Cent. Eur. J. Math. • 12(8) • 2014 • 1214-1228 DOI: 10.2478/s11533-014-0406-1

VERS ITA

Central European Journal of Mathematics

Walsh-Marcinkiewicz means and Hardy spaces

Research Article

Karoly Nagy1*, George Tephnadze2,3^

1 Institute of Mathematics and Computer Sciences, College of Nyiregyhaza, P.O. Box 166, Nyiregyhaza 4400, Hungary

2 Department of Mathematics, Faculty of Exact and Natural Sciences, Tbilisi State University, Chavchavadze St. 1, Tbilisi 0128, Georgia

3 Department of Engineering Sciences and Mathematics, Luleä University of Technology, 971 87 Luleä, Sweden

Received 12 July 2013; accepted 21 November 2013

Abstract: The main aim of this paper is to investigate the Walsh-Marcinkiewicz means on the Hardy space Hp, when 0 < p < 2/3. We define a weighted maximal operator of Walsh-Marcinkiewicz means and establish some of its properties. With its aid we provide a necessary and sufficient condition for convergence of the Walsh-Marcinkiewicz means in terms of modulus of continuity on the Hardy space Hp, and prove a strong convergence theorem for the Walsh-Marcinkiewicz means.

MSC: 42C10

Keywords: Walsh system • Marcinkiewicz means • Maximal operator • Two-dimensional system • Hardy space • Strong convergence • Modulus of continuity

© Versita Sp. z o.o.

1. Introduction

Polntwlse convergence problems are of fundamental Importance In harmonic analysis, and as Is well known they are closely related to studying boundedness of associated maximal operators. In the paper we will be concerned with maximal operators for Walsh series. Let us first recall in brief historical development of the theory.

The a.e. convergence of Walsh-Fejer means o„f was established by Fine in 1955 [4]. This result was generalized later by Schipp [16] for Walsh series and Pal and Simon [15] for bounded Vilenkin series by showing that the corresponding maximal operators o* satisfy the weak type (1,1) inequality. Fujii [5] and Simon [18] verified for Walsh and Vilenkin series that o* is bounded from H1 to L1. Weisz [28] generalized this result for Walsh series and proved the boundedness

* E-mail: nkaroly@nyf.hu f E-mail: giorgitephnadze@gmail.com

Springer

of maximal operator from the martingale space Hp to the space Lp for p > 1/2. Simon [20] gave a counterexample for Walsh series showing failure of boundedness for 0 < p < 1/2, unboundedness at the endpoint p = 1/2 was proved by Goginava [8] (see also [2, 3]). In [22] Tephnadze proved that there exists a martingale f e H1/2 such that the Vilenkin-Fejer means of f are not uniformly bounded in the space L1/2.

In [9, 23, 24] it was shown for Walsh and Vilenkin series that a modified maximal operator op defined by

a* = sup

eN (n + 1)1/p—2 log

,2[1/2+p]/

o <P < 2

where [1/2 + p] denotes the integer part of 1/2 + p, is bounded from the Hardy space Hp to the space Lp. Moreover, it was proved that the sequence {(n + 1)1/p—2 log2[1/2+p](n + 1)}is essential.

Recently, Tephnadze [26] gave a necessary and sufficient condition for the convergence of Walsh-Fejer means in terms of modulus of continuity on the Hardy space Hp. Namely, he proved that if 0 < p < 1/2 and

MHp\ 2*'f) \ 2*№-2)F[1/2+p]

then ||on(f) — f ||hp —0 as n —> oo. Moreover, there exists a martingale f e Hp(G), 0 < p < 1/2, for which

WHp I 2*,f) 0( 2k1/p—2k2[1/2+p]

as k ^ <x,

and ||on(f) — f ||p ^ 0 as n ^ &>.

In [19] Simon proved for Walsh series that, if f e H(G), then

n^™ log N L—

where Sn(f) stands for the n-th partial sum of Walsh-Fourier series of f. The analogous result with respect to Vilenkin systems was established by Gat in [6], and next generalized to Vilenkin-like systems by Blahota [1]. Simon generalized his result to Hp, 0 < p < 1, in [21] showing that there exists a constant Cp such that the inequality

|Sk (f )||

k 2—p

p < CPH

holds for any function f e Hp(G), where 0 < p < 1. For 0 < p < 1/2 and f e Hp(G) Tephnadze [25] proved for the Walsh system that there exists an absolute constant cp, depending only on p, such that

[1/2+p] ,

R (f )||Hp

k 2—2p

For two-dimensional Walsh-Fourier series Weisz [30] proved that the maximal operator M* of the Marcinkiewicz means is bounded from the two-dimensional dyadic martingale Hardy space Hp(G2) to the space Lp(G2) for p > 2/3. Whereas for p = 2/3 Goginava [8] (see also [12]) showed unboundedness of M*. By the interpolation technique it can be shown that M* is not bounded from the Hardy space Hp(G2) to the space weak-Lp(G2) for 0 < p < 2/3. But as it was shown in [10], M* is bounded from H2/3(G2) to weak-L2/3(G2).

Nagy in [14] considered the following modification of M*:

eN log3/2(n +1)'

MM* = sup

and showed that It Is bounded from the Hardy space H2/3(C2) to the space L2/3(G2). He also proved that the sequence {log3/2(n +1)}Is Important In the definition (4) of the maximal operator M*, namely, that the rate of the deviant behaviour of the n-th Marcinkiewicz mean is exactly log3/2(n +1).

The main aim of this paper is to investigate behaviour of the Walsh-Marcinkiewicz means on the Hardy space Hp(G2), when 0 < p < 2/3. In Section 2 we introduce notions and notations, and give more precise historical notes on the investigations of Walsh-Fejer and Walsh-Marcinkiewicz means, this section also illustrates our motivation. In Section 3 we study the rate of the deviant behaviour of the n-th Marcinkiewicz mean. Next, as an application, following ideas of [26] for one-dimensional Fejer means (see conditions (1) and (2)), we give a necessary and sufficient condition for the convergence of Walsh-Marcinkiewicz means in terms of modulus of continuity on Hp(G2), Section 4. With the aid of some useful inequalities given in the proof of our main theorem, in Section 5 we prove a strong convergence theorem for Marcinkiewicz means. That is, we give the two-dimensional version of inequality (3) (see [25]).

2. Definitions and notations

We give a brief introduction to the theory of dyadic analysis, see also [17, 27]. Let N+ denote the set of positive integers, N = N+ U {0}. Denote by Z2 the discrete cyclic group of order 2, that is Z2 = {0,1}, where the group operation is the modulo 2 addition and every subset is open. The Haar measure on Z2 is given so that the measure of a singleton is 1/2. Let G be the complete direct product of countable infinite copies of the compact group Z2. Then elements of G are sequences x = (x0,x^,...,xk,...) with coordinates xk G {0,1}, k G N. The group operation on G is the coordinatewise addition, the measure (denoted by and the topology are the product measure and topology. The compact Abelian group G is called the Walsh group. A base for the neighbourhoods of G can be given in the following way:

Io(x) = C, In(x) = i„(xo.....x„-i) = {y G G : y = (xq.....x„-i,y„,y„+i,... )},

x G G, n G N. These sets are called dyadic intervals. Let 0 = (0 : i G N) G G denote the null element of G, and In = In (0), n G N. Set en = (0,..., 0,1,0,...) G G, the n-th coordinate of which is 1 and the rest are zeros, n G N. For k G N and x G G denote by

rk (x) = (-1)xk

the k-th Rademacher function. If n G N, then we can write n = ^= ni2i, where ni G {0,1}, i G N, i.e. n can be expressed in the number system of base 2. Define the order of n by |n| = max{j G N : nj = 0}, we have 2|n| < n < 2|n|+1. The Walsh-Paley system consists of Walsh-Paley functions:

w„(x) = n^(x))„k = (-1)^=0„kxk, x G G, n G N.

The Dirichlet kernels for the Walsh-Paley system are defined as usually by

D„ = ^ wk, Dq = 0.

One can easily check that (see e.g. [17])

D2„ (x) =

2„ If x G I„, 0 If x (tin,

D„ = w^„krkD2k = w„2>(D2k+i - D2k), „ = „Ï21.

The norm (or quasinorm) of the space Lp is defined by in the usual way for 0 < p < «. The space weak-Lp consists of all measurable functions f for which

¥ ||weak-Lp = sup A^Vp(f > A) < +«.

Denoting by SM(f) = YLi=—1 f('W the partial sums of the Walsh-Fourier series of f, the n-th Fejer means and kernel of the Walsh-Fourier series of f are defined as

an(f) = ! y~Sj(f), Kn = ! y~Dk.

nU nk~0

The a-algebra generated by the dyadic 2-dimensional square /n(x1) x /n(x2) of measure 2—n x 2—n will be denoted by Fnn, n e N. Denote by f = (f(n n) : n e N a one-parameter martingale with respect to Fnn, n e N. The Kronecker product (wnm : n, m e N) of two Walsh systems is called the two-dimensional Walsh system, it consists of functions

Wnm(x1,x2) = Wn(x1) Wm(x2).

For f e Li(G2) the number f(n, m) = Jg2 fwnmdy, n,m e N, is called the (n, m)-th Walsh-Fourier coefficient of f. Let f be a martingale. Denote by Snm(f) the (n, m)-th rectangular partial sum of Walsh-Fourier series of f:

n—1 m— 1

Snm(f ) = VVfM Wj.

i=0 j=0

A bounded measurable function a is called a p-atom if there exists a dyadic 2-dimensional cube /2 such that

fadv = 0, M«< V—Vp(/2), supp a c I2.

The dyadic Hardy martingale spaces Hp(G2), 0 < p < 1, have the following atomic characterization.

Theorem W (Weisz [29]).

A martingale f = (f(n n) : n e N is in Hp(G2), 0 < p < 1, if and only if there exists a sequence (ak : k e N) of p-atoms

and a sequence (vk : k e N) of real numbers such that for every n e N,

y_VkS2n2n (ak ) = f(n,n) (7)

and lvklp < «. Moreover, ||f HHp ~ inf (^«=0 lvklp)^/p, where the infimum is taken over all decompositions of the

form (7)k. 0 p k 0

For f e Hp(G2), 0 < p < 1, the value uHp (1/2n, f) = ||f — S2n2n (f)||H is called the modulus of continuity of f. The Marcinkiewicz-Fejer means of a martingale f are defined as

Mn(f) = - VlSkk(f).

The 2-dimensional Dirichlet kernels and Marcinkiewicz-Fejer kernels are defined as

Dki(x1 ,x2) = Dk(x1)Dt(x2), Kn(x\x2) = n V Dkk(x1,x2).

Consider the following maximal operators:

M*(f) = sup |Mn(f)|, M*p(f) = sup

Mn(f )

n2/p-3

M#(f) = sup |M2n(f)|.

For the maximal operator M# Goginava proved the following theorem.

Theorem G (Goginava [11]).

The maximal operator M# is bounded from the Hardy space H1/2(G2) to the space weak-L^2(G2) and is not bounded from the Hardy space Hp(G2) to the space Lp(G2) for 0 < p < 1/2.

For the martingale f = ^ (f(n,n) — f(n-i,n-i)) the conjugate transforms are defined as

№ = ^ rn(t){f{n,n) - f(n-1,n-1)).

where t e G is fixed. Note that f(°) = f. As it is well known (see [27]),

Hp(G2) = W \\hp(G2)'

^Hp(C2)

3. Properties of the maximal operator MM

Theorem 3.1.

(a) Let 0 < p < 2/3, then the maximal operator M*,p is bounded from the Hardy space Hp(G2) to the space Lp(C2).

(b) Let y : N ^ [1, to) be a nondecreasing function, satisfying the condition

n2/p-3 Urn —r— = +to.

n^TO y(n)

Mn(f )

To prove this theorem we would need the following lemmas of Goginava, Glukhov and Weisz.

Lemma 3.2 (Goginava [10, Lemma 7]).

Let (x 1,x2) e (/¡1 \/¡1+1) x (Im2 \/m2+1) and 0 < Z1 < m2 < N, then

f \Kn{x1 + t\x2 + t2)|d^(t1,t2) <

J In xIn

J1-m2 ^ -^r1

2L'-m ^2r1 D2W2+, (x1 + et1 + er1)

r1=i1+1

YL D2S (x2 + em2 + ^s-,^'1- £ D2S (x1 + et1 + er1 )

r=m2+1 s=/1 r1 ='1 +1

for n > 2N, where xy = ^2.ls=ixses, xiti-1 = 0.

weak-L

Lemma 3.3 (Goginava [10, Lemma 9]).

Let (x\x2) e IN x {Im2 \/m2+1) and 0 < m2 < N. Then

2m2 N—1

f1,X2 + t2)|d^(t\t2) < C ^^(x^ 6m2) for n> 2N.

J/n x/n 2 1

Lemma 3.4 (Glukhov [7]).

There exists a constant c such that

sup ( ^K„[xV2)1 d^(x1,x2) < C.

Lemma 3.5 (Weisz [31]).

Suppose that an operator T is sublinear and p-quasilocal for any 0 < p < 1. If T is bounded from Lx to L^, then

\\Tf \\p < Cp \\f \\Hp for all f G Hp.

Proof of Theorem 3.1. To prove the first part we follow the method of Nagy from [14]. Due to Lemma 3.4, M* p is bounded from the space L^ to the space L^. Let a be an arbitrary p-atom with support I2, and p(I2) = 2-2N. Without loss of generality, we may assume that I2 = IN x IN. By Lemma 3.5, to establish the result it is enough to show that the maximal operator M*,p is p-quasilocal. That is, there exists a constant cp such that

\ |M*,p(a)|Pd^ < cp < oo.

It is evident that Mn(a) = 0 if n < 2N. Therefore, we set n > 2N. As \\a\\^ < 22N/p we immediately have

|Mn(a;x1 ,x2)| ^

n2/p-c 22N/p

11N x/N

n2/p-3 < j-3 f |°(i1.i2)||Kn(x1+ f1,x2 + t2)ld^ ,t2)

/N x /N

C 22N/P r

< -2^/ |Kn (x1 + t1,x2 + t2)|d^(t\t2)

n 2 /p 3 /N x /N

_ C 22N/p r

|MM*p(«)| < ^p—N sup / |Kn(x1+ t1,x2 + t2)|d^(t1,t2).

2( ' n>2NJ/Nx/N

We may write

f_\M*,p(a)\"dv =\ _|M*p(a)|p^ + ( \M*,p(a)\"dv +[_ _ |M*,p(a)|p dp = K + K2 + K3.

J in xIN JIN xIN JIN xIN JIN xIN

First, we estimate K (the estimate of K2 goes analogously). Denote Jt = It \ It+i, t G N. Decompose IN and Jm2 as

N—1 N

„2 „2

/N = [J Jm2, Jm2= U C • (10)

m2=0 q2=m2+1

'/„2+1 (0.....0,xm2=1, 0.....0,x,2 = 1) for m2 <q2 <N,

' Jn (0.....0,xm2=1,0.....01 for q2 = N.

im2,q2 _ /

Now, fix (x1,x2) e /N x N 'q . Using Lemma 3.3, we get

|Mn(o; x1,x2)| c 22N(1/p-1)2

n2/p-3

c 22n(1 /p-1) 2'

n2/p-3

ED2s(x2 + emO < n2/p-3 E2 <

c 22N(1/p-1)2m2+g2 nÜP-

Hence,

;w*,p(o) < sup

c 22N(1/p-1)2m2+g2

< C2N+m2+g2

< cp E E m2g22p(N+m2+g2) d^(x1,x2)

m2=0 g2=m2+1 Inxi ^

N-1 N r N-1

< cp2<P-1>N E E I 22P<m2+g2>du(x2) < cp2<P-1>N E 2m2(2P-1) < cp.

m2=0 g2=m2+1 Im ^ 9 "

Now, consider K3.

N—1 Z1—1

I1 =0 m2=0~*J11 xJm2 I1 =0 m2=0 V ^ XJm2

Consider K3,2 (the estimate of K3,1 goes analogously). Denote

N-1 N-1 r N-. l -:r N-. N-1 r

EM ^»pu = EE / \;M*p(o)\Pd^+EE/ ^»Ru = kU + K3,2.

,1 „ 1 „A 1 xj... 2 ,1 „ 1 „-J 1 xj... 2 '1=0 m2 = '1 J'1 xJm2

K3,2 =

\ \;w*,p(o)\f

J xjm2

To estimate K32m , we use Lemma 3.2 and inequality (9).

i 2 c 22 N 1 Kl ,m2 < cP2 '

3,2 ^ 2N(2-3p) 23N

Since,

we obtain

and hence

m2+1 N ,

2('1-m2)p E 2r1p E / D2Pm2+1 {x1+ ei1+ er1 ) DPs {x2 + em2 + xm,2+1,s-1 ) du(x1,x2)

r1=l1+1 s=m2+1 l1 xJm2

m2 s r

+ 2<l1+m2)p E E Dps {x1+ ei1+ er1) du(x1 ,x2)

s='1 r1=/1+^Jl1 xJm2

s='1 rW+1 ^"l1 x m2

f D2Pm2+1 {x1 + el1+ er1 ) DPps {x2 + em2 + xi^ ) du{x1,x2) < cp2m2<p-1>2s<p-1>

JJl1 xJm2 2

f Dps {x1 + et1 + er1) du(x1 ,x2) < cp2-m22s<p-1>,

J xJm2

kïf < cp2i1p 22m2<P-1> + cp2l1<2P-1>2m2<P-1>

N-1 N-1 N-1 N-1 N-1

K32 < cp E E 2l1P+2m2<P-1> + cp E E 2l1(2P-1)+m2(P-1) < cp E2l1<3P-2> < cp.

l1=0 m2=l1 l1=0 m2=l1 l1=0

This completes the proof of the first part of our theorem.

Now we turn to the second part. Let ( : N —> [1, oo) be a nondecreasing function and ak be a sequence of natural numbers satisfying the condition

{2"k + 1)2/p-3 Urn -—-—— = +00.

k-œ ((2ak + 1)

Set fk (x1,x2) = (D2%+i(x1) - Û2*k (x1))(D2ak+i(x2) - Û2*k (x2)). It is evident that

fk (i,j ) =

1 if i,j = 2ak.....2ak + 1 - 1,

0 otherwise.

Su(fk ; *V2) =

From (5) we conclude

(Di(x1) - DVk (x1)) (D,(x2) - Dv.k (x2)) if i = 2ak.....2ak+1 - 1,

fk(x1,x2) if i > 2ak+1,

0 otherwise.

sup S2»2" (fk)

= \\D2ak+1 - D2«k \\p < 22ak(1-1/p).

Let x = (x1,x2) G G2. By (14) we can write |M2«k+1(fk; x1,x2)| 1

2ak + 1

£ Sjj(fk;x1,x2)

y(2ak +1) (p{2ak + 1)(2ak +1)

|(D2ak +1 (x1 ) - D2ak (x1)) (D2ak + 1(x2) - D2'k (x2)) | ^k (x1 ) W-pk (x2)|

((2ak + 1)(2ak + 1)

((2ak + 1)(2ak +1) ( (2 ak + 1)(2ak +1)'

This yields immediately 1

( (2 ak + 1)(2ak +1)

(J(x\x2) G G2: |M2ak ; x1,x2)| > \ ( ) ((2ak +1)

((2ak + 1)(2ak +1)

(2 ak + 1)2/P-3

> y(2ak + 1)(2ak + 1)22ak(1-Vp) > C ' ((2ak + 1) ^ °° as ^ ^

This completes our theorem.

4. Application 1: necessary and sufficient condition for the convergence in terms of modulus of continuity

Theorem 4.1.

(a) Let 1/2 <p< 2/3, f G Hp (G2) and

2k, f =0 Uki-)1 as k -œ,

then ||Mn(f) - f\\Hp — 0 as n — œ.

(b) Let 0 < p < 2/3, then there exists a martingale f e Hp(G2) such that

MHp[ ^Ujp-s) I as k

and ||Mn(f) — f ||weak-Lp ^ 0 as n ^ «.

Proof. Let 0 < p < 2/3. Combining Theorem 3.1 and (8) we get

||Mn(f)||Hp < cJj\(mM%dt < C^||Mn(W)||pdf

< cn2-3p [\\№\\pHdt = cn—p f Iff dt = cn-p

Jg p Jg

Let 2N < n < 2N+1. Using inequality (15) and Theorem 3.1 we have

||Mn(f ) - f IlHp < \\Mn (f ) - Mn(S2« 2N (f ))\\Hp + \\Mn(S2N 2N (f )) - S2N 2« \\Hp + 1^2« 2« (f ) - f\\

= \\Mn (S2N2N (f ) - f )\Hp + \\S2N2« (f) - f\Hp + \Mn(S2N2N(f)) - S2N2« (f)\\H,

< C ( n2-3p + 1 ) wHp( + |Mn (S2N 2« (f )) - S

>ZN2« (f )|Hp.

We have

2« n 1 2 1 n

Mn(S2«2« (f )) - S2« 2« (f ) = E Skk (S2« 2« (f )) + ^ Skk 2« (f )) - S2« 2« (f )

n k=0 n k=2«+1 2«

= 1 £ Skk(f) + (n - 2«)S2«2«(f) - S2«2« (f) = ^ (M2«(f) - S2«2« (f))

2« 2«

— (S2«2«(M2«(f)) - S2«2«(f)) = — S2«2«(M2«(f) - f).

Let 1/2 < p < 2/3. Combining (8) and Theorem G, and following the steps of estimate (15), we obtain

I ?n \p

||Mn(S2N2N (f)) — S2N2N (f )|Hp < (V) IS2N2N (M2N (f) — f )|Hp < Cp||M2N (f) — f || Hp ^ 0 as N ^ «. Summarising, if

MHp{ 2¡N,H = 0 ^№-3)1 as «

then ||Mn(f) — f Hhp 0 as n -> «.

Now, we prove the second part of the theorem. Denote

a,(x\x2) = 22'(1/p-1>(D2,+1(x1) - 02,(x1))(d2,+1(x2) - 02,(x2)) and W*V2) = £ jg-32)

Clearly,

! ^ ak(x1,x2) If k < A, = 0 If k > A.

S2a2a (ak; x , x ) =

Theorem W yields f e Hp(G2). Now, we write

f - S2"2" (f) = (f(1,1) — S2" 2" (f(1,1))> • • • , f(n,n) - S2"2" (f(n,n)), ■ ■ ■ , f(n+k,n+k) - S2"2" (f(n+k,n+k)), ■ ■ ■ j

/ n+k \

= (0.....0, f(n+1,n+1) - f(n,n).....f(n+k,n+k) - f(n,n), ■■■) = 0.....0..... Yi 22(1/P-32 ,■■■ ,

\ i=n+1 I

k e N+. By Theorem W,

I 1 \ ^ 1 I 1

WHA 2", f - ^ 22i(1/p-3/2) = 0

2" I I— 22i(1/p-3/2) \ 22n(1/P-3/2) I ■

It is easy to check that

f (i,j ) =

2" if (i,j) e {2n.....2n+1 - 1}2, n = 0,1.....

0 if (i,j) e (J{2n.....2n+1 - 1}2■

Hence,

M2"+1(f) = 2""^ E Sjj (f) = 2"TT M2" (f ) + 2""TT S2"+1,2" +1(f)

2n 1 2n

= ~-7 M2n (f) + --- S2n2n (f) + --- W2n2n

2" +1 2 ( ) 2" +1 2 2 ( ) 2" +1 22

2" +1 (f) - f ||weak-Lp > II W2"2" |weak-Lp - 2—, 11(f) - f |LkL - ^^ 11^2" (f) - '

■lp - 2" +1 11 22 11 " 2" +1 Thus,

Um sup ||M2"+i (f) - f ||weak_Lp > c " °(1) > c > 0-This completes the proof. □

5. Application 2: strong convergence theorem

We would need the following lemma from [13].

Lemma 5.1 (Nagy [13]).

Let A,s,l e N, s < I < A, (x1 ,x2) e (is \ 4+1) X (// \ I+), then

K2a (x1,x2) =

0 if there exists i e B1 such that x1 = x2,

0 if for all i e B1, x1 = x2 and there exists m e B2 such that x1 - es - em e //+1, x\ = 1,

2s+m-2 if for all i e B1, xj = x2 and there exists m e B2 such that x1 - es - em e //+1, x1m = 1,

_22s-1 if x1 - es e /+ and for all i e B1, x1 = x2,

where B1 = {I + 1.....A - 1}, B2 = {s + 1.....I}.

Now, we formulate our strong convergence theorem.

Theorem 5.2.

(a) Let 0 < p < 2/3, then there exists an absolute constant cp such that

||Mm(f )|Hp

„3-3p ^ CpU' ||H

for all f e Hp(G2).

(b) Let 0 < p < 2/3 and 0: N+ —> [1,oo) be any non-decreasing function, satisfying the conditions 0(n) |to and

_2k(3-3p)

lim , , = to.

k^TO 0(2k)

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

||Mm(f) |weak-Lp

Proof. The operator Mn is bounded (see Lemma 3.4) from the space L« to the space L«. According to Lemma 3.5 to establish the first statement of the theorem it is enough to prove that

№m(a)HrHn

< C < oo

for every arbitrary p-atom a. Let a be an arbitrary p-atom with support /2 and /2) = 2—2N. Without loss of generality, we may assume that /2 = /N x /N. It is evident that Mn(a) = 0 if n < 2N. Therefore, we set n > 2N. Using ideas of estimate (15) and the Levi theorem, we can write

||Mm(a)||?

„3-3p

1=2N TO

m2/p-3

„=2N - JX«

dy ■

1=2N 1 J1« X1«

n2/p-3

m=2«m J'«X1«

dy + T 1

m=2^ 1 n X 'n

rj2/p-3

= I1 + /2 + /3 + I4. Since Mn is bounded from LTO to LTO, we get

TO 1 f TO || up TO 1

'1 < T \M-(a)\pdy < Cp T JOTOn < Cp T 13-

3p <Cp< TO.

1=2« J'NX'N 1=2« 2 1=2«

„=2« '" j'« X'n Combining (10) and (11) we have

/2 < Cp

TO «-1 « . ,

T T t 11.....

1=2« 12=0 g2=-2+1 X

«-1 «

n2/p-3

dy < cp t 1 t t ln

1=2« 12=0 g2=12+1 'n

22N(1-p)2P(l2+q2) m2-3p

< Cp 2(1-2p)N T ^ T T L2^" dy < Cp2(1-2p)N T rn-p 2(2p-1)N < Cp < TO.

1=2n 12=0 û2=12+1 'n 1=2n

Analogously we can prove that I3 < Cp < то. By decomposition (10) we have

N_1 L1_1

IXJ IV — I L — I f

У H У УJ

v=2N L1=0 m2=0 L

m2/P_3

p то . N-1 N-1 ,

dv + E m ££

m=2N L1 =0 m2^1 JL1

m2/P_3

dv = I41 + I4,2-

We estimate i42 (the estimate of i41 goes analogously). Lemma 3.2 and inequalities (12)&(13) immediately give

m2/p-3

Cp2N(2-3P) m2-3P

dv < cp:2-3; (2L1P22m2(P-1)+2L1(2P-1)2m2(P-1)).

то . N—1 N—1 -^N(2_3p) то i N—1 N-1 oN(2_3p)

1 2___2L1 p+2m:(p_1) + c у ¿Г Г ^__ :L1(2p-1)+m:(p-1)

m '— '— m2-3P p t— m m2-3P

v=2n L1=0 m:=L1 m=2N L1=0 m:=L1

I4,2 < Cp £ m £ £

2N(2-3p)

< c^ WE:L1(3p-2) < Cp^

2N(2-3p)

m3-3p h=2n L1=0

„=2n

„3-3p < Cp.

This completes the proof of the first part.

Now, we prove the second part of our theorem. We use an idea of Goginava from [12]. Let 4>(n) be any non-decreasing function and {nk : k > 0} be a sequence, satisfying the condition

1 23<1-P)<|nk |+1)

Hm ———.— .,, = œ.

ф(2Пк l+1)

Then there exists a subsequence {ак : к > 0} С {пк : к > 0} such that

К !> 4, к > 0,

2_ m < C < то.

23(1-p)!«,!/2

Since,

ф1 /2p (2 !ак !+1)

f(n,n)(x1 ,х )= L- Хк°к(х',х ), where = yiV—W ,

{к: ! ак ! <n}

Ок(х\х2) = 22! ак ! <1/р-1>(0:!ак!+1(х1) - 0:!ак!(х1))(0:!ак!+1(х2) - D^а^х2)).

ак if ! ак ! < n, 0 if ! ак ! > n,

S2n2n (Ok) =

it is easy to show that the martingale f = (f(i,i), f(22),..., f(n,n),...) G Hp(G2). Now, we calculate the Fourier coefficients

f (i,j ) =

'2 ! ак !(1/р-1)/2ф1/2^ : ! ак !+1) if (¿,j) £ |2 ! ак !.....:k !+1 _ к = 0,1,2 ...,

if (i.i) £ U {: ! ак!.....2 !ак!+1 _ 1}2.

Let 2 ! а*! < n < 2 ! ак!+1. Using (17) we can write

2 ! ак!

Mn(f) = 1 ГSjj(f) + 1 y Sjj(f) = III + IV.

n L--n L-

j=2 ! ак +1

It is easy to show that Sj7(f;x1,x2) = 0 for 0 < j < 2 1 "0 !, and Sjj(f;x1,x2) = 2" 1 (1/p—1)/201/2^2 1 "0 1+1)(Dj(x1) —

D2|"0i(x1))(Dj(x2) — D2 "0 i(x2)) for 2 1 "0 1 < j < 2 1 "0 1+1.

Suppose that 2| "s | < j < 2| "s|+1 for some s = 1, 2,..., k. Then we have

Sjjf(x1,x2) = £2| "n |<1/p—1>/201/2^2 | "n+1)(D2|n+1(x1) — D2|„n|(x1)) (D2|n+1(x2) — D2|„,|(x2))

+ 2 | "s |<1/p—1>/201/2p(2| "s|+1)(Dj(x1) — D2 | (x1))(Dj(x2) — D2 | »s|(x2)) (for more details see [12]). For 2 | "s|+1 < j < 2 | "s+^, s = 0.....k — 1, we have

V(x1,x2) = T2 \ ""\ (1/p-1)/201/2p(2\ \+1)(D2 \ n+1 (x1) - D2\ a,\(x1))(D2 \ n+1(x2) - D2\a„\(x2)) (19)

(for more details see [12]). Now, set (x1,x2) e /0 x J0, where

/0 x J0 = /s(x0 = 1,x1 = 1,x2 =1) x /3(x0 = 1,x2 = 1,x22 = 0). Since (see (5) and Lemma 5.1),

D2n (x1) = D2n (x2) = K2n (x1, x2) = 0, n > 4, (20)

from (16), (18) and (19) we obtain

1 k—1 2 | +1

III = 1 V2| "nl<1/p—1>/201/2p(2| "n|+1) V Dv(x1)Dv(x2)

n=0 v=2 | "r^

Applying (18) when s = k, we have

= 1 T2lcnl(1/p-1>/201/2p(2'cnl+1) ^ Dv(x1 )Dv(x2)

n=0 v=2 \ an\

= 1 T2 \ an\(1/p-1)/201/2p(2\ \+1 )(2\ \+1K2\ „\+1 (x1,x2) - 2 \anK2\a,\(x1,x2)) = 0.

|v = n-^ 2 \ \(1/p-1)/20l/2p(2\ \+1 )(D2\ „\ +1 (x1) - D2 \ „\ (x1))(D2 \ „\ +1 (x2) - D2 \ „\ (x2))

. . .............i\+H

21 "k l (1/p—1)/201/2^ 21 "k l" +-^-'- Yi (Dj(x1) — D2 | <k| (x1))(Dj(x2) — D2 | ^ (x2))

j=2 "k +1

= IV1 + IV2.

Combining (16) and (20), we get IV1 = 0 for (x 1,x2) e /0 x J0. Using (6), we get

D2k+1 (x1) D2k+1 (x2) = W2k+1 (x1) W2k+1 (x2) = ±1 , D-2k(x1) D2k(x2) = 0 for k = 0,1,... and (x1 ,x2) e /0 x J0 (we note that (2k)0 = 0). We have

(4/ + 1) |K4,+1(x1,x2)| = '

Y_ D2k+1 (x1 ) D2k+1 (x2) ^ D2k(x1 )D2k(x2)

2/+1 T

> 1 (21)

for (x\x2) e /0 X J0. It is well known that

Dj+2* (x) = D2„ (x) + wJW (x) Dj(x) for j < 2m.

If n is given in the form n = 4/ + 1 then n — 2at\ =4/' + 1 as well. Using (21) we obtain

2Wk\(1/p—1)/201/2p(2lot |+1)

|IV2| =

2\ at\(1 /p—1)/2 01 /2p (2\ at\+1)

2\at\(1/p—1)/2 01/2p (2\at\+1)

—2 \ at\

E ( Dj+2 \ "t\(x1) — D2\ at\(x1))(Dj+2\«t\(x2) — D2\ at\(x2))

j=1 —2 \ at\

E D (x1) DJ (x

|(" — 2\at \) Kn—2\at\(x1,x2)| >

01/2p (2 \ at\+1) 2(\ at\ + 1)(3—1/p)/2 •

Let 0 < p < 2/3, we can write

p 01 /2 (2 \ at\+1)

llM"(f)lweak-Lp > 2( \at\+1)(3p—1)/2 ^

(x1,x2) e C2 : |Mn(f;x1,x2)1 >

c 01/2p (2 \ at \ + 1)

2( \ at\+1)(3—1/p)/2

c01/2(2 \ at \+1) r ,o X ,o; > c01/2(2 \ at \+1)

2( \at\+1)(3p—1)/2 ^l'1 X h] ^ 9( \at \ +1)(3p—1)/2 •

2( at +1)(3p—1)/2

Hence,

l|Mn(f )lweak-Lp 0(")

> 2 1 ^1 llMnf llWe

n=2 \ at\

lMn (f) lweak-Lp

0(") V 0(")

2 \ at\<4n+1<2 \ at\+1

1 c 01/2(2 \ at\+1)

0(") 2(\at\ + 1)(3p—1)/2

2 at <4n+1<2 at +1

1 1 c 23(1—p)( \ at\+1)/2

0T/2(2latl+T) 2( \at\+1)(3p—1)/2 > 01/2(2\at\+1) > ^

2 \ at\<4n+1<2 \ at\+1

as t ^ &>•

This completes the proof.

Acknowledgements

Research is supported by project TAM0P-4.2.2.A-11/1/K0NV-2012-0051 and by Shota Rustaveli National Science Foundation grant DI/9/5-100/13, Function Spaces, Weighted Inequalities for Integral Operators and Problems of Summa-bility of Fourier Series.

The authors thank referees for help and advices which allowed to improve the manuscript.

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