A study of bounded operators on martingale Hardy spaces

Faculty of Engineering Science and Technology

Department of Computer Science and Computational Engineering

A study of bounded operators on martingale Hardy spaces

Giorgi Tutberidze

A dissertation for the degree of Philosophiae Doctor December 2021

A study of bounded operators on martingale Hardy spaces

A dissertation for the degree of Philosophiae Doctor

Faculty of Engineering Science and Technology

Department of Computer Science and Computational Engineering

December 2021

The classical Fourier Analysis has been developed in an almost unbelievable way from the first fundamental discoveries by Fourier. Especially a number of wonderful results have been proved and new directions of such research has been developed e.g. concerning Wavelets Theory, Gabor Theory, Time- Frequency Analysis, Fast Fourier Transform, Abstract Harmonic Analysis, etc.

One important reason for this is that this development is not only important for improving the "State of the art", but also for its importance in other areas of mathematics and also for several applications (e.g. theory of signal transmission, multiplexing, filtering, image enhancement, coding theory, digital signal processing and pattern recogni-tion).

The classical theory of Fourier series deals with decomposition of a function into sinusoidal waves. Unlike these continuous waves the Vilenkin (Walsh) func- tions are rectangular waves. The development of the theory of Vilenkin-Fourier series has been strongly influenced by the classical theory of trigonometric se- ries. Because of this it is inevitable to compare results of Vilenkin series to those on trigonometric series. There are many similarities between these theories, but there exist differences also. Much of these can be explained by modern abstract harmonic analysis, which studies orthonormal systems from the point of view of the structure of a topological group.

The aim of my thesis is to discuss, develop and apply the newest develop- ments of this fascinating theory connected to modern harmonic analysis. In particular, we investigate some strong convergence result of partial sums of Vilenkin-Fourier series. Moreover, we derive necessary and sufficient condi- tions for the modulus of continuity so that norm convergence of subsequences of Fejér means is valid. Furthermore, we consider Riesz and Nörlund logarith- mic means. It is also proved that these results are the best possible in a spe- cial sense. As applications both some well-known and new results are pointed out. In addition, we investigate someTmeans, which are "inverse" summability methods of Nörlund, but only in the case when their coefficients are monotone.

The main body of the PhD thesis consists of seven papers (Papers A – G). We now continue by describing the main content of each of the papers.

In Paper A we investigate some new strong convergence theorems for partial sums with respect to Vilenkin system.

In Paper B 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.We also proved that this result is in a sense sharp.

In Paper C we find necessary and sufficient condition for the modulus of continuity for which subsequences of Fejér means with respect to Vilenkin systems are bounded from the Hardy spaceHpto the Lebesgue spaceLp,for

all0< p <1/2.

In Paper D we prove and discuss some new(Hp, weak−Lp)type inequal- ities of maximal operators of T means with respect to Vilenkin systems with monotone coefficients. We also apply these results to prove a.e. convergence of suchT means. It is also proved that these results are the best possible in a special sense. As applications, both some well-known and new results are pointed out.

In Paper E we prove and discuss some new (H_p, L_p) type inequalities of weighted maximal operators ofT means with respect to the Vilenkin systems with monotone coefficients. We also show that these inequalities are the best possible in a special sense. Moreover, we apply these inequalities to prove strong convergence theorems of such T means. We also show that these results are the best possible in a special sense. As applications, both some well-known and new results are pointed out.

In Paper F we derive a new strong convergence theorem of Riesz logarithmic means of the one-dimensional Vilenkin-Fourier (Walsh-Fourier) series. The corresponding inequality is pointed out and it is also proved that the inequality is in a sense sharp, at least for the case with Walsh-Fourier series.

In Paper G we investigate(Hp, Lp)- type inequalities for weighted maximal operators of Nörlund logaritmic means, for 0 < p < 1. Moreover, we apply these inequalities to prove strong convergence theorems of such Nörlund logaritmic means.

These new results are put into a more general frame in an Introduction, where, in particular, a comparison with some new international research and broad view of such interplay between applied mathematics and engineering problems is presented and discussed.

This PhD thesis is composed of seven papers [A] – [G] and a matching Introduc- tion. In the Introduction the papers [A] – [G] are discussed and put into a more general frame. The Introduction is also of independent interest since it contains a brief discussion on the important definitions and notations in the theory of Fourier analysis and martingale Hardy spaces.

A very brief presentation of the main content of the seven papers can be found in the Abstract above and in a more general context at the end of the Introduction.

List of Papers

Paper A: G. Tutberidze. “A note on the strong convergence of partial sums with respect to Vilenkin system”.

J. Contemp. Math. Anal., 54 (2019), no.6, 319–324.

Paper B: L. E. Persson, G. Tephnadze, G. Tutberidze, “On the boundedness of subsequences of Vilenkin-Fejér means on the martingale Hardy spaces”.

Operators and Matrices, 14 (2020), no.1, 283–294.

Paper C: G. Tutberidze. “Modulus of continuity and boundedness of subse- quences of Vilenkin- Fejér means in the martingale Hardy spaces”. 13 pagesSubmitted for publication.

Paper D: G. Tutberidze. “Maximal operators ofT means with respect to the Vilenkin system”.Nonlinear Studies, 27 (2020), no.4, 1–11.

Paper E: G. Tutberidze. “Sharp(Hp, Lp)type inequalities of maximal opera- tors ofTmeans with respect to Vilenkin systems with monotone co- efficients”.Submitted for publication.

Paper F: D. Lukkassen, L.E. Persson, G. Tephnadze, G. Tutberidze. “Some inequalities related to strong convergence of Riesz logarithmic means of Vilenkin-Fourier series”. J. Inequal. Appl., 2020, DOI:

https://doi.org/10.1186/s13660-020-02342-8(17 pages).

Paper G: G. Tephnadze, G. Tutberidze. “A note on the maximal operators of the Nörlund logaritmic means of Vilenkin-Fourier series”. Trans. A.

Razmadze Math. Inst., 174, (2020), no.1, 107–112.

It is a pleasure to express my warmest thanks to my supervisors Professors Lars-Erik Persson, Dag Lukkassen and George Tephnadze for the attention to my work, their valuable remarks and suggestions and for their constant support and help.

I am also grateful to Professor Natasha Samko for helping me with several practical and professional things. I am also indepted to PhD student Harpal Singh for supporting me in various important complementary ways. Moreover, I appreciate very much the warm and friendly atmosphere at the Department of Mathematics at UiT The Arctic University of Norway, which helped me to do my research more effectively.

I thank my Georgian colleagues from The University of Georgia for their interest in my research and for many fruitful discussions.

I am also grateful to Professor Vakhtang Tsagareishvili from Tbilisi State University, for his support.

I also want to pronounce that the agreement about scientific collaboration and PhD education between The University of Georgia and UiT The Arctic University of Norway has been very important. Especially, I express my deepest gratitude to Professor Roland Duduchava at The University of Georgia for his creative and hard work to realize this important agreement.

I thank Shota Rustaveli National Science Foundation for financial support, which was very important to do my research in the frame of this project.

Finally, I thank my family for their love, understanding, patience and long lasting constant support.

Introduction

1.1 Preliminaries

1.1.1 Vilenkin groups and functions

Denote byN⁺the set of positive integers,N:=N⁺∪ {0}.Letm:= (m₀, m₁, . . .) be a sequence of positive integers not less than 2. Denote by

Zm_k :={0,1, . . . , mk−1}

the additive group of integers modulomk.

Define the groupGmas the complete direct product of the groupsZm_kwith the product of the discrete topologies ofZm_k.

The direct productµof the measures

µ_k(j) := 1/m_k (j ∈Z_mk) is the Haar measure onGmwithµ(Gm) = 1.

If sup_n∈Nmn < ∞, then we call Gm a bounded Vilenkin group. If the generating sequencemis not bounded, thenGmis said to be an unbounded Vilenkin group.

In this PhD thesis we discuss bounded Vilenkin groups, i.e. the case when sup_n∈Nm_n <∞.

The elements ofG_mare represented by sequences x:= (x0, x1, . . . , xj, . . .) xj∈Zm_j

.

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

M₀:= 1, M_k+1:=m_kM_k (k∈N), then everyn∈Ncan be uniquely expressed as

n=

∞

X

j=0

n_jM_j,

wherenj ∈Zm_j (j∈N⁺)and only a finite number ofn⁰_js differ from zero.

Vilenkin group can be metrizable with the following metric:

ρ(x, y) :=|x−y|:=

∞

X

k=0

|xk−yk| Mk+1

, (x∈Gm).

It is easy to give a base for the neighborhoods ofGm: I₀(x) : =G_m,

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

e_n:= (0, . . . ,0, x_n = 1,0, . . .)∈G_m (n∈N). If we defineI_n:=I_n(0),forn∈NandI_n:=G_m \I_n,then

I_N =

N−1

[

s=0

I_s\Is+1=

N−2

[

k=0 N−1

[

l=k+1

I_N^k,l

! [

N−1

[

k=1

I_N^k,N

! ,

where

I_N^k,l:=











IN(0, . . . ,0, xk6= 0,0, ...,0, xl6= 0, xl+1, . . . , xN−1, . . .), for k < l < N,

IN(0, . . . ,0, xk6= 0, xk+1= 0, . . . , xN−1= 0, xN, . . .), for l=N.

The norm (or quasi-norm when0< p <1) of the Lebesgue spaceL_p(G_m) (0< p <∞)is defined by

kfk_p:=

Z

G_m

|f|^pdµ 1/p

.

The spaceweak−Lp(Gm)consists of all measurable functionsf, for which kfk_weak−L

p:= sup

λ>0

λ{µ(f > λ)}^1/p<+∞.

The norm of the space of continuous functionsC(Gm)is defined by kfk_C := sup

x∈Gm

|f(x)|< c <∞.

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

ψ∈Pn

kf−ψk_p,

wherePnis set of all Vilenkin polynomials of order less thann∈N.

The modulus of continuity of functions in Lebesgue spacesf ∈L_p(G_m)and continuous functionsf ∈C(G_m)are defined by

ωp

1 Mn

, f

:= sup

h∈In

kf(· −h)−f(·)k_p and

ωC

1 Mn

, f

:= sup

h∈In

kf(· −h)−f(·)k_C,

respectively.

Next, we introduce onGmorthonormal systems, which are called Vilenkin systems.

At first, we define the complex-valued function rk(x) : Gm → C, the generalized Rademacher functions, by

rk(x) := exp (2πixk/mk), i²=−1, x∈Gm, k∈N . Now, define Vilenkin systemsψ:= (ψ_n:n∈N)onG_mas:

ψ_n(x) :=

∞

Y

k=0

rⁿ_k^k(x), (n∈N).

The Vilenkin systems are orthonormal and complete inL₂(G_m)(for details see e.g. [1] , [61] and [108]).

It is well-known that for alln∈N,

|ψ_n(x)| = 1,

ψn(x+y) = ψn(x) ψn(y), ψn(−x) = ψn^∗(x) =ψ_n(x), ψ_n(x−y) = ψ_n(x) ψ_n(y),

ψ_n

b+k(x) = ψsψn(x), (s, n∈N, x, y∈Gm). Specifically, we call this system the Walsh-Paley system whenm= 2.

1.1.2 Partial sums and Fejér means with respect to the Vilenkin systems

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

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

fb(n) :=

Z

G_m

f ψ_ndµ, (n∈N), Snf :=

n−1

X

k=0

fb(k)ψk, (n∈N⁺), σnf : = 1

n

n−1

X

k=0

Skf, (n∈N⁺),

D_n :=

n−1

X

k=0

ψ_k, (n∈N⁺),

Kn : = 1 n

n−1

X

k=0

Dk, (n∈N⁺).

respectively.

It is easy to see that

S_nf(x) = Z

G_m

f(t)

n−1

X

k=0

ψ_k(x−t)dµ(t)

= Z

Gm

f(t)Dn(x−t)dµ(t)

= (f ∗D_n) (x).

It is well-known that (for details see e.g. [1] , [61] and [108]) that for anyn∈N and1≤sn≤mn−1the following equalities holds:

Dj+M_n=DM_n+ψM_nDj =DM_n+rnDj, j ≤(mn−1)Mn,

DM_n−j(x) = DM_n(x)−ψ_Mn−1(−x)Dj(−x)

= D_Mn(x)−ψ_Mn−1(x)D_j(x), j < M_n.

DM_n(x) =

Mn x∈In

0 x /∈In (1.1)

Ds_nM_n=DM_n s_n−1

X

k=0

ψkM_n=DM_n s_n−1

X

k=0

r^k_n (1.2)

and

D_n=ψ_n





∞

X

j=0

D_Mj

m_j−1

X

k=m_j−nj

r^k_j



.

By using (1.1) we immediately get that kDM_nk1= 1<∞.

It is obvious that

σnf(x) = 1 n

n−1

X

k=0

(Dk∗f) (x)

= Z

Gm

f(t)Kn(x−t)dµ(t)

= (f∗Kn) (x), whereKnare the so called Fejér kernels.

It is well-known that (for details see e.g. [42]) for everyn > t, t, n ∈ Nwe have the following equality:

KM_n(x) =







M_t

1−rt(x), x∈I_t\It+1, x−x_te_t∈I_n,

M_n+1

2 , x∈I_n, 0, otherwise.

Moreover,

snMnKs_nM_n=

s_n−1

X

l=0 l−1

X

i=0

rⁱ_n

!

MnDM_n+

s_n−1

X

l=0

r^l_n

!

MnKM_n.

The next equality of Fejér kernels is very important for our further inves- tigations (for details see Blahota and Tephnadze [26]). In particular, ifn = Pr

i=1sn_iMn_i, wheren1 > n2 > · · · > nr ≥ 0and1 ≤ sn_i < mn_i for all 1≤i≤ras well asn^(k)=n−Pk

i=1sn_iMn_i, where0< k≤r, then nKn=

r

X

k=1





k−1

Y

j=1

rn^snjj



sn_kMn_kKs_nkM_nk +

r−1

X

k=1





k−1

Y

j=1

rn^snjj



n^(k)Ds_nkM_nk.

It is well-known that

kK_nk₁< c <∞.

We define the maximal operatorsS^∗andσ^∗of partial sums and Féjer means by

S^∗f := sup

n∈N|Snf|, σ^∗f := sup

n∈N|σnf|.

Moreover, we define the restricted maximal operatorsSe_#^∗ andσe^∗_#of partial sums and Féjer means by

Se_#^∗f := sup

n∈N|SM_nf|, eσ_#^∗f := sup

n∈N|σM_nf|.

1.1.3 Character ρ(n) and Lebesgue constants with respect to Vilenkin systems

Let us define

hni:= min{j∈N:n_j6= 0} and |n|:= max{j∈N:n_j 6= 0}, that isM_|n|≤n≤M_|n|+1.Set

ρ(n) :=|n| − hni, for all n∈N.

For the natural numbersn=P∞

j=1njMjandk=P∞

j=1kjMjwe define n+kb :=

∞

X

i=0

(ni⊕ki)Mi+1

and

n−kb :=

∞

X

i=0

(ni ki)Mi+1, where

a_i⊕b_i:= (a_i+b_i)modmi, a_i, b_i∈Z_mi and is the inverse operation for⊕.

For the natural numbern=P∞

j=1njMj,we define functionsvandv^∗by v(n) :=

∞

X

j=1

|δj+1−δj|+δ₀, v^∗(n) :=

∞

X

j=1

δ_j^∗,

where

δj=sign(nj) =sign( nj) and δ_j^∗=| nj−1|δj. Then-th Lebesgue constant is defined in the following way:

Ln:=kDnk₁.

For the trigonometric system it is important to note that the results of Fejér and Szego, latter on proved in [121] gives an explicit formula for the Lebesgue constants. The most properties of the Lebesgue constants with respect to the Walsh-Paley system were obtained by Fine in [36]. In [108], p. 34, the two- sided estimate is proved. In [76], Lukomskii presented the lower estimate with sharp constant 1/4. Malykhin, Telyakovskii and Kholshchevnikova [77] (see also Astashkin and Semenov [8]) improved the estimation above and proved sharp estimate with factor 1. A new and shorter proof which improved upper bound and provide a similar lower bound can be found in [23]. In particular, forλ:= sup_n∈Nand for anyn=P∞

i=1niMiandmnwe have the following two sided estimate:

1

4λv(n) + 1

λ²v^∗(n)≤Ln ≤v(n) +v^∗(n). (1.3) Moreover, it yields that (see Memic, Simon and Tephnadze [79]):

1 nMn

Mn−1

X

k=1

v(k)≥ 2

λ². (1.4)

From the inequality (1.3) it immediately follows that for anyn ∈ N there exists an absolute constantc,such that

kDnk₁≤clogn.

For example, if we takeqn_k = M2n_k +M2n_k−2+M2+M0, we have the following two-sided inequality

nk

2λ ≤ D_qnk

₁≤λn_k, λ:= sup

n∈Nm_n.

1.1.4 Definition and examples of Nörlund and T means and its maximal operators

Let{qk : k ∈ N}be a sequence of nonnegative numbers. Then-th Nörlund means for the Fourier series off is defined by

tnf := 1 Qn

n

X

k=1

q_n−kSkf, (1.5)

where

Q_n :=

n−1

X

k=0

q_k.

A representation

tnf(x) = Z

G

f(t)An(x−t)dµ(t) plays a central role in the sequel, where

An:= 1 Q_n

n

X

k=1

q_n−kDk

is the so-called Nörlund kernel.

In Moore [80] (see also Tephnadze [130]) it was found necessary and suffi- cient conditions for regularity of Nörlund means. In particular, if{qk:k≥0}is a sequence of nonnegative numbers,q0>0and

n→∞lim Qn=∞,

then the summability method (1.5) generated by{qk : k≥0}is regular if and only if

n→∞lim q_n−1

Q_n = 0.

In addition, if the sequence {qk : k ∈ N} is non-increasing, then the summability method generated by{qk :k∈N}is regular, but if the sequence {qk : k ∈ N} is non-decreasing, then the summability method generated by {qk:k∈N}is not always regular.

Let{qk :k≥0}be a sequence of non-negative numbers. Then-thTmean Tnfor a Fourier series offis defined by

Tnf := 1 Qn

n−1

X

k=0

qkSkf,

whereQn:=Pn−1

k=0qk.It is obvious that Tnf(x) =

Z

G_m

f(t)Fn(x−t)dµ(t),

whereF_n:= _Q¹

n

n

P

k=1

q_kD_kis called the kernel ofT means.

We always assume that{qk:k≥0}is a sequence of non-negative numbers andq0 >0.Then the summability method (1.1.4) generated by{qk :k ≥0}is regular if and only iflim_n→∞Qn=∞.

Lettn be Nörlund means with monotone and bounded sequence{qk :k∈ N}, such that

q:= lim

n→∞qn> c >0.

If the sequence{qk:k∈N}is non-decreasing, then we get that nq₀≤Q_n≤nq.

In the case when the sequence{qk :k∈N}is non-increasing, we have that nq≤Qn≤nq0.

In both cases we can conclude that q_n−1

Q_n =O 1

n

, when n→ ∞.

One of the most well-known summability methods which is an example of Nörlund andT means are the so called Fejér means, which is given when {qk= 1 :k∈N}as follows:

σnf := 1 n

n

X

k=1

Skf.

The(C, α)-means (Cesàro means) of the Vilenkin-Fourier series are defined by

σ^α_nf := 1 A^α_n

n

X

k=1

A^α−1_n−kS_kf,

where

A^α₀ := 0, A^α_n := (α+ 1)...(α+n)

n! .

It is well-known that (see e.g. Zygmund [186])

A^α_n =

n

X

k=0

A^α−1_n−k,

A^α_n−A^α_n−1=A^α−1_n , A^α_nvn^α.

We also consider the "inverse"(C, α)-meansU_n^α, which is an example of a T-mean:

U_n^αf := 1 A^α_n

n−1

X

k=0

A^α−1_k Skf, 0< α <1.

LetV_n^αdenote theT mean, where

q0= 0, qk =k^α−1:k∈N⁺ ,that is

V_n^αf := 1 Qn

n−1

X

k=1

k^α−1Skf, 0< α <1.

Then-th Nörlund logarithmic meanLnand the Riesz logarithmic meanRn

are defined by

L_nf := 1 l_n

n−1

X

k=1

Skf n−k,

Rnf := 1 ln

n−1

X

k=1

S_kf k , respectively, where

ln:=

n−1

X

k=1

1 k.

The kernels of the Nörlund logarithmic meanPn and the Riesz logarithmic meanYnare, respectively, defined by

Pnf := 1 ln

n−1

X

k=1

D_kf n−k,

Ynf := 1 l_n

n−1

X

k=1

Dkf k .

Up to now we have considered Nörlund and T means in the case when the sequence{qk : k ∈ N}is bounded but now we consider Nörlund andT summabilities with unbounded sequence{qk :k∈N}.

Letα∈R⁺, β∈N⁺and

log^(β)x:=

βtimes

z }| { log...logx.

If we define the sequence{qk :k∈N}by n

q0= 0 and qk = log^(β)k^α:k∈N⁺o ,

then we get the class of Nörlund meansκ^α,β_n with non-decreasing coefficients:

κ^α,β_n f := 1 Q_n

n

X

k=1

log^(β)(n−k)^αS_kf.

First we note thatκ^α,β_n are well-defined for everyn∈N⁺. It is obvious that n

2 log^(β)n^α

2^α ≤Qn≤nlog^(β)n^α. It follows that

q_n−1 Qn

≤ clog^(β)(n−1)^α nlog^(β)n^α

= O

1 n

→0, as n→ ∞.

If we define the sequence{q_k :k∈N}byn

q₀= 0, q_k = log^(β)k^α:k∈N⁺o , then we get the class ofTmeansB^α,β_n with non-decreasing coefficients:

B_n^α,βf := 1 Qn

n−1

X

k=1

log^(β)k^αSkf.

We note thatB_n^α,βare well-defined for everyn∈N.

It is obvious thatⁿ₂log^(β)n₂^α_α ≤Q_n≤nlog^(β)n^α→0, as n→ ∞.

Let us define the maximal operatorst^∗ andT^∗ of Nörlund andT means, respectively, by

t^∗f := sup

n∈N|tnf|, T^∗f := sup

n∈N|Tnf|.

The well-known examples of maximal operators of Nörlund andT means are maximal operator of Cesáro meansσ^α,∗,Nörlund logarithmic meanL^∗and Reisz logarithmic meanR^∗which are defined by:

σ^α,∗f := sup

n∈N|σ_n^αf|, L^∗f := sup

n∈N|Lnf|, R^∗f := sup

n∈N|Rnf|.

We also define some new maximal operatorsκ^α,β,∗andβ^α,∗as follows:

κ^α,β,∗f := sup

n∈N

κ^α,β_n f , β^α,∗f := sup

n∈N|β^α_nf|.

1.1.5 Weak-type and strong-type inequalities and a.e convergence

The convolution of two functionsf, g∈L₁(G_m)is defined by (f∗g) (x) :=

Z

Gm

f(x−t)g(t)dt (x∈Gm). It is easy to see that

(f∗g) (x) = Z

Gm

f(t)g(x−t)dt (x∈Gm).

It is well-known (for details see e.g. [1] , [61] and [108]) that iff ∈ Lp(Gm), g∈L1(Gm)and1≤p <∞,thenf∗g∈Lp(Gm)and

kf∗gk_p≤ kfk_pkgk₁,

In classical Fourier analysis (see e.g. [186]), a pointx∈(−∞,∞)is called a Lebesgue point of an integrable functionf if it yields that

h→0lim 1 h

Z x+h x

|f(t)−f(x)|dµ(t) = 0.

OnG_mwe have the following definition of Lebesgue point: A pointxon the Vilenkin group is called Lebesgue point off ∈L₁(G_m),if

n→∞lim Mn

Z

I_n(x)

f(t)dt=f(x) a.e. x∈Gm. It is well-known that iff ∈L₁(G_m),then

n→∞limS_Mnf(x) =f(x) a.e. on G_m,

where SM_n is the Mn-th partial sum with respect to the Vilenkin system (for details see e.g. [1], [61] and [108]).

We introduce the operatorWAby

WAf(x) :=

A−1

X

s=0

Ms m_s−1

X

r_s=1

Z

I_A(x−rse_s)

|f(t)−f(x)|dµ(t).

A pointx∈Gmis a Vilenkin-Lebesgue point off ∈L1(Gm),if lim

A→∞W_Af(x) = 0.

In most applications the a.e. convergence of{Tn:n∈N}can be established forfin some dense class ofL1(Gm).In particular, the following result plays an important role for studying this type of questions (see e.g. the books [61], [108]

and [186]).

Lemma 1.1.1.Letf ∈L1andTn:L1→L1be some sub-linear operators and T^∗:= sup

n∈N|Tn|. If

Tnf →f a.e. for everyf ∈S,

where the setSis dense in the space L₁and the maximal operatorT^∗is bounded from the spaceL₁to the spaceweak−L₁,that is

sup

λ>0

λµ{x∈Gm: |T^∗f(x)|> λ} ≤ kfk₁, then

Tnf →f, a.e. for everyf ∈L1(Gm).

Remark 1.1.2. Since the Vilenkin function ψm is constant on In(x) for every x ∈Gmand0≤ m < Mn,it is clear that each Vilenkin function is a complex- valued step function, that is, it is a finite linear combination of characteristic functions

χ(E) =

1, x∈E, 0, x /∈E.

On the other hand, notice that, by (1.2), it yields that

χ(I_n(t)) (x) = 1 Mn

Mn−1

X

j=0

ψ_j(x−t), x∈I_n(t),

for eachx, t∈Gmandn∈N. Thus each step function is a Vilenkin polynomial.

Consequently, we obtain that the collection of step functions coincides with a collection of Vilenkin polynomialsP. Since the Lebesgue measure is regular it follows from the Lusin theorem that given f ∈ L1 there exist Vilenkin polynomialsP1, P2...,such thatPn → f a.e. whenn → ∞.This means that the Vilenkin polynomials are dense in the spaceL1.

1.1.6 Basic notations concerning Walsh groups and functions Let us define byQ2 the set of rational numbers of the form p2⁻ⁿ, where 0≤p≤2ⁿ−1for somep∈Nandn∈N.

Anyx∈[0,1]can be written in the form x=

∞

X

k=0

x_k2^−(k+1),

where eachxk = 0or1. For eachx∈[0,1]\Q2there is only one expression of this form. We shall call it the dyadic expansion ofx.Whenx ∈ Q2 there are two expressions of this form, one which terminates in0’s and one which terminates in1’s.By the dyadic expansion of anx∈Q2we shall mean the one

which terminates in0’s.Notice that1 Q2so the dyadic expansion ofx= 1 terminates in1’s.

Ifm_k= 2, for allk∈N,we have dyadic group G₂=

∞

Y

j=0

Z₂,

which is called the Walsh group

Rademacher functions are defined by:

ρn(x) := (−1)^xn. We define Walsh functionswnby

wn :=

∞

Y

k=0

ρⁿ_k^k.

LetL⁰represent the collection of a.e. finite, Lebesgue measurable functions fromG2into[−∞,∞]. For0< p <∞letL^prepresent the collection off ∈L⁰ for which

kfkp:=

Z

G₂

|f|^p 1/p

is finite. Moreover, letL^∞represent the collection off ∈L⁰for which kfk_∞:= inf{y∈R:|f(x)| ≤yfor a.e. x∈G2} is finite. It is well known thatL^pis a Banach space for each1≤p≤ ∞.

Iff ∈ L1(G2),then we can establish the Fourier coefficients, the partial sums of the Fourier series, the Fejér means, the Dirichlet and Fejér kernels with respect to the Walsh systemwin the usual manner:

fb^w(k) : = Z

G2

f αkdµ, (k∈N), S^wf : =

n−1

X

k=0

fb(k)wk, (n∈N⁺, S₀^wf := 0), D^w_n : =

n−1

X

k=0

w_k, (n∈N⁺).

We state well-known equalities for Dirichlet kernels (for details see e.g. [61]

and [108]):

D₂^wn(x) =

2ⁿ, if x∈In

0, if x /∈In

and

D^w_n =w_n

∞

X

k=0

n_kr_kD₂^wk =w_n

∞

X

k=0

n_k(D^w₂k+1−D^w₂k), for n=

∞

X

i=0

n_i2ⁱ. Next we sketch the graph of some Dirichlet kernels onG2:

The most properties of Lebesgue constants with respect to the Walsh-Paley system were obtained by Fine in [36]. Moreover, in [108], p. 34, the two-sided estimate

V(n)

8 ≤Ln≤V(n) was proved, wheren=P∞

j=1nj2^jandV(n)is defined by V (n) :=

∞

X

j=1

|nj+1−nj|+n0.

Iff ∈L₁(G₂),then the Fejér meansσ_n^wand Fejér kernelsK_n^wwith respect to the Walsh systemware, respectively, defined by

σ^w_nf : = 1 n

n−1

X

k=0

S_k^wf, (n∈N⁺),

K_n^w : = 1 n

n−1

X

k=0

D_k^w, (n∈N⁺).

Then-th Nörlund logarithmic meanL^α_n and the Riesz logarithmic meanR^α_n with respect to the Walsh systemψ(Walsh systemw) are defined by

L^w_nf := 1 l_n

n−1

X

k=1

S^w_kf

n−k, (n∈N⁺), (n∈N⁺), respectively, where

l_n:=

n−1

X

k=1

1 k.

The kernels of the Nörlund logarithmic meanP_n^αand the Riesz logarithmic meanY_n^αare, respectively, defined by

P_n^wf := 1 ln

n−1

X

k=1

D^w_kf

n−k, (n∈N⁺), Y_n^wf := 1

ln n−1

X

k=1

D^w_kf

k (n∈N⁺).

1.1.7 On martingale Hardy spaces for 0 < p≤1 Theσ-algebra generated by the intervals

{In(x) :x∈Gm} will be denoted byzⁿ(n∈N).

A sequence f = f⁽ⁿ⁾:n∈N

of integrable functions f⁽ⁿ⁾ is said to be a martingale with respect to theσ-algebraszⁿ(n∈N)if (for details see e.g.

Weisz [173] and Burkholder [31])

1) fniszⁿmeasurable for alln∈N, 2) SM_nfm=fnfor alln≤m.

The martingalef = f⁽ⁿ⁾, n∈N

is said to beLp-bounded (0< p≤ ∞) if f⁽ⁿ⁾∈Lpand

kfk_p:= sup

n∈Nkfnk_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. If1 ≤ p ≤ ∞and f ∈L_p(G_m),thenf = f⁽ⁿ⁾, n∈N

isL_p-bounded and

n→∞lim kSMnf −fk_p= 0

and consequentlykFk_p=kfk_p(see [90]). The converse of the latest statement holds also if1 < p ≤ ∞(see [90]): for an arbitraryLp-bounded martingale f = f⁽ⁿ⁾, n∈N

there exists a functionf ∈Lp(Gm)for whichf⁽ⁿ⁾=SM_nf.

Ifp= 1,then there exists a functionf ∈L1(Gm)of the preceding type if and only iff is uniformly integrable (see [90]), namely, if

y→∞limsup

n∈N

Z

{|fn|>y}

|fn(x)|dµ(x) = 0.

Thus the mapf →f := (SM_nf :n∈N)is isometric fromLponto the space ofLp-bounded martingales when1< p≤ ∞.Consequently, these two spaces can be identified with each other. Similarly, the spaceL1(Gm)can be identified with the space of uniformly integrable martingales.

Analogously, the martingalef = f⁽ⁿ⁾, n∈N

is said to beweak −Lp- bounded (0< p≤ ∞) iff⁽ⁿ⁾∈Lpand

kfk_weak−L

p:= sup

n∈Nkfnk_weak−L

p<∞.

The maximal functionf^∗of a martingalefis defined by f^∗:= sup

n∈N

f⁽ⁿ⁾

.

In the casef ∈L1(Gm),the maximal functionsf^∗are also given by f^∗(x) := sup

n∈N

1

|In(x)|

Z

I_n(x)

f(u)dµ(u) .

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

kfk_H

p:=kf^∗k_p<∞.

Vilenkin-Fourier coefficients of the martingalef = f⁽ⁿ⁾:n∈N

must be defined in a slightly different manner:

fb(i) := lim

k→∞

Z

G_m

f^(k)ψ_idµ.

Investigation of the classical Fourier analysis, definition of several variable Hardy spaces and real Hardy spaces and related theorems of atomic decompo- sitions of these spaces can be found in Fefferman and Stein [34] (see also Later [73], Torchinsky [156], Wilson [175]).

A bounded measurable functionais a p-atom if there exist an intervalI such that

Z

I

adµ= 0, kak_∞≤µ(I)^−1/p, supp(a)⊂I.

Explicit constructions ofp-atoms can be found in the papers [18] and [19] by Blahota, Gát and Goginava.

Next, we note that the Hardy martingale spacesH_p(G_m)for0< p≤1have atomic characterizations:

The following useful lemma was proved by Weisz [171, 173] (see also Persson, Tephnadze and Weisz [105]):

Lemma 1.1.3.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 everyn∈N,

∞

X

k=0

µ_kS_Mna_k =f⁽ⁿ⁾, a.e., where

∞

X

k=0

|µ_k|^p<∞.

Moreover,

kfk_H

pvinf

∞

X

k=0

|µk|^p

!1/p

,

where the infimum is taken over all decomposition of f = f⁽ⁿ⁾:n∈N of the form (1.1.3).

Explicit constructions ofHp martingales can be found in the papers [104], [105], [124], [125], [128], [131], [136], [138], [141], [142], [145], [149] and [150].

By using atomic characterization it can be easily proved that the following Lemmas hold:

Lemma 1.1.4.Suppose that an operatorT is sub-linear and for some0< p0≤1 Z

−

I

|T a|^p0dµ≤cp<∞

for everyp0-atoma, whereIdenotes the support of the atom. IfTis bounded from Lp₁toLp₁,(1< p1≤ ∞),then

kT fk_p

0 ≤cp₀kfk_H

p0.

Moreover, ifp0<1,then we have the weak (1,1) type estimate λµ{x∈G_m: |T f(x)|> λ} ≤ kfk₁ for allf ∈L1.

A proof of Lemma 1.1.4 can be found in Weisz [171] (see also Persson, Tephnadze and Weisz [105]).

Lemma 1.1.5.Suppose that an operatorT is sub-linear and for some0< p₀≤1 sup

λ>0

λ^p0µ

x∈⁻I :|T f|> λ

≤c_p0<+∞

for everyp0-atoma, whereIdenote the support of the atom. IfT is bounded from Lp₁toLp₁,(1< p1≤ ∞),then

kT fk_weak−L

p0

≤c_p0kfk_H

p0

. Moreover, ifp0<1,then

λµ{x∈G_m: |T f(x)|> λ} ≤ kfk₁, for allf ∈L1.

The best approximation off ∈Lp(Gm) (1≤p≤ ∞)is defined as En(f, Lp) := inf

ψ∈Pⁿkf−ψk_p,

wherePnis set of all Vilenkin polynomials of order less thann∈N.

The concept of modulus of continuity ωH_p in martingale Hardy spaceHp

(p >0)is defined by ωH_p

1 M_n, f

:=kf −SM_nfk_H

p.

We need to understand the meaning of the expressionf −SM_nf, where f is a martingale andSM_nf is function. Hence, we give an explanation in the following remark: