Some inequalities related to strong convergence of Riesz logarithmic means

R E S E A R C H Open Access

Some inequalities related to strong

convergence of Riesz logarithmic means

D. Lukkassen¹, L.E. Persson^1,2*, G. Tephnadze³and G. Tutberidze^4,1

*Correspondence:

larserik6pers@gmail.com

1UiT The Arctic University of Norway, Narvik, Norway

2Department of Mathematics and Computer Science, Karlstad University, Karlstad, Sweden Full list of author information is available at the end of the article

Abstract

In this paper 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.

MSC: 26D10; 26D20; 42B25; 42C10

Keywords: Inequalities; Vilenkin systems; Walsh system; Riesz logarithmic means;

Martingale Hardy space; Strong convergence

1 Introduction

Concerning definitions used in this introduction we refer to Sect.2. Weisz [47] proved the boundedness of the maximal operator of Fejér meansσ^ψ,∗ with respect to bounded Vilenkin systems from the martingale Hardy spaceHp(Gm) to the spaceLp(Gm), forp> 1/2.

Simon [31] gave a counterexample, which shows that boundedness does not hold for 0 <

p< 1/2. The corresponding counterexample forp= 1/2 is due to Goginava [14]. Moreover, Weisz [50] proved the following result.

Theorem W The maximal operator of Fejér meansσ^ψ,∗is bounded from the Hardy space H1/2(Gm)to the space weak-L1/2(Gm).

In[35]and[36]it was proved that the maximal operatorσ_p^ψ,∗defined by

σ_p^ψ,∗:=sup

n∈N

|σn^ψ|

(n+ 1)^1/p–2log^2[1/2+p](n+ 1),

where0 <p≤1/2and[1/2 +p]denotes the integer part of1/2 +p,is bounded from the Hardy space H_p(G_m)to the space L_p(G_m).Moreover, for any nondecreasing functionϕ: N+→[1,∞)satisfying the condition

n→∞lim

(n+ 1)^1/p–2log^2[1/2+p](n+ 1)

ϕ(n) = +∞, (1.1)

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there exists a martingale f ∈Hp(Gm),such that

sup

n∈N

σ_n^ψf ϕ(n)

p

=∞.

For Walsh–Kaczmarzi system some analogical results were proved in [16] and [37].

Weisz [47] considered the norm convergence of the Fejér means of a Vilenkin–Fourier series and proved the following result.

Theorem W1(Weisz) Let p> 1/2and f ∈H_p(G_m).Then there exists an absolute constant c_p,depending only on p,such that for all k= 1, 2, . . .and f ∈H_p(G_m)the following inequality holds:

σ_k^ψf

p≤cpfHp(Gm).

Moreover, in [34] it was proved that the assumptionp> 1/2 in Theorem W1 is essential.

In fact, the following is true.

Theorem T1 There exists a martingale f∈H1/2(Gm)such that sup

n∈N

σ_n^ψf

1/2= +∞. Theorem W1 implies that

1 n^2p–1

n k=1

σ_k^ψf^pp

k^2–2p ≤cpf^p_Hp(Gm), 1/2 <p<∞,n= 1, 2, . . . . If Theorem W1 holds for 0 <p≤1/2, then we would have

1 log^[1/2+p]n

n k=1

σ_k^ψf^pp

k^2–2p ≤cpf^p_Hp(Gm), 0 <p≤1/2,n= 2, 3, . . . . (1.2) For the Walsh system in [38] and for the bounded Vilenkin systems in [37] were proved that (1.2) holds, though Theorem T1 is not true for 0 <p< 1/2.

Some results concerning summability of the Fejér means of a Vilenkin–Fourier series can be found in [10,12,16,25,28,30].

The Riesz logarithmic means with respect to the Walsh system was studied by Simon [31], Goginava [15], Gát, Nagy [13] and for Vilenkin systems by Gát [11] and Blahota, Gát [3], Persson, Ragusa, Samko, Wall [26]. Moreover, in [27] it was proved that the maximal operator of the Riesz logarithmic means of a Vilenkin–Fourier series is bounded from the martingale Hardy spaceH_p(G_m) to the spaceL_p(G_m) whenp> 1/2 and is not bounded from the martingale Hardy spaceH_p(G_m) to the spaceL_p(G_m) when 0 <p≤1/2.

In [35] and [36] it was proved that the Riesz logarithmic means has better properties than the Fejér means. In particular, one considered the maximal operatorR^ψ,∗p of a Riesz logarithmic meansR^ψ,∗p defined by

R^ψ,∗_p :=sup

n∈N

|R^ψn|log(n+ 1) (n+ 1)^1/p–2log^2[1/2+p](n+ 1),

where 0 <p≤1/2 and [1/2 +p] denotes the integer part of 1/2 +p, which is bounded from the Hardy spaceHp(Gm) to the spaceLp(Gm).

Moreover, this result is sharp in the following sense: For any nondecreasing function ϕ:N+→[1,∞) satisfying the condition

n→∞lim

(n+ 1)^1/p–2log^2[1/2+p](n+ 1)

ϕ(n)log(n+ 1) =∞, (1.3)

there exists a martingalef∈Hp(Gm), such that

sup

n∈N

R^ψnf ϕ(n)

p

=∞.

The main aim of this paper is to derive a new strong convergence theorem of the Riesz logarithmic means of one-dimensional Vilenkin–Fourier (Walsh–Fourier) series (see The- orem1). The corresponding inequality is pointed out. The sharpness is proved in Theo- rem2, at least for the case with Walsh–Fourier series.

The paper is organized as follows: In Sect.2some definitions and notations are pre- sented. The main results are presented and proved in Sect.3. Section4is reserved for some concluding remarks and open problems.

2 Definitions and notations

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

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

Denote by

Zm_k:={0, 1, . . .mk– 1}

the additive group of integers modulom_k.

Define the groupG_mas the complete direct product of the groupZ_mjwith the product of the discrete topologies of theZmj.

The direct productμof the measures μ_k

{j}

:= 1/mk (j∈Zm_k)

is a Haar measure onG_mwithμ(G_m) = 1.

Ifsup_n∈Nm_n<∞, then we callG_ma bounded Vilenkin group. If the generating sequence mis not bounded, thenGmis said to be an unbounded Vilenkin group. In this paper we discuss only bounded Vilenkin groups.

The elements ofGmare represented by the sequences x:= (x0,x1, . . . ,xj, . . .) (xk∈Zm_k).

It is easy to give a base for the neighborhood ofG_m, namely I0(x) :=Gm,

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

DenoteIn:=In(0) forn∈NandIn:=Gm\In. Let

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

It is evident that

IM= _M–2

k=0 mk–1

x_k=1 M–1

l=k+1 ml–1

x_l=1

Il+1(xkek+xlel) ∪ _M–1

k=1 mk–1

x_k=1

IM(xkek) . (2.1)

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

M0:= 1, Mk+1:=mkMk (k∈N),

then everyn∈Ncan be uniquely expressed asn=_∞

k=0njMj, wherenj∈Zmj(j∈N) and only a finite number of thenjdiffer from zero. Let|n|:=max{j∈N;nj = 0}.

The norm (or quasi-norm whenp< 1) of the spaceL_p(G_m) is defined by fp:=

Gm

|f|^pdμ 1/p

(0 <p<∞).

The space weak-L_p(G_m) consists of all measurable functionsf for which fweak-Lp(Gm):=sup

λ>0

λ^pμ(f >λ) < +∞.

Next, we introduce onGman orthonormal system which is called the Vilenkin system.

Let us define complex valued function r_k(x) :G_m →C, the generalized Rademacher functions, as

r_k(x) :=exp(2πix_k/m_k)

i²= –1,x∈G_m,k∈N . Now, define the Vilenkin systemψ:= (ψ_n:n∈N) onG_mas

ψ_n(x) :=

∞ k=0

rⁿ_k^k(x) (n∈N).

The Vilenkin systems are orthonormal and complete inL2(Gm) (for details see e.g. [1]).

Specifically, we call this system Walsh–Paley ifm_k= 2, for allk∈N. In this case we have the dyadic groupG2=_∞

j=0Z2, which is called the Walsh group and the Vilenkin system coincides with the Walsh functions defined by (for details see e.g. [17] and [29])

w_n(x) :=

∞ k=0

r_k^nk(x) =r_|n|(x)(–1)^{|n|–1k=0 nkxk} (n∈N),

wherenk= 0∨1 andxk= 0∨1.

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

If f ∈L1(Gm), 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 Vilenkin systemψ(Walsh systemw) in the usual manner:

f^α(k) :=

Gm

fα_kdμ (αk=wkorψ_k) (k∈N),

S^α_nf :=

n–1 k=0

f(k)αk (αk=wkorψ_k)

n∈N+,S^α₀f := 0 ,

σ_n^αf :=1 n

n–1 k=0

S_k^αf (α=worψ) (n∈N+),

D^α_n:=

n–1 k=0

α_k (α=worψ) (n∈N+),

K_n^α:= 1 n

n–1 k=0

D^α_k (α=worψ) (n∈N+).

It is well known that (see e.g. [1]) sup

n∈N

Gm

K_n^αdμ≤c<∞, whereα=worψ. (2.2)

Theσ-algebra generated by the intervals{In(x) :x∈Gm}will be denoted byn(n∈N).

Denote byf = (f⁽ⁿ⁾,n∈N) a martingale with respect ton(n∈N) (for details see e.g.

[5,23,46]). The maximal function of a martingalef is defend by f^∗=sup

n∈N

f⁽ⁿ⁾.

In the casef ∈L₁(G_m), the maximal functions are also given by

f^∗(x) =sup

n∈N

1

|In(x)|

In(x)

f(u)μ(u) .

For 0 <p<∞the Hardy martingale spacesHp(Gm) consist of all martingales for which fHp(Gm):=f^∗

p<∞.

If f ∈L1(Gm), then it is easy to show that SMnf isnmeasurable and the sequence (SMnf :n∈N) is a martingale. Iff = (f⁽ⁿ⁾,n∈N) is a martingale, then the Vilenkin–Fourier (Walsh–Fourier) coefficients must be defined in a slightly different manner, namely

f(i) := lim

k→∞

Gm

f^(k)(x)α_i(x)dμ(x), whereα=worψ.

The Vilenkin–Fourier coefficients off ∈L₁(G_m) are the same as those of the martingale (SMnf :n∈N) obtained fromf.

In the literature, there is the notion of the Riesz logarithmic means of a Fourier series.

Thenth Riesz logarithmic means of the Fourier series of an integrable functionfis defined by

R^α_nf := 1 ln

n k=1

S^α_kf

k , whereα=worψ, with

ln:=

n k=1

1 k.

The kernels of Riesz‘s logarithmic means are defined by L^α_n:= 1

ln

n k=1

D^α_k

k , where (α=worψ).

For the martingalef we consider the following maximal operators:

σ^α,∗f:sup

n∈N

σ_n^αf (α=worψ), R^∗f :=sup

n∈N

R^α_nf (α=worψ), R^α,∗f:=sup

n∈N

|R^α_nf|

log(n+ 1) (α=worψ), R^α,∗_p f:=sup

n∈N

log(n+ 1)|R^α_nf|

(n+ 1)^1/p–2 (α=worψ).

A bounded measurable functionais ap-atom, if there exists an intervalI, such that

I

a dμ= 0, a∞≤μ(I)^–1/p, supp(a)⊂I.

In order to prove our main results we need the following lemma of Weisz (for details see e.g. Weisz [49]).

Proposition 1 A martingale f = (f⁽ⁿ⁾,n∈N)is in H_p(G_m) (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 a real numbers such that for every n∈N

∞ k=0

μ_kSMnak=f⁽ⁿ⁾ (2.3)

and ∞

k=0

|μ_k|^p<∞.

Moreover,fHp(Gm)inf(_∞

k=0|μ_k|^p)^1/p,where the infimum is taken over all decomposi- tions of f of the form(2.3).

By using atomic characterization (see Proposition 1) it can be easily proved that the following statement holds (see e.g. Weisz [50]).

Proposition 2 Suppose that an operator T is sub-linear and for some0 <p0≤1

¯I

|Ta|^p0dμ≤cp<∞

for every p0-atom a,where I denotes the support of the atom.If T is bounded from Lp1 to L_p1(1 <p₁≤ ∞),then

Tfp0≤c_p0fHp0(Gm). (2.4)

Let us define classical Hardy spaces (see e.g. [44]). Let Hp(D), p > 0 be the one- dimensional complex quasi-Banach space of analytic functionsf on the unit discD:=

(z:|z|< 1) for which fHp(D)=sup

r<1

1 2π

[–π,π]

f

re^itpdt 1/p

.

Now, we define real Hardy spaces. A real-valued distributionsf(t)∈D(T) belongs to Hp(T) whereT= (–π,π] if and only if there exists a functionF(z)∈Hp(D) with the prop- ertiesIm(F(0)) = 0 andf(t) =lim_r→1ReF(re^it) in the sense of distributions. Equipped with quasi-norm f(z)Hp(T)=F(z)Hp(D) the class obviously becomes a real quasi-Banach space with quite the same properties asHp(D). Atomic decomposition of classical Hardy spaces and real Hardy spaces can be found e.g. in Fefferman and Stein [6] (see also Later [19], Torchinsky [44], Wilson [51]).

3 Main results

Our first main result reads as follows.

Theorem 1 Let0 <p< 1/2and f ∈Hp(Gm).Then there exists an absolute constant cp, depending only on p,such that the inequality

∞ n=1

log^pnR^ψ_nf^p_Hp(Gm)

n^2–2p ≤c_pf^p_Hp(Gm) (3.1)

holds,where R^ψnf denotes the nth Riesz logarithmic mean with respect to the Vilenkin–

Fourier series of f.

For the proof of Theorem1we will use the following lemmas.

Lemma 1(see [38]) Let x∈IN(xkek+xlel), 1≤xk≤mk–1, 1≤xl≤ml–1,k= 0, . . . ,N–2, l=k+ 1, . . . ,N– 1.Then

IN

K_n^ψ(x–t)dμ(t)≤cMlMk

nM_N , when n≥MN.

Let x∈IN(xkek), 1≤xk≤mk– 1,k= 0, . . . ,N– 1.Then

IN

K_n^ψ(x–t)dμ(t)≤cMk

MN

, when n≥M_N.

Lemma 2(see [39]) Let x∈IN(xkek+xlel), 1≤xk≤mk–1, 1≤xl≤ml–1,k= 0, . . . ,N–2, l=k+ 1, . . . ,N– 1.Then

I_N

n j=MN+1

|K_j^ψ(x–t)|

j+ 1 dμ(t)≤cMkMl

M²_N .

Let x∈IN(xkek), 1≤xk≤mk– 1,k= 0, . . . ,N– 1.Then

IN

n j=MN+1

|K_j^ψ(x–t)|

j+ 1 dμ(t)≤cMk

MN

l_n.

Proof By using an Abel transformation, the kernels of the Riesz logarithmic means can be rewritten as (see also [39])

L^ψ_n = 1 ln

n–1 j=1

K_j^ψ j+ 1+Kn^ψ

ln

. (3.2)

Hence, according to (2.2) we get sup

n∈N

Gm

L^α_ndμ≤c<∞, whereα=worψ

and it follows thatR^ψ_nis bounded fromL_∞toL_∞. By Proposition2, the proof of Theorem1 will be complete, if we show that

∞ n=1

log^pn

I¯|R^ψna|^pdμ

n^2–2p ≤cp<∞, for 0 <p< 1/2, (3.3) for everyp-atoma, whereIdenotes the support of the atom.

Letabe an arbitrary p-atom with supportIandμ(I) =M^–1_N. We may assume thatI=IN. It is easy to see thatR^ψna=σ_n^ψ(a) = 0, whenn≤MN. Therefore we suppose thatn>MN.

Sincea∞≤cM²_N if we apply (3.2), then we can conclude that R^ψ_na(x)

=

IN

a(t)L^ψ_n(x–t)dμ(t)

≤ a∞

IN

L^ψ_n(x–t)dμ(t)

≤cM_N^1/p ln

IN

n–1 j=MN+1

|K_j^ψ(x–t)|

j+ 1 dμ(t) +cM^1/p_N

l_n

IN

K_n^ψ(x–t)dμ(t). (3.4)

Letx∈IN(xkek+xlel), 1≤xk≤mk– 1, 1≤xl≤ml– 1,k= 0, . . . ,N– 2,l=k+ 1, . . . ,N– 1.

From Lemmas1and2it follows that R^ψ_na(x)≤cM_lM_kM_N^1/p–2

log(n+ 1) . (3.5)

Letx∈IN(xkek), 1≤xk≤mk– 1,k= 0, . . . ,N– 1. Applying Lemmas1and2we can conclude that

R^ψ_na(x)≤M^1/p–1_N Mk. (3.6)

By combining (2.1) and (3.4)–(3.6) we obtain

IN

R^ψ_na(x)^pdμ(x)

=

N–2

k=0 N–1

l=k+1

mj–1 xj=0,j∈{l+1,...,N–1

I_N^k,l

R^ψ_na^pdμ+

N–1

k=0

I_N^k,N

R^ψ_na^pdμ

≤c

N–2

k=0

N–1 l=k+1

ml+1. . .mN–1

MN

(MlMk)^pM^1–2p_N log^p(n+ 1) +

N–1

k=0

1 MN

M_k^pM_N^1–p

≤ cM^1–2p_N log^p(n+ 1)

N–2

k=0 N–1

l=k+1

(MlMk)^p M_l +

N–1

k=0

M^p_k M^p_N

≤ cM^1–2p_N

log^p(n+ 1)+cp. (3.7)

It is easy to see that ∞

n=MN+1

1

n^2–2p ≤ c

M_N^1–2p, for 0 <p< 1/2. (3.8)

By combining (3.7) and (3.8) we get ∞

n=MN+1

log^pn

IN|Rna|^pdμ n^2–2p

≤ ∞ n=MN+1

cpM^1–2p_N n^2–p + cp

n^2–p

+c_p

≤cpM^1–2p_N ∞ n=MN+1

1 n^2–2p +

∞ n=MN+1

1

n^2–p +cp≤Cp<∞.

It means that (3.3) holds true and the proof is complete.

Our next main result shows in particular that the inequality in Theorem1is in a special sense sharp at least in the case of Walsh–Fourier series (cf. also Problem2in the next section).

Theorem 2 Let0 <p< 1/2andΦ:N→[1,∞)be any nondecreasing function,satisfying the condition

n→∞lim Φ(n) = +∞. (3.9)

Then there exists a martingale f ∈Hp(G2)such that ∞

n=1

log^pnR^w_nf^ppΦ(n)

n^2–2p =∞, (3.10)

where R^w_nf denotes the nth Riesz logarithmic means with respect to Walsh–Fourier series of f.

Proof It is evident that if we assume thatΦ(n)≥cn, wherecis some positive constant then

log^pnΦ(n)

n^2–2p ≥n^1–2plog^pn→ ∞, asn→ ∞,

and also (3.10) holds. So, without loss of generality we may assume that there exists an increasing sequence of positive integers{α_k:k∈N}such that

Φ α_k

=o α_k

, ask→ ∞. (3.11)

Let{α_k:k∈N} ⊆ {α_k :k∈N}be an increasing sequence of positive integers such that α₀≥2 and

∞ k=0

1

Φ^1/2(2^2αk)<∞, (3.12)

k–1 η=0

2^2αη/p

Φ^1/2p(2^2αη)≤ 2^2αk–1/p+1

Φ^1/2p(2^2αk–1), (3.13)

2^2αk–1/p+1

Φ^1/2p(2^2αk–1)≤ 1 128αk

2^{2αk(1/p–2)}

Φ^1/2p(2^2αk). (3.14)

We note that under condition (3.11) we can conclude that 2^2αη/p

Φ^1/2p(2^2αη)≥ 2^2αη

Φ(2^2αη) 1/2p

→ ∞, asη→ ∞

and it immediately follows that such an increasing sequence{α_k:k∈N}, which satisfies conditions (3.12)–(3.14), can be constructed.

Let

f^(A)(x) :=

{k;2αk<A}

λ_ka_k,

where

λk= 1 Φ^1/2p(2^2αk)

and

ak= 2^{2αk(1/p–1)}(D₂^2αk+1–D₂^2αk).

From (3.12) and Lemma1we can conclude thatf= (f⁽ⁿ⁾,n∈N)∈H_p(G₂).

It is easy to show that

f^w(j) =

⎧⎨

⎩

2^{2αk(1/p–1)}

Φ^1/2p(2^2αk), ifj∈ {2^2αk, . . . , 2^2αk+1– 1},k∈N, 0, ifj∈/_∞

k=1{2^2αk, . . . , 2^2αk+1– 1}.

(3.15)

Forn=s

i=12ⁿⁱ,n₁<n₂<· · ·<n_swe denote A0,2:=

n∈N:n= 2⁰+ 2²+

sn

i=3

2ⁿⁱ

.

Let 2^2αk≤j≤2^2αk+1– 1 andj∈A0,2. Then

R^w_jf =1 lj

2^2αk–1 n=1

Snf n +1

lj

j n=2^2αk

Snf

n :=I+II. (3.16)

Letn< 2^2αk. Then from (3.13), (3.14) and (3.15) we have S_n^wf(x)≤

k–1 η=0

2^2αη+1–1 v=2^2αη

f^w(v)≤ k–1

η=0 2^2αη+1–1

v=2^2αη

2^{2αη(1/p–1)} Φ^1/2p(2^2αη)

≤ k–1

η=0

2^2αη/p

Φ^1/2p(2^2αη)≤ 2^2αk–1/p+1

Φ^1/2p(2^2αk–1)≤ 1 128α_k

2^{2αk(1/p–2)} Φ^1/2p(2^2αk). Consequently,

|I| ≤ 1 lj

2^2αk–1 n=1

|S^w_nf(x)| n

≤ 1 l₂2αk

1 128α_k

2^{2αk(1/p–2)} Φ^1/2p(2^2αk)

2^2αk–1 n=1

1 n≤ 1

128α_k

2^{2αk(1/p–2)}

Φ^1/2p(2^2αk). (3.17) Let 2^2αk≤n≤2^2αk+1– 1. Then we have the following:

S^w_nf = k–1

η=0 2^2αη+1–1

v=2^2αη

f^w(v)w_v+ n–1 v=2^2αk

f^w(v)w_v

= k–1

η=0

2^{2αη(1/p–1)} Φ^1/2p(2^2αη)

D^w

2^2αη+1–D^w

2^2αη

+ 2^{2αk(1/p–1)} Φ^1/2p(2^2αk)

D^w_n–D^w

2^2αk

.

This gives

II=1 lj

2^2αk+1

n=2^2αk

1 n

_k–1

η=0

2^{2αη(1/p–1)} Φ^1/2p(2^2αη)

D^w

2^2αη+1–D^w

2^2αη

+1 l_j

2^{2αk(1/p–1)} Φ^1/2p(2^2αk)

j n=2^2αk

(D^w_n–D^w

2^2αk) n

:=II1+II2. (3.18)

Letx∈I₂(e₀+e₁)∈I₀\I₁. We use well-known equalities for Dirichlet kernels (for details see e.g. [17] and [29]): recall that

D^w₂n(x) =

⎧⎨

⎩

2ⁿ, ifx∈In,

0, ifx∈/I_n, (3.19)

and

D^w_n=w_n ∞

k=0

n_kr_kD^w₂k=w_n ∞

k=0

n_k

D^w₂k+1–D^w₂k

, forn= ∞

i=0

n_i2ⁱ, (3.20)

so we can conclude that

D^w_n(x) =

⎧⎨

⎩

w_n, ifnis odd number, 0, ifnis even number.

Sinceα₀≥2,k∈Nwe obtain 2αk≥4, for allk∈Nand if we apply (3.19) we get

II1= 0 (3.21)

and

II2=1 l_j

2^{2αk(1/p–1)} Φ^1/2p(2^2αk)

(j–1)/2

n=2^2αk–1

w2n+1

2n+ 1=1 l_j

2^{2αk(1/p–1)}r1

Φ^1/2p(2^2αk)

(j–1)/2

n=2^2αk–1

w2n

2n+ 1.

Letx∈I2(e0+e1). Then, by the definition of Walsh functions, we get w_4n+2=r₁w_4n= –w_4n

and

|II2|= 1 lj

2^{2αk(1/p–1)} Φ^1/2p(2^2αk)

(j–1)/2

n=2^2αk–1

w_2n 2n+ 1

= 1 l_j

2^{2αk(1/p–1)} Φ^1/2p(2^2αk)

w_j–1 j +

(j–1)/4

n=2^2αk–2+1

w4n–4

4n– 3+ w4n–2

4n– 1

= 1 lj

2^{2αk(1/p–1)} Φ^1/2p(2^2αk)

wj–1

j +

(j–1)/4

n=2^2αk–2+1

w4n–4

4n– 3– w4n–2

4n– 1

≥ c log(2^2αk+1)

2^{2αk(1/p–1)} Φ^1/2p(2^2αk)

wj–1

j –

(j–1)/4

n=2^2αk–2+1

|w4n–4| 1

4n– 3– 1 4n– 1

≥ 1 4αk

2^{2αk(1/p–1)} Φ^1/2p(2^2αk)

1 j –

(j–1)/4

n=2^2αk–2+1

1

4n– 3– 1

4n– 1 . (3.22)

By a simple calculation we can conclude that

(j–1)/4

n=2^2αk–2+1

1

4n– 3– 1 4n– 1

=

(j–1)/4

n=2^2αk–2+1

2 (4n– 3)(4n– 1)

≤

(j–1)/4

n=2^2αk–2+1

2

(4n– 4)(4n– 2)=1 2

(j–1)/4

n=2^2αk–2+1

1 (2n– 2)(2n– 1)

≤1 2

(j–1)/4

n=2^2αk–2+1

1

(2n– 2)(2n– 2)=1 8

(j–1)/4

n=2^2αk–2+1

1 (n– 1)(n– 1)

≤1 8

(j–1)/4

n=2^2αk–2+1

1

(n– 1)(n– 2)=1 8

(j–1)/4

l=2^2αk–2+1

1 n– 2– 1

n– 1

≤1 8

1

2^2αk–2– 1– 4 j– 5

≤1 8

1 2^2αk–2– 1–4

j

. Since 2^2αk≤j≤2^2αk+1– 1, whereα_k≥2, we obtain

2

2^2αk– 4≤ 2 2⁴– 4=1

6 and

|II2| ≥ 1 4αk

2^{2αk(1/p–1)} Φ^1/2p(2^2αk)

1 j –1

8 1

2^2αk–2– 1–4 j

(3.23)

≥ 1 4αk

2^{2αk(1/p–1)} Φ^1/2p(2^2αk)

3

2j– 1

2^2αk+1– 8

≥ 1 4αk

2^{2αk(1/p–1)} Φ^1/2p(2^2αk)

3 4

1 2^2αk –1

2 1 2^2αk– 4

≥ 1 4α_k

2^{2αk(1/p–1)} Φ^1/2p(2^2αk)

1 4

1 2^2αk +1

2 1 2^2αk –1

2 1 2^2αk– 4

= 1 4α_k

2^{2αk(1/p–1)} Φ^1/2p(2^2αk)

1 4

1

2^2αk – 2 2^2αk(2^2αk– 4)

≥ 1 4αk

2^{2αk(1/p–1)} Φ^1/2p(2^2αk)

1 4

1 2^2αk –1

6 1 2^2αk

≥ 1 48αk

2^{2αk(1/p–2)} Φ^1/2p(2^2αk)≥ 1

64αk

2^{2αk(1/p–2)} Φ^1/2p(2^2αk).

By combining (3.14), (3.16)–(3.23) for∈I2(e0+e1) and 0 <p< 1/2 we find that R^w_jf(x)≥ |II2|–|II1|–|I|

≥ 1 64αk

2^{2αk(1/p–2)} Φ^1/2p(2^2αk)– 1

128αk

2^{2αk(1/p–2)} Φ^1/2p(2^2αk)= 1

128αk

2^{2αk(1/p–2)} Φ^1/2p(2^2αk). Hence,

R^w_jf^p

weak-Lp(G2)

≥ 1 128α_k^p

2^2αk(1–2p) Φ^1/2(2^2αk)μ

x∈G2:R^w_jf≥ 1 128αk

2^{2αk(1/p–2)} Φ^1/2p(2^2αk)

1/p

≥ 1 128α_k^p

2^2αk(1–2p) Φ^1/2(2^2αk)μ

x∈I2(e0+e1) :R^w_jf≥ 1 128αk

2^{2αk(1/p–2)} Φ^1/2p(2^2αk)

≥ 1 128α_k^p

2^2αk(1–2p) Φ^1/2(2^2αk)

μ

x∈I₂(e₀+e₁)

> 1 516α_k^p

2^2αk(1–2p)

Φ^1/2(2^2αk). (3.24) Moreover,

∞ j=1

R^w_jf^p_weak-Lp(G2)log^p(j)Φ(j) j^2–2p

≥

{j∈A0,2:2^2αk<j≤2^2αk+1–1}

R^w_jf^p_weak-Lplog^p(j)Φ(j) j^2–2p

≥ c α_k^p

2^2αk(1–2p) Φ^p/2(2^2αk)

{j∈A0,2:2^2αk<j≤2^2αk+1–1}

log^p(j)Φ(j) j^2–2p

≥cΦ(2^2αk)log^p(2^2αk) α_k^p

2^2αk(1–2p) Φ^1/2(2^2αk)

{j∈A0,2:2^2αk<j≤2^2αk+1–1}

1 j^2–2p

≥Φ^1/2 2^2αk

→ ∞, ask→ ∞.

The proof is complete.

4 Final remarks and open problems

In this section we present some final remarks and open problems, which might be inter- esting for further research. The first problem reads as follows.

Problem 1 For anyf∈H1/2, is it possible to find strong convergence theorems for Riesz meansR^w_m, whereα=worα=ψ?

Remark1 Similar problems for Fejér means with respect to Walsh and Vilenkin systems can be found in [2,4,40] (see also [45] and [48]). Our method and estimations of Riesz and Fejér kernels (see Lemmas1and2) do not give an opportunity to prove even similar

strong convergence result as for the case of Fejer means. In particular, for anyf ∈H1/2is it possible to prove the following inequality:

1 logn

n k=1

R^α_kf^1/2_1/2

k ≤cf^1/2_H1/2, whereα=worα=ψ?

It is interesting to generalize Theorem2for Vilenkin systems.

Problem 2 For 0 <p< 1/2 and any nondecreasing functionΦ:N→[1,∞) satisfying the conditionslim_n→∞Φ(n) = +∞, is it possible to find a martingalef ∈Hp(Gm) such that

∞ n=1

log^pnR^ψnf^ppΦ(n) n^2–2p =∞,

whereR^ψ_nf denotes thenth Riesz logarithmic means with respect to the Vilenkin–Fourier series off?

Problem 3 Is it possible to find a martingalef ∈H_1/2, such that sup

n∈N

R^α_nf

1/2=∞, whereα=worα=ψ?

Remark2 For 0 <p< 1/2, divergence in the spaceLp of Riesz logarithmic means with respect to Walsh and Vilenkin systems of martingalef ∈H_pwas already proved in [27].

Problem 4 For anyf ∈Hp (0 <p≤1/2), is it possible to find necessary and sufficient conditions for the indiceskjfor which

R^α_kjf–f

Hp→0, asj→ ∞, whereα=worα=ψ?

Remark3 Similar problem for partial sums and Fejer means with respect to Walsh and Vilenkin systems can be found in Tephnadze [41,42] and [43].

Problem 5 Is it possible to find necessary and sufficient conditions in terms of the one- dimensional modulus of continuity of martingalef ∈H_p(0 <p≤1/2), for which

R^α_jf–f

Hp→0, asj→ ∞, whereα=worψ?

Remark4 Approximation properties of some summability methods in the classical and real Hardy spaces were considered by Oswald [24], Kryakin and Trebels [18], Storoienko [32,33] and for martingale Hardy spaces in Fridli, Manchanda and Siddiqi [9] (see also [7,8]), Nagy [20–22], Tephnadze [41–43].