Sharp Hp-Lp type inequalities of weighted maximal operators of Vilenkin-Nörlund means and its applications

R E S E A R C H Open Access

Sharp H _p -L _p type inequalities of weighted maximal operators of Vilenkin-Nörlund means and its applications

Lasha Baramidze¹, Lars-Erik Persson^2,3*, George Tephnadze^1,2and Peter Wall²

*Correspondence: [email protected]

2Department of Engineering Sciences and Mathematics, Luleå University of Technology, Luleå, 971 87, Sweden

3UiT, The Artic University of Norway, P.O. Box 385, Narvik, 8505, Norway Full list of author information is available at the end of the article

Abstract

We prove and discuss some newH_p-L_ptype inequalities of weighted maximal operators of Vilenkin-Nörlund means with monotone coefficients. It is also proved that these inequalities are the best possible in a special sense. We also apply these results to prove strong summability for such Vilenkin-Nörlund means. As applications, both some well-known and new results are pointed out.

MSC: 42C10; 42B25

Keywords: inequalities; Vilenkin system; Vilenkin group; Nörlund means; martingale Hardy space;Lpspaces; maximal operator; Vilenkin-Fourier series

1 Introduction

The definitions and notations used in this introduction can be found in our next section.

In the one-dimensional case the weak (, )-type inequality for maximal operator of Fejér meansσ^∗f :=sup_n∈N|σ_nf|can be found in Schipp [] for Walsh series and in Pál, Simon []

for bounded Vilenkin series. Fujji [] and Simon [] verified thatσ^∗is bounded fromH

toL_. Weisz [] generalized this result and proved boundedness ofσ^∗from the martingale spaceHp to the Lebesgue spaceLp forp> /. Simon [] gave a counterexample, which shows that boundedness does not hold for  <p< /. In the casep= / a counterexample with respect to Walsh system was given by Goginava [] and for the bounded Vilenkin system was proved by Tephnadze []. Weisz [] proved that the maximal operator of the Fejér meansσ^∗is bounded from the Hardy spaceH_/to the space weak-L_/.

Weisz [] proved that the maximal operator of Cesàro meansσ^α,∗f :=sup_n∈N|σ_n^αf|is bounded from the martingale spaceHpto the spaceLpforp> /(+α). Goginava [] gave a counterexample, which shows that boundedness does not hold for  <p≤/( +α). Simon and Weisz [] showed that the maximal operatorσ^α,∗( <α< ) of the (C,α) means is bounded from the Hardy spaceH/(+α) to the space weak-L/(+α). In [] and [] it was also proved that the maximal operator

σ_p^α,∗:=sup

n∈N

σ_n^αf/

(n+ )^{/p–α–}log(+α)[p+α(+α)](n+ )

©2016 Baramidze et al. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, pro- vided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

is bounded from the Hardy spaceHpto the Lebesgue spaceLp, where  <p≤/( +α).

Moreover, the rate of the weights{(n+ )^{/p–α–}log(+α)[p+α(+α)](n+ )}^∞n= innth Cesàro mean is given exactly.

It is well known that Vilenkin systems do not form bases in the spaceL(Gm). Moreover, there is a function in the Hardy space H(Gm), such that the partial sums off are not bounded inL-norm. Simon [] (for unbounded Vilenkin systems in the case whenp=  see [] and for  <p<  another proof was pointed out in []) proved that there exists an absolute constantcp, depending only onp, such that

 log^[p]n

n k=

Skf^pp

k^–p ≤cpf^pHp ( <p≤) ()

for allf ∈Hpandn∈N+, where [p] denotes the integer part ofp. In [] for Walsh sys- tem and in [] with respect to bounded Vilenkin system it was proved that sequence {/k^–p}^∞_k=( <p< ) in () cannot be improved.

In [] it was proved that there exists an absolute constantcp, depending only onp, such that

 log^[/+p]n

n k=

σkf^pp

k^–p ≤cpf^p_Hp ( <p≤/,n= , , . . . ). () An analogous result for (C,α) ( <α< ) means whenp= /( +α) was generalized in [] and when  <p< /( +α) it was proved in []. In particular, the following inequality:

 log[α/(+α)+p]n

n k=

σ_k^αf^pp

k^{–(+α)p} ≤cpf^p_Hp

 <p≤/( +α),n= , , . . .

holds.

Móricz and Siddiqi [] investigated the approximation properties of some special Nör- lund mean of theLpfunction in norm. For more information on Nörlund means, see the paper of Blahota and Gát [] and Nagy [] (see also [, ], and []).

In [] forp= /( +α) and in [] for  <p< /( +α) there was proved that for every f ∈H_p and for every Nörlund meant_nf, generated by the non-increasing sequence{q_n: n≥}, satisfying the conditions

n^α/Qn=O(), asn→ ∞, ()

and

(qn–qn+)/n^α–=O(), asn→ ∞, ()

there exists an absolute constantc_α,psuch that

 log[α/(+α)+p]n

n k=

t_kf^pp

k^{–(+α)p} ≤cα,pf^p_Hp (n= , , . . . ) ()

and sup

n∈N|tnf|/

(n+ )^{/p–α–}log(+α)[p+α(+α)](n+ )

Hp

≤cα,pfHp. ()

In [] it was proved that in the endpoint casep= /( +α) both () and () conditions are sharp in a special sense.

In this paper we investigate the case when  <p< /( +α) and prove inequalities () and () forf∈Hpand Vilenkin-Nörlund means with non-increasing coefficients, but with weaker conditions than () and (), which give possibility to prove analogous results for the wider class of Vilenkin-Nörlund means when  <p< /( +α). 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 definitions and notations are presented in Section . The main results can be found in Section . For the proofs of the main results we need some lemmas, both well known, but also some new ones of independent interest. These results are presented in Section . The detailed proofs are given in Section . Some well-known and new consequences of our main results are presented in Section .

2 Definitions and notations

Denote byN+ the set of the positive integers,N:=N+∪ {}. Letm:= (m_,m_, . . .) be a sequence of the positive integers not less than . Denote by

Z_mk:={, , . . . ,m_k– }

the additive group of integers modulomk.

Define the groupGmas the complete direct product of the groupsZmiwith the product of the discrete topologies of theZmj.

The direct productμof the measures μ_k

{j}

:= /mk (j∈Zm_k)

is the Haar measure onGmwithμ(Gm) = .

In this paper we discuss bounded Vilenkin groups,i.e.the case whensup_nm_n<∞. The elements ofG_mare represented by the sequences

x:= (x_,x, . . . ,xj, . . .) (x_j∈Zmj).

It is easy to give a base for the neighborhood ofGm: I(x) :=Gm, In(x) :={y∈Gm|y=x, . . . ,yn–=xn–}, wherex∈Gm,n∈N.

DenoteIn:=In() forn∈N+, andIn:=Gm\In.

IN= ^N–

k=

m_k–

s_k=

N–

l=k+

m_l–

s_l=

Il+(skek+slel) ∪

N–

k=

m_k–

s_k=

IN(skek)

. ()

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

M_:= , M_k+:=m_kM_k (k∈N),

then everyn∈Ncan be uniquely expressed asn=_∞

j=njMj, wherenj∈Zmj(j∈N+) and only a finite number of thenjdiffer from zero.

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

rk(x) :=exp(πixk/mk)

i^= –,x∈Gm,k∈N . Now, define the Vilenkin systemψ:= (ψ_n:n∈N) onG_mas

ψ_n(x) :=

∞ k=

rⁿ_k^k(x) (n∈N).

Specifically, we call this system a Walsh-Paley system whenm≡.

The norms (or quasi-norms) of the spaces Lp(Gm)and weak-Lp(Gm) ( <p<∞) are, respectively, defined by

f^pp:=

Gm

|f|^pdμ, f^p_weak-Lp:=sup

λ>

λ^pμ(f >λ).

The Vilenkin system is orthonormal and complete inL_(G_m) (see []).

Next, we introduce analogs of the usual definitions in Fourier-analysis. Iff ∈L_(G_m) we can define the Fourier coefficients, the partial sums of the Fourier series, the Dirichlet kernels with respect to the Vilenkin system in the usual manner:

f(n) :=

Gm

fψ_ndμ (n∈N),

Snf :=

n–

k=

f(k)ψk, Dn:=

n–

k=

ψ_k (n∈N+),

respectively.

Recall that D_Mn(x) =

M_n, ifx∈I_n,

, ifx∈/I_n, ()

and

Dn=ψ_{n ∞}

j=

DMj mj–

k=mj–nj

r^k_j

, ()

forn=_∞

i=niMi.

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 seee.g.[]).

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

n∈N

f⁽ⁿ⁾.

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

p<∞.

Iff = (f⁽ⁿ⁾,n∈N) is a martingale, then the Vilenkin-Fourier coefficients must be defined in a slightly different manner:

f(i) := lim

k→∞

Gm

f^(k)ψ_idμ.

Let{q_k:k≥}be a sequence of nonnegative numbers. Thenth Nörlund means for a Fourier series off is defined by

t_nf :=  Qn

n k=

q_n–kS_kf, ()

where Qn:=

n–

k=

qk.

A representation tnf(x) =

Gm

f(t)Fn(x–t)dμ(t)

plays a central role in the sequel, where

Fn:=  Qn

n k=

qn–kDk

is the so-called Nörlund kernel.

We say that the Nörlund meantnis of (N,α) type if n^α

Qn

=O(), asn→ ∞ ()

and for anyε> , we have Qn

n^α+ε →, asn→ ∞. ()

For our further investigation it is much more convenient to replace condition () by its equivalent one:

Q_n≤cαn^αϕ_n, where lim

j→∞

ϕ_j

j^ε = , for everyε> . ()

We always assume thatq>  andlim_n→∞Qn=∞. In this case it is well known that the summability method generated by{qk:k≥}is regular if and only if

n→∞lim q_n–

Qn

= .

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

Ifqn≡, then we get thenth Fejér mean and the Fejér kernel

σ_nf :=  n

n k=

Skf, Kn:=  n

n k=

Dk,

respectively.

Lett,n∈N. It is well known that (see [])

K_Mn(x) =

⎧⎪

⎨

⎪⎩

, ifx–x_te_t∈/I_n,x∈I_t\I_t+,

Mt

–rt(x), ifx–xtet∈In,x∈It\It+, (Mn+ )/, ifx∈In.

()

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

σ_n^αf :=  A^α_n

n k=

A^α–_n–kSkf,

where

A^α_:= , A^α_n:=(α+ )· · ·(α+n)

n! , α= –, –, . . . .

For the martingalef we consider the following maximal operators:

σ^∗f :=sup

n∈N|σ_nf|, σ^α,∗f :=sup

n∈N

σ_n^αf, t^∗f:=sup

n∈N|tnf|.

We also consider the following weighted maximal operators:

σ_p^∗:=sup

n∈N|σ_nf|/

(n+ )^/p–log^{[p+/]}(n+ )

( <p≤/), σ_p^α,∗:=sup

n∈N

σ_n^αf/

(n+ )^{/p–α–}log(+α)[p+α(+α)](n+ )  <p≤/( +α) , t_p^∗:=sup

n∈N|tnf|/

(n+ )^{/p–α–}log(+α)[p+α(+α)](n+ )  <p≤/( +α) .

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

I

a dμ= , a∞≤μ(I)^–/p, supp(a)⊂I.

3 The main results

Our sharpHp-Lpinequality reads as follows.

Theorem 

(a) Letf ∈Hp,where <p< /( +α)for some <α≤,and{qk:k∈N}be a sequence of non-increasing numbers satisfying conditions()and().Then the maximal operator

∼t^∗_p,α:=sup

n∈N

|t_nf| (n+ )^{/p––α}

is bounded from the martingale Hardy spaceH_pto the Lebesgue spaceL_p,i.e. the following inequality holds:

sup

n∈N|t_nf|/

(n+ )^{/p––α}

p≤c_α,pfHp. ()

(b) Let <p< /( +α)for some <α≤,and{ n:n∈N+}be any non-decreasing sequence,satisfying the condition

n→∞lim

(n+ )^{/p––α}

n

=∞. ()

Then the inequality()is sharp in the sense that there exist a Nörlund mean with non-increasing sequence{qk:k∈N}satisfying the conditions()and()and a martingalef∈HPsuch that

sup

k∈N

^tM_M^nk+fk

nk+weak-Lp

fkHp

=∞.

Our new result concerning strong summability of Nörlund means with non-increasing sequences reads as follows.

Theorem  Let f ∈H_p,where <α< ,  <p< /( +α),and let{q_n:n≥}be a sequence of non-increasing numbers,satisfying conditions()and().Then there exists an absolute constant cα,p,depending only onαand p,such that the inequality

∞ k=

tkf^pp

k^{–(+α)p} ≤cα,pf^pHp

holds.

4 Lemmas

We need the following well-known lemma of Weisz [].

Lemma  Suppose that an operator T isσ-linear and for some <p≤and

– I

|Ta|^pdμ≤cp<∞,

for every p-atom a,where I denotes the support of the atom.If T is bounded from L_∞ to L∞,then

Tfp≤cpfHp.

The next results are due to Blahota, Persson, and Tephnadze [].

Lemma  Let snMn<r≤(sn+ )Mn,where≤sn≤mn– .Then for every Nörlund mean, without any restriction on the generative sequence{qk:k∈N}we have the following equal- ity:

QrFr=QrDsnMn–ψsnMn–

snMn–

l=

(qr–snMn+l–qr–snMn+l+)lKl

–ψ_snMn–(s_nM_n– )qr–K_snMn–+ψ_snMnQr–snMnFr–snMn. We also need the following new lemmas of independent interest.

Lemma  Let <α≤and{qn:n≥}be a sequence of non-increasing numbers satisfying conditions()and().Then

|Q_nF_n| ≤cα

_|n|

j=

M_j^αϕ_j|K_Mj|

,

where

j→∞lim ϕ_j

j^ε = , for everyε> . ()

Lemma  Let <α≤and{qn:n≥}be a sequence of non-increasing numbers,satisfy- ing conditions()and().If r≥M_N,then

IN

Fr(x–t)dμ(t)≤cαM^α_lϕ_lMk

r^αMN ≤cαM^α_lϕ_lMk

M^+α_N , x∈Il+(skek+slel), where

≤sk≤mk– , ≤sl≤ml–  (k= , . . . ,N– ,l=k+ , . . . ,N– ) and

IN

Fr(x–t)dμ(t)≤cαMk

M_N , x∈IN(skek), where

≤sk≤mk–  (k= , . . . ,N– ).

5 Proofs

Proof of Lemma Let  <α≤ and{q_k:k≥}satisfy the conditions () and (). Since n^αϕ_n≥Qn≥nqn–

we obtain

q_n–≤n^α–ϕ_n, ()

whereϕ_nsatisfies condition ().

By using an Abel transformation we get

Qn= n–

j=

(qj–qj+)j+qn–(n– ) +q ()

and n–

j=

|qj–qj+|j≤Qn≤n^αϕ_n. ()

Suppose that

|qj–qj+| ≥j^α–φjδj,

for allj∈N, whereδ_jis any function, such that

j→∞limδ_j=∞. ()

Under condition () there exists an increasing sequence{α_k:k≥}, such thatα_k+≥

αkand δ_αk↑ ∞.

Hence,

α_k++

j=α_k

|qj–qj+|j≥cφα_kδα_k α_k++

j=α_k

j^α–

≥cφα_kδα_k

α_k++

α_k

x^α–dx

≥ cφα_kδα_k

α x^{α αk+}

α_k

≥cφα_kδα_kα_k^α. ()

By combining () and () we get

Qα_k++

(αk++ )^α(φα_k++ )≥

α_k++

j= |qj–qj+|j

(αk++ )^α(φα_k++ )≥cδα_k→ ∞, ask→ ∞. This is a contradiction with condition (), that is,

Qn

n^αφ_n =O(), asn→ ∞. ()

It follows that

|qj–qj+| ≤j^α–φj. ()

It is easy to see that

Qk|DsMn| ≤cM^α_nφ_n|DsMn| ()

and

(sM_n– )q_k–|K_sMn–| ≤cφ_kk^α–M_n|K_sMn–| ≤cM^α_nφ_n|K_sMn–|. () Let

n=snMn+snMn+· · ·+snrMnr, n>n>· · ·>nr, and

n^(k)=snk+Mnk++· · ·+snrMnr, ≤snl≤ml– ,l= , . . . ,r.

By combining ()-() and Lemma  we have

|Q_nF_n|

≤cαM_n^α_φ_n|DsnMn|+cα snMn–

l=

n^()+lα–φ_n()+l|lKl| +cαM_n^α_φ_n|K_sn

Mn–|+cα|Q_n()F_n()|. By repeating this processrtimes we get

|QnFn|

≤cα

r k=

M^α_nkφ_nk|Ds_nkM_nk|+

s_nkM_nk–

l=

n^(k)+lα–

φ_n(k)+l|lKl|+M_n^α_kφ_nk|Ks_nkM_nk–| :=I+II+III.

By combining (), (), and () we find that

I≤cα

|n|

k=

M_k^αφ_k|DMk| ≤cα

|n|

k=

M^α_kφ_k|KMk|

and

III≤cα

r k=

M^α–_nk φ_nk|MnkKs_nkM_nk–Ds_nkM_nk| ≤cα

r k=

M^α_kφ_nk|KMk|.

Moreover,

II =cα

r k=

nk

A=

sAMA–

l=sA–MA–

n^(k)+lα–

φ_n(k)+l|lK_l|

=cα

r k=

nk+

A=

sAMA–

l=s_A–M_A–

n^(k)+lα–

φ_n(k)+l|lK_l|

+cα

r k=

nk

A=n_k++

sAMA–

l=s_A–M_A–

n^(k)+lα–

φ_n(k)+l|lK_l|

≤cα

r k=

M_n^α–_k+φ_nk+

n_k+

A=

s_AM_A–

l=sA–MA–

|lKl|

+cα

r k=

n_k

A=nk++

M^α–_A φ_A

s_AM_A–

l=sA–MA–

|lKl|:=II+II.

By applying () forIIwe get

II≤cα

r k=

M_n^α–_k+φ_nk+

nk+

A=

sAMA–

l=sA–MA–

A j=

Mj|KMj|

≤cα n

k=

M_k^α–φ_k k A=

MA

A j=

Mj|KMj|

≤cα n

k=

M_k^α–φ_k k

j=

Mj|KMj|

=cα n

j=

Mj|KMj|

n

k=j

φ_kM^α–_k ≤cα n

j=

φ_jM^α_j|KMj|.

By using () forIIwe have similarly

II≤cα

r k=

n_k

A=nk++

M_A^α–φ_A A

j=

Mj|KMj|

≤cα n

A=

M_A^α–φ_A A

j=

M_j|K_Mj| ≤cα n

j=

φ_jM^α_j|K_Mj|.

The proof is complete by combining the estimates above.

Proof of Lemma Letx∈Il+(skek+slel), ≤sk≤mk– , ≤sl≤ml– . Then, by applying (), we have

KMn(x) = , whenn>l>k.

Suppose thatk<n≤l. Moreover, by using () we get KMn(x)≤cMk.

Letn≤k<l. Then

KMn(x)= (Mn+ )/≤cMk.

If we now apply Lemma  and () we can conclude that

QrFr(x)≤cα

l A=

M_A^αϕ_AKMA(x)≤cα

l A=

M^α_Aϕ_AMk≤cαM^α_lϕ_lMk. ()

Letx∈Il+(skek+slel), for some ≤k<l≤N– . Sincex–t∈Il+(skek+slel), fort∈IN

andr≥MN from () we obtain

IN

Fr(x–t)dμ(t)≤cαM^α_lϕ_lM_k

r^αM_N . ()

Letx∈I_N(s_ke_k),k= , . . . ,N– . Then, by applying () and () we have

IN

F_r(x–t)dμ(t)≤  Qn

n m=

q_n–m

IN

D_m(x–t)dμ(t)

≤  Qn

n m=

qn–m

cM_k MN ≤cM_k

MN

. ()

By combining () and () we complete the proof of Lemma .

Proof of Theorem According to Lemma  the proof of the first part of Theorem  will be complete, if we show that

IN

^∼t^∗,pa(x)^pdμ(x) <∞,

for every /( +α–ε)-atoma. We may assume thatais an arbitraryp-atom with support I,μ(I) =M^–_N andI=IN. It is easy to see thattn(a) = , whenn≤MN. Therefore, we can suppose thatn>MN.

By using Lemma  we easily see that^∼t^∗,pis bounded fromL∞toL∞. Letx∈IN. Since a∞≤M^/p_N we obtain

t_na(x)≤

IN

a(t)F_n(x–t)dμ(t)

≤ a∞

IN

Fn(x–t)dμ(t)≤M^/p_N

IN

Fn(x–t)dμ(t).

Letx∈I_l+(s_ke_k+s_le_l), ≤k<l<N. From Lemma  we get tna(x)≤cαM^α_lϕlMkM_N^/p

n^αMN ≤cαϕlM^α_lMk

n^αM^–/p_N ≤cαϕlM_l^αMk

M_N^{+α–/p} . ()

Letx∈IN(skek), ≤k<N. According to Lemma  we have tna(x)≤cαM^/p_N Mk

M_N ≤cαM_N^/p–Mk. ()

By combining () and ()-() we obtain

IN

t^∼∗,pa^pdμ

=

N–

k=

mk–

sk=

N–

l=k+

ml–

sl=

Il+(skek+slel)

sup

n>MN

tna n^{/p––α}

^pdμ +

N–

k=

mk–

s_k=

IN(s_ke_k)

sup

n>MN

tna n^{/p––α}

^pdμ

≤cα N–

k=

N–

l=k+

(m_k– )(m_l– ) Ml+

ϕ_lM^α_lMk

p

+cα N–

k=

(m_k– ) MN

M^αp_NM^p_k

≤cα N–

k=

N–

l=k+

(ϕlM^α_lMk)^p Ml+

+cα N–

k=

M_k^p M^(–αp)_N

≤cα N–

k=

M_k^p N–

l=k+

ϕ_l^p

M^–αp_l +cα≤ N–

k=

M^p_k ϕ_k^p M_k^–αp +cα

≤cα N–

k=

ϕ_k^p

M^{–p(α+)}_k +cα≤cα N–

k=

ϕ^p_k

k(–p(+α))+cα≤cα<∞.

The proof of part (a) is complete.

Under condition () there exist positive integersnksuch that

k→∞lim

(M_nk+ )^{/p––α}

Mnk+

=∞,  <p< /( +α).

Lettnbe Nörlund mean with non-increasing sequence{qk:k∈N}satisfying () and condition (), but in the restricted form

cα(Mn_k+ )^{/p––α}

ϕ_{Mnk+ Mnk+} → ∞, ask→ ∞.

Set

fk:=DMnk+–DMnk. Then

fk(i) =

, i=Mn_k, . . . ,Mn_k+– ,

, otherwise, ()

and

Sifk=

⎧⎪

⎨

⎪⎩

Di–DMnk, i=Mn_k+ , . . . ,Mn_k+– , fk, i≥Mn_k+,

, otherwise.

()

Moreover,

fkHp≤λM_n^–/p_k , ()

whereλ=sup_nmn. By using () we get

|tMnk+fk|

Mnk+

= q|SMnk+| QMnk+ Mnk+

=q_|D_Mnk+–D_Mnk| QMnk+ Mnk+

= q_|ψ_Mnk|

QMnk+ Mnk+ ≥ cα

M^α_nk Mnk+ϕ_Mnk+. Hence,

μ

x∈Gm:|tMnk+fk(x)|

Mnk+

≥ cα

M^α_nk Mnk+ϕ_Mnk+

= . ()

By applying () we have

cα

M_^α_nk Mnk+ϕMnk+(μ{x∈Gm:^{|tMnk+fk(x)|}

Mnk+ ≥_Mα ^q

nk Mnk+ϕMnk+})^/p fkHp

≥ cαM^{/p––α}_nk

Mnk+ϕ_Mnk+ ≥cα(M_nk+ )^{/p––α}

Mnk+ϕ_Mnk+ → ∞, ask→ ∞.

The proof is complete.

Proof of Theorem According to Lemma  the proof of Theorem  will be complete, if we show that

∞ m=

tma^pp

m^{–(+α)p} ≤cα<∞,

for everyp-atoma. Analogously to the first part of Theorem  we can assume thatn>MN

andabe an arbitraryp-atom, with supportI,μ(I) =MN, andI=IN. Letx∈IN. Sincea∞≤cM^/p_N if we apply Lemma  we obtain

IN

|tma|^pdμ≤

IN

Fm^p_a^p_∞dμ≤cα,pϕ^p_m

IN

a^p_∞dμ≤cα,pϕ^p_m. Hence

∞ m=MN+

IN|tma|^pdμ m^{–(+α)p} ≤cα,p

∞ m=MN+

ϕ^p_m

m^{–(+α)p} ≤ cα,pϕ^p_N

N(–(+α)p) ≤cα,p<∞.