hardy space H1, such that the partial sums of f are not bounded in L1-norm.

Supported by TÁMOP 4.2.2.A-11/1/KONV-2012-0051 and by Shota Rustaveli Na- tional Science Foundation grant no. 52/54 (Bounded operators on the martingale Hardy spaces).

1

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Onneweer [6] showed that if the modulus of continuity off ∈L₁[0,1)satisfies the condition

(2) ω1(δ, f) =o

1 log (1/δ)

, asδ →0,

then its Vilenkin-Fourier series converges in L₁-norm. He also proved that condition (2) can not be improved.

In [8] (see also [9]) it was proved that if f ∈H₁ and

(3) ω_H1

1 Mn

, f

=o 1

n

, asn→ ∞,

thenS_kf converge to f in L₁-norm. Moreover, there was showed that con- dition (3) can not be improved.

It is also known that any subsequence S_nk is bounded from L₁ to L₁ if and only if n_k has uniformly bounded variation and as a corollary the subsequence S₂ⁿ of partial sums is bounded from Hardy space H_p to the Hardy spaceH_p,for allp >0.

In this paper we improve the upper bound in (1) and also prove a new similar lower bound by using a completely different new method. By apply- ing this results we also find the characterizations of boundedness (or even the ratio of divergence of the norm) of the subsequence of partial sums of the Vilenkin-Fourier series of H₁ martingales in terms of n^,s variation. We also derive a relationship of the ratio of convergence of the partial sum of the Vilenkin series with the modulus of continuity of a martingale. The conditions given in the paper are in a sense necessary and sufficient.

Our main results (Theorem 1) is presented and proved in Section 3. The mentioned applications especially Theorems 2 and 3 can be in Section 4.

Section 2 is reserved for necessary definitions, notations and some Lemmas (Lemmas 2 and 3 are new).

2. Preliminaries

LetN₊ denote the set of the positive integers,N:=N₊∪ {0}.

Letm:= (m₀, m₁, . . .)denote a sequence of the positive numbers not less than 2.

Denote by

Z_mk :={0,1, . . . , m_k−1}

the additive group of integers modulo m_k, k∈N.

Define the group G_m as the complete direct product of the group Z_mk with the product of the discrete topologies ofZ_mk‘s.

The direct product µof the measures

µ_k({j}) := 1/m_k,(j∈Z_mk) is the Haar measure on Gm,withµ(Gm) = 1.

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In this paper we discuss bounded Vilenkin groups only, that is sup

n∈N

mn<∞.

The elements ofG_m are represented by sequences x:= (x₀, x₁, . . . , x_k, . . .), (x_k∈Z_mk). It is easy to give a base for the neighbourhood ofGm :

I₀(x) :=G_m,

In(x) :={y∈Gm |y0=x0, . . . , yn−1 =xn−1},(x∈Gm, n∈N).

Denote In:=In(0),for n∈NandI⁻n:=Gm\In.

The norm (or quasi-norm) of the spacesL_p(G_m) is defined by kfk_p:=

Z

Gm

|f|^pdµ 1/p

(0< p <∞).

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

M₀:= 1, M_k+1 :=m_kM_k (k∈N), then every n∈ N can be uniquely expressed as n= P^∞

k=0

n_kM_k, where n_k ∈ Z_mk (k ∈ N) and only a finite number of n_k‘s differ from zero. Let |n| :=

max{k∈N: n_k 6= 0}.

For the natural numbern=P∞

j=0n_jM_j,we define

δ_j :=sign(n_j) =sign(⊖n_j), δ_j^∗:=|⊖n_j−1|δ_j, where⊖is the inverse operation for

a_k⊕b_k= (a_k+b_k) mod m_k. We define functionsv and v^∗ by

v(n) :=

X∞ j=0

|δ_j+1−δ_j|+δ₀, v^∗(n) :=

X∞ j=0

δ^∗_j,

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

r_k(x) := exp (2πıx_k/m_k), ı² =−1, x∈G_m, k∈N . Letx∈Z_mn.It is well-known that

(4)

mXn−1 k=0

r^k_n(x) =

0 x_n6= 0, m_n x_n= 0.

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Now, define the Vilenkin systemsψ:= (ψ_n:n∈N) on G_m as:

ψn(x) :=

Y∞ k=0

rⁿ_k^k(x), (n∈N).

Specifically, we call this system the Walsh-Paley one if m≡2.

The Vilenkin systems are orthonormal and complete in L₂(G_m) (see e.g.

[1, 10]).

Next we introduce analogues of the usual definitions in Fourier-analysis.

If f ∈ L1(Gm) we can establish the Fourier coefficients, the partial sums, the Dirichlet kernels, with respect to Vilenkin systems in the usual manner:

fb(n) :=

Z

Gm

f ψ_ndµ, (k∈N),

S_nf :=

n−1X

k=0

fb(k)ψ_k, (k∈N), and

D_n:=

n−1X

k=0

ψ_k, (k∈N). Letn∈N. Then

(5) D_Mn(x) =

n−1Y

k=0

mXk−1 s=0

r_k^s(x)

!

=

M_n x∈I_n, 0 x /∈In, and

(6) Dn=ψn



 X∞ j=0

DMj

mj−1

X

u=mj−nj

r^u_j



.

The σ-algebra generated by the intervals {In(x) :x∈Gm} is denoted by ̥_n(n∈N). Let f := f⁽ⁿ⁾, n∈N

be a martingale with respect to

̥_n(n∈N).(for details see e.g. [12]).

The maximal function of a martingalef is defined by f^∗ := sup

n∈N

f⁽ⁿ⁾.

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

n∈N

1

|I_n(x)|

Z

In(x)

f(u)µ(u)

For0< p <∞ the Hardy martingale spacesH_pconsist of all martingales for which

kfk_H

p :=kf^∗k_p<∞.

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The martingalef = f⁽ⁿ⁾, n∈N

is said to be L_p-bounded (0 < p≤ ∞) if f⁽ⁿ⁾ ∈L_p and

kfk_p := sup

n∈N

f⁽ⁿ⁾

p<∞.

Iff ∈L₁(G_m),then it is easy to show that the sequenceF = (S_Mnf :n∈N) is a martingale. This type of martingales is called regular. If 1 ≤ p ≤ ∞ and f ∈L_p(G_m)thenf = f⁽ⁿ⁾, n∈N

isL_p-bounded and

n→∞lim kS_Mnf−fk_p = 0,

consequently kFk_p = kfk_p,(see [5]). The converse of the latest statement holds also if 1 < p ≤ ∞ (see [5]): for an arbitrary L_p-bounded martingale f = f⁽ⁿ⁾, n∈N

there exists a functionf ∈L_p(G_m)for whichf⁽ⁿ⁾ =S_Mnf.

If p = 1,then there exists a function f ∈ L₁(G_m) of the preceding type if and only iff is uniformly integrable (see [5]) namely if

y→∞limsup

n∈N

Z

{|^f(n)|^>y}

f⁽ⁿ⁾(x)dµ(x) = 0.

Thus the map f → f := (S_Mnf :n∈N) is isometric from L_p onto the space of Lp-bounded martingales when1< p≤ ∞.Consequently, these two spaces can be identified with each other. Similarly, the space L₁(G_m) can be identified with the space of uniformly integrable martingales.

A bounded measurable function ais a p-atom if there exists an interval I such that Z

I

adµ= 0, kak_∞≤µ(I)^−1/p, supp(a)⊂I.

Iff = 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)ψ_idµ.

The best approximation off ∈L_p(G_m)(1≤p∈ ∞) is defined as E_n(f, L_p) := inf

ψ∈pn

kf−ψk_p,

wherep_n is the set of all Vilenkin polynomials of order less than n∈N. The integrated modulus of continuity off ∈L_p is defined by

ωp

1 M_n, f

:= sup

h∈In

kf(·+h)−f(·)k_p.

The concept of modulus of continuity inH_p (0< p≤1)can be defined in the following way:

ω_Hp 1

M_n, f

:=kf −S_Mnfk_Hp.

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Watari [11] showed that there are strong connections between ω_p

1 Mn

, f

, E_Mn(f, L_p) and

kf −S_Mnfk_p, p≥1, n∈N. In particular,

(7) 1

2ω_p 1

M_n, f

≤ kf −S_Mnfk_p≤ω_p 1

M_n, f

and 1

2kf−S_Mnfk_p ≤E_Mn(f, L_p)≤ kf−S_Mnfk_p.

The Hardy martingale spacesH_p (G_m)for 0< p≤1have atomic charac- terizations (see [12], [13]):

Lemma 1. A martingale f = f⁽ⁿ⁾, n∈N

∈H_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 everyn∈N,

(8)

X∞ k=0

µ_kS_Mna_k=f⁽ⁿ⁾, a.e.

X∞ k=0

|µ_k|^p <∞.

Moreover,

kfk_Hp ∽inf X∞ k=0

|µ_k|^p

!1/p

,

where the infimum is taken over all decomposition off of the form (8).

For the proof of main result we also need the following new Lemmas of independent interest:

Lemma 2. Let k, s∈N andx∈G_m. Then

sXk−1 u=1

r^u_k(x)

= cos (πs_kx_k/m_k) sin (π(s_k−1)x_k/m_k) sin (πx_k/m_k) ı +sin (πs_kx_k/m_k) sin (π(s_k−1)x_k/m_k)

sin (πx_k/m_k) .

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Proof. Since

sXk−1 u=1

r^u_k(x) =

sXk−1 u=1

cos

2πux_k m_k

+

sXk−1 u=1

ısin

2πux_k m_k

,

if we apply the following well-known identities (9)

Xn k=1

coskx= sin^nx₂ cos^(n+1)x₂ sin^x₂ and

(10)

Xn k=1

sinkx= sin^nx₂ sin^(n+1)x₂ sin^x₂ .

we immediately get the proof.

Lemma 3. Let k,N, 2≤s_k≤m_k and x_k= 1. Then

sXk−1 n=1

r_kⁿ(x)

= sin (π(s_k−1)x_k/m_k) sin (πx_k/m_k) ≥1.

Proof. Since

sin (π(m_k−1)/m_k)

sin (π/m_k) = sin (π/m_k) sin (π/m_k) = 1, if we take graph of sinxinto accout we obtain that

sin (π(s_k−1)/m_k)

sin (π/m_k) ≥1, for 2≤s_k≤m_k. Letx_k = 1.By using Lemma 2 we get that

(11)

sXk−1 u=1

r^u_k(x)

=

cos²(πs_kx_k/m_k) sin²(π(s_k−1)x_k/m_k) sin²(πx_k/m_k)

+ sin²(πs_kx_k/m_k) sin²(π(s_k−1)x_k/m_k) sin²(πx_k/m_k)

1/2

= sin (π(s_k−1)x_k/m_k)

sin (πx_k/m_k) = sin (π(s_k−1)/m_k) sin (π/m_k) ≥1.

The proof is complete.

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3. The main result Our main result reads:

Theorem 1. Let n=P∞

j=0n_jM_j. Then

(12) 1

4λv(n) + 1

λ²v^∗(n)≤L_n≤v(n) +v^∗(n), where λ:= sup_n∈Nm_n.

Proof. First we choose indices 0 ≤ ℓ₁ ≤ α₁ < ℓ₂ ≤ α₂ < ... < ℓ_s ≤ α_s <

ℓ_s+1=∞,such thatα_j+ 1< ℓ_j+1,forj= 1,2, ..., s, n_k= 0, for0< k < ℓ₁, n_k ∈ {1,2, ..., m_k−1}, for ℓ_j ≤k ≤ α_j and n_k = 0,for α_j < k < ℓ_j+1. According to (6) we have that

(13) Dn=ψn

X∞ k=0

DMk

mXk−1 u=1

r_k^u

!

−ψn

X∞ k=0

DMk

mkX−nk−1 u=1

r^u_k

!

=ψ_n



 Xs j=1

αj

X

k=ℓj

D_Mk

mXk−1 u=1

r^u_k



−ψ_n



 Xs j=1

αj

X

k=ℓj

D_Mk

⊖nXk−1 u=1

r_k^u



 :=I−II.

Since

(14) M_k−1 =

k−1X

j=0

(m_j−1)M_j if we apply again (6) we get that

D_Mk−1 =ψ_Mk−1





k−1X

j=0

D_Mj

mj−1

X

u=1

r^u_j



.

Hence,

(15) I =ψ_n



 Xs j=1





αj

X

k=0

D_Mk

mXk−1 u=1

r^u_k−

ℓj−1

X

k=0

D_Mk

mXk−1 u=1

r^u_k









=ψ_n



 Xs j=1

D_Mαj+1−1

ψ_Mαj+1−1 −D_Mℓj−1 ψ_Mℓj−1

!



=ψ_n



 Xs j=1

D_Mαj+1−ψ_Mαj+1−1 ψM_αj+1−1

−D_Mℓj −ψ_Mℓj−1 ψM_ℓj−1

!



=ψ_n



 Xs j=1

D_Mαj+1

ψ_Mαj+1−1 − DM_ℓj

ψ_Mℓj−1

!



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and

kIk₁ ≤ Xs j=1

D_Mαj+1

1+D_Mℓj

1

= 2s≤v(n).

Moreover,

kIIk₁≤ Xs j=1

αj

X

j=ℓj

|⊖n_j −1|δ_jD_Mj

1

= Xs j=1

αj

X

j=ℓj

|⊖n_j−1|δ_j ≤v^∗(n).

The proof of the upper estimate in 1 follows by combining the last two estimates.

Letx∈I_k+1(x_ke_k),where1≤x_k≤n_k−1ande_k:= (0, . . . ,0,1,0, . . .)∈ G_m, where only thek-th coordinate is one, the others are zero. Then, by the definition of Vilenkin functions, if we apply (14) and equalities x₀ = x₁ = ...=x_k−1= 0,we find that

(16) ψ_Ml−1(x) =

l−1Y

t=0

r^m_t ^t−1(x)

=

l−1Y

t=0

e^{(2πıxt(mt−1))/mt} =

l−1Y

t=0

e⁰ = 1, for any 0≤l≤k.

Let ℓ_j ≤k ≤α_j and x∈ I_k+1(x_ke_k), where1 ≤x_k ≤n_k−1. Then, in view of (5) and (15) we get that

I =−ψ_n(x) D_Mℓj (x) ψ_Mℓj−1(x) +ψ_n(x)

Xj−1 l=1

D_Mαl+1(x)

ψM_αl+1−1(x) − D_Mℓl (x) ψM_ℓl−1(x)

!!

=ψ_n(x) −M_ℓj+ Xj−1

l=1

(M_αl+1−M_ℓl)

! . By using Lemma 2 we have that

II=ψn(x) DMk(x)

mk−nXk−1 u=1

r^u_k(x)

!

+ψn(x)



 Xk−1 l=ℓj

DMl(x)

⊖nXl−1 u=1

r_l^u(x) + Xj−1 s=0

αs

X

l=ℓs

DMl(x)

⊖nXl−1 u=1

r_l^u(x)





=ψ_n(x)M_kcos (π(⊖n_k)x_k/m_k) sin (π(⊖n_k−1)x_k/m_k)

sin (πx_k/m_k) ı

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+ψ_n(x)M_ksin (π(⊖n_k)x_k/m_k) sin (π(⊖n_k−1)x_k/m_k) sin (πx_k/m_k)

+ψn(x)

k−1X

l=ℓj

M_l(⊖n_l−1) +ψn(x) Xj−1 s=0

αs

X

l=ℓs

M_l(⊖n_l−1). Letx∈I_k+1(e_k) andλ:= sup_n∈Nm_n. Sincex_k= 1 and

sin (π(⊖n_k)x_k/m_k) sin (π(⊖n_k−1)x_k/m_k)

sin (πx_k/m_k) ≥0

if we apply Lemma 3 we obtain that

|I−II| ≥

≥ M_kcos (π(⊖n_k)x_k/m_k) sin (π(⊖n_k−1)x_k/m_k) sin (πx_k/m_k)

2

+

M_ksin (π(⊖n_k)x_k/m_k) sin (π(⊖n_k−1)x_k/m_k) sin (πx_k/m_k)

2!1/2

≥ M_ksin (π(⊖n_k−1)x_k/m_k)

sin (πx_k/m_k) ≥M_k ≥ M_k|⊖n_k−1|

λ .

Let x ∈ Iαj+2 xαj+1eαj+1

, where 1 ≤ xαj+1 ≤ mαj+1 −1. Then, by using (6) if we invoke equalities (13), (15) and (16) we get that

|D_n|=

=

Xj k=1

D_Mαk+1

ψ_Mαk+1−1 − D_Mℓk ψ_Mℓk−1

!

−



 Xj k=1

αk

X

l=ℓk

D_Ml

ml−nXl−1 u=1

r^u_l





= Xj k=1



(M_αk+1−M_ℓk)−

αk

X

l=ℓk

|⊖n_l−1|M_l





≥ Xj k=1



(M_αk+1−M_ℓk)−

αk

X

l=ℓk

(m_l−2)M_l





= Xj k=1



(M_αk+1−M_ℓk)−

αk

X

l=ℓk

M_l+1+ 2

αk

X

l=ℓk

M_l





= Xj k=1

αk

X

l=ℓk

M_l≥Mαj.

Hence,

≥ Xs

l=0 αl

X

k=ℓl+1

Z

Ik+1(ek)

M_k|⊖n_k−1|

λ dµ

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+ Xs j=0

m_αj+1−1

X

x_αj+1=1

Z

I_αj+2

x_αj+1e_αj+1

M_αjdµ

≥ Xs l=0

αl

X

k=ℓl

M_k|⊖n_k−1|

λ

1 M_k+1 +

Xs j=0

mαj+1−1 Mαj

M_αj+2

≥ Xs

l=0 αl

X

k=ℓl

|⊖n_k−1|

λ² + Xs j=0

1 2λ ≥ 1

λ²v^∗(n) + 1 4λv(n).

The proof is complete.

The next result for Vilenkin systems is known (see e.g. [1]) but it also follows from our result.

Corollary 1. Let q_n=M_2n+M_2n−2+. . .+M₂. Then n

2λ ≤ kDqnk₁ ≤λn, where λ:= sup_n∈Nm_n.

Proof. First we observe that

(17) v(qn) = 2n.

By using Theorem 1 we get that kD_qnk₁ ≥ 1

4λv(q_n) = n 2λ. Moreover, since

v^∗(q_n) = Xn j=0

(m_2j −2)≤(λ−2) Xn j=0

1≤(λ−2)n if we apply (17) we readily obtain that

kD_qnk₁ ≤v^∗(q_n) +v(q_n)≤(λ−2)n+ 2n=λn.

The proof is complete.

Finally, we mention that the following well-known results for the Walsh systems (see the book [7]) also follows directly from our main result.

Corollary 2. For the Walsh system the inequality 1

8v(n)≤L_n≤v(n), holds.

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4. Applications

First we use our main result to find a characterizations for the boundedness (or even the ratio of divergence of the norm) of a subsequence of partial sums of the Vilenkin-Fourier series ofH₁ martingales.

Theorem 2. a)Let f ∈ H₁ and M_k < n ≤ M_k+1. Then there exists an absolute constant c such that

kS_nfk_H1 ≤c(v(n) +v^∗(n))kfk_H1.

b) Let{Φ_n:n∈N} be any non-decreasing and non-negative sequence sat- isfying condition

n→∞limΦ_n=∞ and {n_k ≥2 :k∈N} be a subsequence such that

k→∞lim

v(n_k) +v^∗(n_k) Φnk

=∞.

Then there exists a martingalef ∈H₁ such that sup

k

S_nkf Φ_nk

1

→ ∞, whenk→ ∞.

Proof. In view of Theorem 1 we can conclude that kS_nfk₁ ≤L(n)kfk₁≤L(n)kfk_H1

≤c(v(n) +v^∗(n))kfk_H1. Let us consider the following martingale:

f_#:= (S_MkS_nf, k≥1)

= (S_M0f, . . . , S_Mkf, . . . , S_nf, . . . , S_nf, . . .) .

It is easy to see that

kS_nfk_H1 ≤ kf_#k_H

1 ≤ sup

0≤l≤k

|S_Mlf|

1

+kS_nfk₁

≤ kfk_H1+kS_nfk₁

≤ kfk_H1+c(v(n) +v^∗(n))kfk_H1

≤c(v(n) +v^∗(n))kfk_H1.

b) Under the conditions of Theorem 2 there exists an increasing sequence {α_k:k∈N₊} ⊂ {n_k :k∈N₊} of the positive integers such that

(18)

X∞ k=1

1

Φ_|αk| <∞.

Let

f⁽ⁿ⁾:= X

{k:|αk|<n}

λ_ka_k,

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where

(19) λ_k= 1

Φ_|αk|, a_k=DM|αk|+1−DM|αk|.

By combining (18) and Lemma 1 we conclude that the martingale f ∈H1. It is easy to see that

(20) fb(j)

=







1

Φ|αk|, if j∈

M_|αk|, . . . , M_|αk|+1−1 , k ∈N 0, if j /∈ S^∞

k=0

M_|αk|, . . . , M_|αk|+1−1 . It follows that

Sαkf =

k−1X

i=1

D_M

|αi|⁺¹ −D_M

|αi|

Φ_|αi| +

D_αk−D_M

|αk|

Φ_|αk| .

Hence, if we invoke (18) for sufficiently largek we can conclude that

kS_αkfk₁≥ kDαkk₁ Φ_|αk| −

DM

|αk|

1

Φ_|αk| −

k−1X

i=1

DM

|αi|⁺¹ −DM

|αi|

1

Φ_|αi|

≥ kDαkk₁ Φ_|αk| −2

Xk i=1

1 Φ_|αi|

≥ c1(v(α_k) +v^∗(α_k))

Φ_|αk| −c2 → ∞,whenk→ ∞.

The proof is complete.

At first we prove the following estimation:

Corollary 3. Let f ∈ H1 and M_k < n ≤ M_k+1. Then there exists an absolute constant c such that

(21) kS_nf−fk_H1 ≤c(v(n) +v^∗(n))ω_H1 1

M_k, f

.

Proof of Theorem 3. By using Theorem 2 and obvious estimates we find that kS_nf −fk_H

1 ≤ kS_nf−S_Mkfk_H

1 +kS_Mkf−fk_H

1

=kS_n(S_Mkf −f)k_H

1+kS_Mkf−fk_H

1

≤(v(n) +v^∗(n) + 1)ω_H1 1

M_k, f

≤c(v(n) +v^∗(n))ω_H1 1

M_k, f

.

Thus, the proof is complete.

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Next we use Corollary 3 to derive necessary and sufficient conditions for the modulus of continuity of martingale Hardy spaces H_p, for which the partial sums of Vilenkin-Fourier series convergence in Lp-norm. We also point out the sharpness of this result.

Theorem 3. a) Letf ∈H1 and {n_k:k∈N} be a sequence of non-negative integers such that

ω_H1 1

M_|nk|, f

=o

1

v(n_k) +v^∗(n_k)

, ask→ ∞.

Then

kS_nkf −fk_H

1 →0, when k→ ∞.

b) Let {n_k:k≥1} be sequence of non-negative integers such that sup

k∈N

(v(nk) +v^∗(nk)) =∞.

Then there exists a martingalef ∈H1 and a sequence{α_k:k∈N} ⊂ {n_k :∈

N} for which

ω_H1 1

M_|αk|, f

=O

1

v(α_k) +v^∗(α_k)

and

(22) lim sup

k→∞

kS_αkf −fk₁ > c >0 whenk→ ∞.

Proof. The proof of part a) follows immediately from (21) in Corollary 3.

Under the conditions of part b) of Theorem 3, there exists a sequence {α_k:k∈N} ⊂ {n_k:k∈N} such that

(23) v(α_k) +v^∗(α_k)↑ ∞ whenk→ ∞ and

(24) (v(α_k) +v^∗(α_k))² ≤v(α_k+1) +v^∗(α_k+1). Let

f⁽ⁿ⁾ := X

{k:|αk|<n}

λ_ka_k,

where

λ_k = 1

v(α_k) +v^∗(α_k), a_k =DM|αk|+1−DM|αk|.

By combining (23), (24) and Lemma 1 we conclude that the martingale f ∈H₁.

It is easy to see that

(25) fb(j)

=





1

v(αk)+v^∗(αk), if j∈

M_|αk|, . . . , M_|αk|+1−1 , k ∈N, . . . 0, if j /∈ S^∞

k=0

M_|αk|, . . . , M_|αk|+1−1 .

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It follows that (26) S_αkf =

k−1X

i=1

D_M

|αi|+1 −D_M

|αi|

v(α_i) +v^∗(α_i) +

D_αk−D_M

|αk|

v(α_k) +v^∗(α_k). Since

SMnf =f⁽ⁿ⁾, for f =

f⁽ⁿ⁾:n∈N

∈Hp

and

S_Mkf⁽ⁿ⁾:k∈N

= (S_MkS_Mnf, k∈N)

= S_M0f, . . . , S_Mn−1f, S_Mnf, S_Mnf, . . .

=

f⁽⁰⁾, . . . , f⁽ⁿ⁻¹⁾, f⁽ⁿ⁾, f⁽ⁿ⁾, . . . we obtain that

f−S_Mnf =

f^(k)−S_Mkf :k∈N is a martingale for which

(27) (f−SMnf)^(k)=

0, k= 0, . . . . , n, f^(k)−f⁽ⁿ⁾, k≥n+ 1, According to Lemma 1 we get that

kf −SMnfk_H1

≤ X∞ i=n+1

1 v(α_i) +v^∗(α_i)

=O

1

v(αn) +v^∗(αn)

when n→ ∞.

By combining (5), (25) and (26) with Theorem 1 we obtain that kf−Sαkfk1

≥ k D_M

|αk|⁺¹−D_αk v(α_k) +v^∗(α_k) +

X∞ i=k+1

D_M

|αi|⁺¹−D_M

|αi|

v(α_i) +v^∗(α_i) k1

≥ kDαkk1

v(α_k) +v^∗(α_k) −

kD_M

|αk|⁺¹k₁ v(α_k) +v^∗(α_k) −

X∞ i=k+1

kD_M

|αi|⁺¹−D_M

|αi|k₁ v(α_i) +v^∗(α_i)

≥c− 1

v(α_k) +v^∗(α_k) −3 X∞ i=k+1

1 v(α_i) +v^∗(α_i)

≥c− 3

v(α_k) +v^∗(α_k). Hence,

lim sup

k→∞

kS_αkf −fk₁ > c >0ask→ ∞.

The proof is complete.

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This known results can be found in [8].

Corollary 4. Let f ∈H₁ and

ωH1

1 Mn, f

=o 1

n

, whenn→ ∞.

Then

kS_kf−fk_H1 →0, when k→ ∞.

b) Then there exists a martingale f ∈H₁ for which

ω_H1 1

M_n, f

=O 1

n

whenn→ ∞

and

kS_kf −fk₁ 90 whenk→ ∞.

References

[1] G. N. Agaev, N. Ya. Vilenkin, G. M. Dzhafarly and A. I. Rubinstein,Multiplicative systems of functions and harmonic analysis on zero-dimensional groups, Baku, Ehim, 1981 (in Russian).

[2] B. I. Golubov, A. V. Efimov and V. A. Skvortsov, Walsh series and transforms.

(Russian) Nauka, Moscow, 1987, English transl. in Mathematics and its Applications (Soviet Series), 64. Kluwer Academic Publishers Group, Dordrecht, 1991.

[3] S. F. Lukomskii,Lebesgue constants for characters of the compact zero-dimensional Abelian group, East. J. Appr., 15 (2010), no 2, 219-231.

[4] O. A. Lukyanenko, On Lebesgue constants for Vilenkin system, "Mathematics. Me- chanics", Saratov State University, (2005), no. 7, 70-73 (in Russian).

[5] J. Neveu,Discrete-parameter martingales, North-Holland Mathematical Library, Vol.

10. North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York, 1975.

[6] C.W. Onneweer,OnL-convergence of Walsh-Fourier series, Internat. J. Math. Sci. 1 (1978), 47-56.

[7] F. Schipp, W.R. Wade, P. Simon and J. Pál,Walsh series, An Introduction to Dyadic Harmonic Analysis, Akadémiai Kiadó, (Budapest-Adam-Hilger (Bristol-New-York)), 1990.

[8] G. Tephnadze,On the partial sums of Vilenkin-Fourier series, Izv. Nats. Akad. Nauk Armenii Mat. 49 (2014), no. 1, 60–72; translation in J. Contemp. Math. Anal. 49 (2014), no. 1, 23-32.

[9] G. Tephnadze, Martingale Hardy Spaces and Summability of the One Dimensional Vilenkin-Fourier Series, PhD thesis, Department of Engineering Sciences and Math- ematics, Luleå University of Technology, Oct. 2015 (ISSN 1402-1544).

[10] N. Ya. Vilenkin,A class of complete orthonormal systems, Izv. Akad. Nauk. U.S.S.R., Ser. Mat., 11 (1947), 363-400.

[11] C. Watari, Best approximation by Walsh polynomials, Tohoku Math. J., 15 (1963), 1-5.

[12] F. Weisz, Martingale Hardy spaces and their applications in Fourier Analysis, Springer, Berlin-Heidelberg-New York, 1994.

[13] F. Weisz, Hardy spaces and Cesàro means of two-dimensional Fourier series, Bolyai Soc. Math. Studies, (1996), 353-367.

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I. Blahota, Institute of Mathematics and Computer Sciences, College of Nyíregyháza, P.O. Box 166, Nyíregyháza, H-4400, Hungary.

E-mail address: blahota@nyf.hu

L.E. Persson, Department of Engineering Sciences and Mathematics, Luleå University of Technology, SE-971 87 Luleå, Sweden and Narvik University College, P.O. Box 385, N-8505, Narvik, Norway.

E-mail address: larserik@ltu.se

G. Tephnadze, Department of Mathematics, Faculty of Exact and Natural Sciences, Tbilisi State University, Chavchavadze str. 1, Tbilisi 0128, Georgia and Department of Engineering Sciences and Mathematics, LuleåUniversity of Technology, SE-971 87 Luleå, Sweden.

E-mail address: giorgitephnadze@gmail.com