A note on the maximal operators of Vilenkin-Nörlund means with non-increasing coefficients

DOI: 10.1556/012.2016.1342

A NOTE ON THE MAXIMAL OPERATORS OF VILENKIN–N ¨ORLUND MEANS WITH NON-INCREASING

COEFFICIENTS^∗

N. MEMI ´C¹, L. E. PERSSON^2,3 and G. TEPHNADZE^2,4

1Department of Mathematics, University of Sarajevo, Zmaja od Bosne 33-35, Sarajevo, Bosnia and Herzegovina

e-mail: [email protected]

2Department of Engineering Sciences and Mathematics, Lule˚a University of Technology, SE-971 87 Lule˚a, Sweden

e-mail: [email protected]

3Narvik University College, P.O. Box 385, N-8505, Narvik, Norway

4Department of Mathematics, Faculty of Exact and Natural Sciences, Iv. Javakhishvili Tbilisi State University, Chavchavadze str. 1, Tbilisi 0128, Georgia

e-mail: [email protected]

Communicated by A. Kro´o

(Received March 16, 2015; accepted June 19, 2015) Abstract

In [14] we investigated some Vilenkin–N¨orlund means with non-increasing coefficients.

In particular, it was proved that under some special conditions the maximal operators of such summabily methods are bounded from the Hardy spaceH_1/(1+α)to the spaceweak- L1/(1+α), (0< α51). In this paper we construct a martingale in the space H1/(1+α), which satisfies the conditions considered in [14], and so that the maximal operators of these Vilenkin–N¨orlund means with non-increasing coefficients are not bounded from the Hardy spaceH_1/(1+α) to the spaceL_1/(1+α). In particular, this shows that the conditions under which the result in [14] is proved are in a sense sharp. Moreover, as further applications, some well-known and new results are pointed out.

1. Introduction and statement of the main result

Denote by N+ the set of the positive integers, N:=N+∪ {0}. Letm:=

(m0, m1, . . .) be a sequence of the positive integers not less than 2. Denote by Zmk :={0,1, . . . , mk−1} the additive group of integers modulo mk.

Define the groupGm as the complete direct product of the groups Zmi, with the product of the discrete topologies ofZmj.

2010Mathematics Subject Classification.Primary 42C10, 42B25.

Key words and phrases. Vilenkin system, Vilenkin group, Vilenkin–Fourier series, Ces´aro means, Fej´er means, N¨orlund means, martingale Hardy space, Lp spaces, weak- Lpspaces, maximal operator.

∗The research was supported by a Swedish Institute scholarship, provided within the framework of the SI Baltic Sea Region Cooperation/Visby Programme.

⃝c 2016Akad´emiai Kiad´o, Budapest

The direct productµof the measuresµ_k( {j})

:= 1/m_k (j∈Zmk) is the Haar measure on Gm withµ(Gm) = 1.

In this paper we discuss bounded Vilenkin groups, i.e. the case when sup_nmn<∞.

The elements ofGmare represented by sequencesx:= (x0, x1, . . . , xj, . . .) (x_j ∈Z_mj).

It is easy to give a base for the neighborhoods ofG_m:

I₀(x) :=G_m, I_n(x) :={y∈G_m|y₀ =x₀, . . . , y_n−1=x_n−1}, wherex∈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 in the following way:

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

j=0n_jM_j, where nj ∈Zmj (j∈N+) and only a finite number ofnj‘s differ from zero.

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 systemψ:= (ψn: n∈N) on Gm as:

ψn(x) :=

∏∞ k=0

rⁿ_k^k(x), (n∈N).

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

The Vilenkin system is orthonormal and complete inL₂(G_m) (see [17]).

The norm (or quasi-norm) of the space Lp(Gm) and weak-Lp(Gm) (0< p <∞) are respectively defined by

∥f∥^p_p:=

∫

Gm

|f|^pdµ, ∥f∥^p_weak-Lp := sup

λ>0

λ^pµ(f > λ) <+∞.

If f ∈L1(Gm) we can respectively define the Fourier coefficients, the partial sums of the Fourier series, the Dirichlet kernels with respect to the Vilenkin system in the usual manner:

fb(n) :=

∫

Gm

f ψ_ndµ, (n∈N),

S_nf :=

n−1

∑

k=0

fb(k)ψ_k, D_n:=

n−1

∑

k=0

ψ_k, (n∈N+)

Recall that

(1) D_Mn(x) =

{M_n, if x∈I_n, 0, if x /∈I_n. The σ-algebra generated by the intervals {

In(x) : x∈Gm

} will be de- noted byzn(n∈N). Denote byf =

(

)

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

n∈N

|

|

For 0< p <∞ the Hardy martingale spaces H_p(G_m) consist of all mar- tingales for which

∥f∥_Hp :=∥f^∗∥_p<∞.

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→∞

∫

Gm

f^(k)ψ_idµ.

A bounded measurable functionais a p-atom (p >0), if there exists an intervalI, such that

∫

I

adµ= 0, ∥a∥_∞5µ(I)^−1/p, supp(a)⊂I.

We also need the following auxiliary result (see [19]):

Lemma 1. A martingale f =

(

)

(2)

∑∞ k=0

µ_kSMna_k=f⁽ⁿ⁾,

∑∞ k=0

|µ_k|^p<∞.

Moreover,∥f∥_Hp vinf

(

)

Let{q_k: k=0} be a sequence of nonnegative numbers. The n-th N¨or- lund mean for a Fourier series off is defined by

(3) t_nf = 1

Qn

∑n

k=1q_n−kS_kf,

whereQ_n:=∑_n−1

k=0q_k.

We always assume that q₀>0 and lim_n→∞Q_n=∞. In this case it is well-known that the summability method generated by{q_k: k=0}is regu- lar if and only if

nlim→∞

q_n−1 Q_n = 0.

Concerning this fact and related basic results, we refer to [6].

The (C, α)-means (Ces´aro means) of the Vilenkin–Fourier series are de- fined by

σ_n^αf = 1 A^α_n

∑n

k=1A^α_n⁻₋¹_kS_kf, where

A^α₀ = 0, A^α_n= (α+ 1). . .(α+n)

n! , α̸=−1,−2, . . . Whenα= 1 the Ces´aro means coincide with the Fej´er means

σnf = 1 n

∑n

k=1Skf.

For the martingalef we consider the following maximal operators:

t^∗f := sup

n∈N|tnf|, σ^∗f := sup

n∈N|σnf|, σ^α,∗f := sup

n∈N

σ^α_nf.

In the one-dimensional case the result with respect to the a.e. conver- gence of Fej´er is due to P´al and Simon [11] (c.f. also [2]) for bounded Vilenkin series. Weisz [20] proved that the maximal operator of the Fej´er means σ^∗ is bounded from the Hardy spaceH_1/2 to the space weak-L_1/2. Simon [12]

gave a counterexample, which shows that boundedness does not hold for 0< p <1/2. A counterexample forp= 1/2 was given in [16].

In [4] Goginava investigated the behaviour of Ces´aro means of Walsh–

Fourier series in detail. The a.e. convergence of Ces´aro means of f ∈L₁ was proved in [5]. Furthermore, Simon and Weisz [13] showed that the maximal

operator σ^α,∗ (0< α <1) of the (C, α) means is bounded from the Hardy space H_1/(1+α) to the space weak-L_1/(1+α). Moreover, Goginava [3] gave a counterexample, which shows that boundedness does not hold for 0< p5 1/(1 +α).

M´oricz and Siddiqi [7] investigated the approximation properties of some special N¨orlund means of Walsh–Fourier series of L_p functions in norm. In the two-dimensional case approximation properties of N¨orlund was consid- ered by Nagy (see [8]–[10]). In [1] and [15] it was proved strong convergence theorems for N¨orlund means of Vilenkin–Fourier series with monotone coef- ficients. Moreover, there was also shown boundedness of weighted maximal operators of such N¨orlund means on martingale Hardy spaces. Recently, in [14] it was proved that the following is true:

Theorem A. a) Let 0< α51. Then the maximal operator t^∗ of summability method (3) with non-increasing sequence {q_k: k=0}, satis- fying the conditions

(4) n^αq0

Q_n =O(1), |qn−qn+1|

n^α−2 =O(1), as n→ ∞, is bounded from the Hardy space H_1/(1+α) to the space weak-L_1/(1+α).

b)Let0< α51,05p <1/(1 +α)and{q_k: k=0}be a non-increasing sequence, satisfying the condition

(5) q₀

Q_n = c

n^α, (c >0).

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

n∈N∥t_nf∥_weak-Lp=∞.

c)Let {q_k: k=0} be a non-increasing sequence, satisfying the condition

(6) lim

n→∞

q0n^α

Q_n =∞, (0< α51).

Then there exists an martingale f ∈H_1/(1+α), such that sup

n∈N∥tnf∥_{weak-L1/(1+α)} =∞.

In this paper we complement this result by proving sharpness of both conditions of (4). Our main result reads:

Theorem 1. Let 0< α51 and {q_k : k=0} be a non-increasing se- quence, satisfying the conditions

(7) lim

n→∞

|qn−qn+1|

n^α−2 =c, (c >0), and

(8) n^αq₀

Qn =c, (c >0, n∈N).

Then there exists a martingalef ∈H_1/(1+α), such that

n∈Nsup∥t_nf∥_1/(1+α) =∞.

The proof can be found in the Section 2 and some applications and final remark in the Section 3.

2. Proof of Theorem 1

Proof. Under the condition (7), there exists an increasing sequence {n_k: k∈N}of positive integers such that

(9) M_2n^α_k+1

QM2nk+1

> c_α>0, k∈N.

Let {α_k : k∈N} ∈ {n_k : k∈N} be an increasing sequence of positive integers such that:

∑∞ k=0

1/α^1/(1+α)_k <∞, (10)

λ

k−1

∑

η=0

M_α^1+α_η

α_η < M_α^1+α_k α_k (11)

and

(12) 32λM_α^1+α_k−1

αk−1

< M_[α^α+1

k/2]

αk

,

whereλ= sup_nmn and [α_k/2] denotes the integer part of α_k/2.

We note that such increasing sequence{αk: k∈N}which satisfies con- ditions (10)–(12) can be constructed.

Let the martingalef :=

(

)

(13) f⁽ⁿ⁾ = ∑

{k:αk<n}

λ_kθα_k,

where

(14) λ_k = λ

α_k and θαk = M_α^α

k

λ

(DM_αk+1−DM_αk

).

Since

SMAθ_k =

{θ_k, if α_k< A, 0, if α_k=A, supp(θk) =Iαk,

∫

I_αk

θkdµ= 0, ∥θk∥_∞5M_α^1+α_k = (suppθk)^1+α,

if we apply Lemma 1 and (10) we can conclude thatf ∈H_1/(1+α). Moreover, it is easy to see that

(15) fb(j) =









M_αk^α

αk , if j∈{

Mα_k, . . . , Mα_k+1−1}

, k= 0,1,2. . . , 0, if j /∈ ∪^∞

k=1

{M_αk, . . . , M_αk+1−1} .

Lets= 0, . . . , k−1. We can write that tM_αk+Msf

= 1

QM_αk+Ms

M∑_αk

j=0

q_jS_jf + 1 QM_αk+Ms

M_αk∑+Ms

j=M_αk+1

q_jS_jf

:=I+II.

LetMαs 5 j5Mαs+1, wheres= 0, . . . , k−1. Moreover, D_j −D_Mαs 52j5λM_αs, (s∈N)

so that, according to (1) and (15), we have that

|S_jf| (16)

=

M_αs∑₋₁₊₁−1 v=0

fb(v)ψ_v+

j−1

∑

v=M_αs

fb(v)ψ_v 5

^s∑⁻¹

η=0

M_αη+1∑−1 v=M_αη

M_α^α_η α_η ψ_v

+M_α^α_s

α_s

|

|

= ^s∑⁻¹

η=0

M_α^α_η α_η

(D_Mαη+1−D_Mαη)

+M_α^α_s

α_s

|

|

5λ

s−1

∑

η=0

M_α^α+1_η

α_η +λM_α^α+1_s α_s

5 λM_α^α+1_s αs

+λM_α^α+1_s

αs 5 2λM_α^α+1_k−1 α_k−1 .

LetMαs−1+1+ 15j5Mαs, where s= 1, . . . , k. Analogously to (16) we find that

|Sjf|=

M_αs−1+1−1

∑

v=0

fb(v)ψv

=

s−1

∑

η=0

M_αη+1∑−1 v=Mαη

M_α^α_η α_η ψv

= ∑^s−1

η=0

M_α^α_η αη

(D_Mαη+1−D_Mαη)5 2λM_α^α+1_k−1 αk−1

.

Hence,

|I|5 1 Q_Mαk+Ms

M∑_αk

j=0

qj|Sjf| (17)

5 2λM_α^α+1_k−1 α_k−1

1 Q_Mαk+Ms

M∑_αk

j=0

qj

5 2λM_α^α+1_k−1 α_k−1 .

Letx∈I_s/I_s+1. Since

(18) Dj+Mn =DMn+ψMnDj =DMn+rnDj, when j < Mn,

if we now apply Abel transformation, (15) and inequalities of (8) and (9) we get that

|II|= 1 QM_αk+Ms

M_α^α_k α_k

M_αk∑+Ms

j=M_αk+1

qM_αk+Ms−j

(Dj−DM_αk

)

= 1

Q_Mαk+Ms

M_α^α_k α_k

Ms

∑

j=1

q_Ms−j(

D_j+Mαk −D_Mαk)

= 1

Q_Mαk+Ms

ψ_MαkM_α^α

k

α_k

Ms

∑

j=1

q_Ms−jD_j

= M_α^α

k

α_kQ_Mαk+Ms

Ms

∑

j=1

q_Ms−jj

= c α_k

Ms

∑

j=1

(qMs−j−qMs−j−1

)j²

= cM_s² α_k

Ms

∑

j=[Ms/2]

qMs−j−qMs−j−1

= cM_s² α_k

[M∑s/2]

j=0

|qj−qj+1|

= cM_s² α_k

[M∑s/2]

j=0

j^α−2

= cM_s^α−1M_s²

α_k = cM_s^α+1 α_k .

Let [α_k/2]< s5α_k. Therefore, it yields that

∫

Gm

tM_αk+Msf(x)^1/(1+α)dµ(x) (19)

=|II| − |I|= cM_s^1+α

αk −4λM_α^α+1_k−1

αk−1 = cM_s^1+α αk

.

By combining (17) and (19) we get that

∫

Gm

|t^∗f|^1/(1+α)dµ

=

α_k

∑

s=[αk/2]+1

∫

Is/Is+1

t_Mαk+Msf^1/(1+α)dµ

=c

α_k

∑

s=[αk/2]

M_s

M_sα^1/(1+α)_k =c

α∑_k−3 s=[αk/2]

1 α^1/(1+α)_k

= c α^1/(1+α)_k

αk

∑

s=[αk/2]

1= cαk

α^1/(1+α)_k

=cα^α/(1+α)_k → ∞, as k→ ∞.

The proof is complete.

3. Applications and final remark

Remark1. We note that under the both conditions of (7) in Theorem 1 the conditions (4) in Theorem A can still be fulfilled. So our main result shows that under the both conditions of (7) in part a) of Theorem A are in a sence sharp and the pointp= 1/(1 +α) is the smallest number for which we have boundedness from the Hardy space H_1/(1+α) to the space weak- L_1/(1+α).

Our main result Theorem 1 immediately implies the following results of Goginava [3] and Tephnadze [16]:

Corollary 1 (Goginava). The maximal operator of the (C, α)-means σ^α,∗ is not bounded from the Hardy space H_1/(1+α) to the space L_1/(1+α), where 0< α51.

Corollary 2 (Tephnadze). The maximal operator of the Fej´er means σ^∗ is not bounded from the Hardy spaceH_1/2 to the spaceL_1/2.

Letθ_n^αdenote the N¨orlund mean, where{q₀= 0,q_k=k^α−1 : k=1}, that is

θ_n^αf = 1 Qn

∑n

k=1(n−k)^α−1Skf.

It is easy to see that

|q_n−q_n+1| n^α−2 = 1

n^α−2 (n^α

n − (n+ 1)^α n+ 1

) (20)

5 1 n^α−2

(n^α

n − n^α n+ 1

)

= 1

n^α−2 n^α n(n+ 1) 5 1

n^α−2 2

n^2−α =O(1), as n→ ∞. Since

Qn:=

n−1

∑

k=0

k^α−1=

n−1

∫

1

x^α−1dx=cn^α

we obtain that

(21) n^αq₀

Qn

=O(1), as n→ ∞.

By combining inequalities (20) and (21) we get the following new result:

Corollary 3. The maximal operator of the θ^α_n-means θ^α,∗:= sup

n∈N|θnf|

is not bounded from the martingale Hardy space H_1/(1+α) to the Lebesgue spaceL_1/(1+α), where 0< α51.

Acknowledgment. We thank the referee for some valuable remarks, which have improved the final version of this paper.

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