Miskolc Mathematical Notes HU e-ISSN 1787-2413 Vol. 19 (2018), No. 1, pp. 303–317 DOI: 10.18514/MMN.2018.2347
CONVERGENCE OF CES ´ARO MEANS WITH VARYING PARAMETERS OF WALSH-FOURIER SERIES
ANAS AHMAD ABU JOUDEH AND GY ¨ORGY G ´AT Received 06 June, 2017
Abstract. In 2007 Akhobadze [1] (see also [2]) introduced the notion of Ces`aro means of Fourier series with variable parameters. In the present paper we prove the almost everywhere conver- gence of the the Ces`aro.C; ˛n/means of integrable functionsn˛nf !f, whereN˛;K3n! 1forf 2L1.I /, whereI is the Walsh group for every sequence˛D.˛n/,0 < ˛n< 1. This theorem for constant sequences˛that is,˛˛1was proved by Fine [3].
2010Mathematics Subject Classification: 42C10
1. INTRODUCTION AND MAIN RESULTS
We follow the standard notions of dyadic analysis introduced by the mathem- aticians F. Schipp, P. Simon, W. R. Wade (see e.g. [9]) and others. Denote by NWD f0; 1; :::g;P WDNn f0g, the set of natural numbers, the set of positive integers andIWDŒ0; 1/the unit interval. Denote by.B/D jBjthe Lebesgue measure of the setB.BI /. Denote byLp.I /the usual Lebesgue spaces andk:kpthe correspond- ing norms.1p 1/. Set
JWD p
2n;pC1 2n
Wp; n2N
the set of dyadic intervals and for given x 2I and let In.x/ denote the interval In.x/2 J of length 2 n which contains x .n2N/. Also use the notion In WD In.0/ .n2N/. Let
xD
1
X
nD0
xn2 .nC1/
be the dyadic expansion of x2I, where xnD0 or1and if x is a dyadic rational number.x2 f2pn Wp; n2Ng/we choose the expansion which terminates in0’s. The
Research is supported by the Hungarian National Foundation for Scientific Research (OTKA), grant no. K111651 and by project EFOP-3.6.1-16-2016-00022 supported by the European Union, co-financed by the European Social Fund.
c 2018 Miskolc University Press
notion of the Hardy space H.I / is introduced in the following way [9]. A func- tiona2L1.I /is called an atom, if eitheraD1or a has the following properties:
suppaIa;kak1 jIaj 1;R
IaD0, for someIa 2J. We say that the functionf belongs toH, iff can be represented asf DP1
iD0iai;whereai’s are atoms and for the coefficients.i/the inequalityP1
iD0jij<1is true. It is known thatH is a Banach space with respect to the norm
kfkH WDinf
1
X
iD0
jij;
where the infimum is taken over all decompositionsf DP1
iD0iai 2H. Set the definition of thenth (n2N) Walsh-Paley function at pointx2I as:
!n.x/WD
1
Y
jD0
. 1/xjnj; whereN3nDP1
nD0nj2j.nj 2 f0; 1g.j 2N//. It is known (see [8] or [10]) that the system.!n; n2N/is the character system of.I;C/, where the group operation Cis the so-called dyadic or logical addition onI. That is, for anyx; y2I
xCyWD
1
X
nD0
jxn ynj2 .nC1/: Denote by
f .n/O WD Z
I
f !nd ; DnWD
n 1
X
kD0
!k; Kn1WD 1 nC1
n
X
kD0
Dk
the Fourier coefficients, the Dirichlet and the Fej´er or.C; 1/kernels, respectively. It is also known that the Fej´er or.C; 1/means off is
n1f .y/WD 1 nC1
n
X
kD0
Skf .y/D Z
I
f .x/Kn1.yCx/d .x/
D 1 nC1
n
X
kD0
Z
I
f .x/Dk.yCx/d .x/; .n2N; y2I /:
It is known [9] that forn2N; x2I it holds D2n.x/D
(2n;ifx2In
0 ;ifx…In
and also that
Dn.x/D!n.x/
1
X
kD1
D2k.x/nk. 1/xk;
wherenDP1
iD1ni2i,niD f0; 1g.i 2N/.
Denote by Kn˛n the kernel of the summability method .C; ˛n/ and call it the .C; ˛n/kernel or the Ces`aro kernel for˛n2Rn f 1; 2; : : :g
Kn˛nD 1 A˛nn
n
X
kD0
A˛n kn 1Dk; where
A˛knD.˛nC1/.˛nC2/:::.˛nCn/
kŠ :
It is known [12] thatA˛nnDPn
kD0A˛kn 1; A˛kn A˛knC1D ˛nA
˛n k
kC1 :The.C; ˛n/ Ces`aro means of integrable functionf is
n˛nf .y/WD 1 A˛nn
n
X
kD0
A˛n kn 1Skf .y/D Z
I
f .x/Kn˛n.yCx/d .x/:
In [3] Fine proved the almost everywhere convergence n˛nf !f for all in- tegrable function f with constant sequence ˛nD˛1> 0. On the rate of conver- gence of Ces`aro means in this constant case see the paper of Fridli [4]. For the two-dimensional situation see the paper of Goginava [7].
Comment1. With respect to other locally constant orthonormal sysytems for in- stance it was a question of Taibleson [8] open for a long time, that does the Fej´er- Lebesgue theorem, that is the a.e. convergencen1f !f hold for all integrable functionf with respect to the character system of the group of2-adic integers. In 1997 G´at answered [1] this question in the affirmative. Zheng and G´at generalized this result [9,2] for more general orthonormal systems.
Set two variable functionP .n; ˛/WDP1
iD0ni2i ˛ forn2N; ˛2R. For instance P .n; 1/Dn. Also set for sequences˛ D.˛n/ and positive reals K the subset of natural numbers
N˛;KWD
n2NW P .n; ˛n/ n˛n K
:
We can easily remark that for a sequence ˛ such that 1 > ˛n˛0> 0 we have N˛;K DNfor someK depending only on˛0. We also remark that2n2N˛;K for every˛D.˛n/,0 < ˛n< 1andK1.
In this paperC denotes an absolute constant andCK another one which may de- pend only onK. The introduction of.C; ˛n/means due to Akhobadze investigated [1] the behavior of theL1-norm convergence of n˛nf !f for the trigonometric system. In this paper it is also supposed that1 > ˛n> 0for alln.
The main aim of this paper is to prove :
Theorem 1. Suppose that1 > ˛n> 0. Letf 2L1.I /. Then we have the almost everywhere convergencen˛nf !f provided thatN˛;K 3n! 1.
The method we use to prove Theorem 1 is to investigate the maximal operator ˛f WDsupn2N˛;Kjn˛nfj. We also prove that this operator is a kind of type.H; L/
and of type.Lp; Lp/for all1 < p 1. That is,
Theorem 2. Suppose that1 > ˛n> 0. Letjfj 2H.I /. Then we have k˛fk1CKkjfjkH:
Moreover, the operator˛is of type.Lp; Lp/for all1 < p 1. That is, k˛fkpCK;pkfkp
for all1 < p 1.
For the sequence˛nD1Theorem2is due to Fujii [5]. For the more general but constant sequence˛nD˛1see Weisz [11].
Basically, in order to prove Theorem1we verify that the maximal operator ˛f .˛D.˛n//is of weak type.L1; L1/. The way we get this, the investigation of kernel functions, and its maximal function on the unit intervalI by making a hole around zero and some quasi locality issues (for the notion of quasi-locality see [9]).
To have the proof of Theorem2is the standard way. We need several Lemmas in the next section.
2. PROOFS
Lemma 1. Forj; n2N; j < 2nwe have
D2n j.x/DD2n.x/ !2n 1.x/Dj.x/:
Proof.
D2n.x/D
2n 1
X
kD0
!k.x/D
2n j 1
X
kD0
!k.x/C
2n 1
X
kD2n j
!k.x/DD2n jC
2n 1
X
kD2n j
!k.x/:
We have to prove :
2n 1
X
kD2n j
!k.x/D!2n 1.x/Dj.x/:
Fork < j; kDkn 12n 1C:::Ck121Ck0we have
!2n 1.x/!k
D!2n 1C:::C21C20.x/!kn 12n 1C:::Ck0.x/
D!.1Ckn 1.mod 2//2n 1C:::C.1Ck0.mod 2//20.x/
D!.1 kn 1.mod 2//2n 1C:::C.1 k0.mod 2//20.x/
D!2n 1C2n 2C:::C20 .kn 12n 1C:::Ck0/.x/D!2n 1 k.x/:
Thus,
!2n 1.x/Dj.x/D!2n 1.x/
j 1
X
kD0
!k.x/D
j 1
X
kD0
!2n 1 k.x/D
2n 1
X
kD2n j
!k.x/:
This completes the proof of Lemma1.
Introduce the following notations: fora; n; j 2Nletn.j /WDPj 1
iD0ni2i, that is, n.0/D0; n.1/Dn0 and for2Bn < 2BC1, letjnj WDB,nDn.BC1/. Moreover, introduce the following functions and operators forn2Nand1 > ˛n> 0
Tn˛a WD 1 A˛na
2jnj 1
X
jD0
A˛n ja 1Dj;
TQn˛a WD 1 A˛naD2B
2B 1
X
jD0
A˛na 1
.B/Cj
C.1 ˛a/
2B 1
X
jD0
A˛na 1
.B/Cj
jC1 n.B/CjC1
ˇ ˇ ˇKj1
ˇ ˇ
ˇCA˛na 12Bˇ ˇK21B 1
ˇ ˇ; tn˛af .y/WD
Z
I
f .x/Tn˛a.yCx/d .x/;
Q
tn˛af .y/WD Z
I
f .x/TQn˛a.yCx/d .x/:
Now, we need to prove the next Lemma which means that maximal operator supn;ajQtn˛ajis quasi-local. This lemma together with the next one are the most im- portant tools in the proof of the main results of this paper.
Lemma 2. Let1 > ˛a> 0,f 2L1.I /such that suppf Ik.u/,R
Ik.u/f d D0 for some dyadic intervalIk.u/. Then we have
Z
InIk.u/
sup
n;a2NjQtn˛afjd Ckfk1: Moreover,ˇ
ˇTn˛aˇ
ˇ QTn˛a.
Proof. It is easy to have that forn < 2k and x2Ik.u/we haveTQn˛a.yCx/D TQn˛a.yCu/and
Z
Ik.u/
f .x/TQn˛a.yCx/d .x/
D QTn˛a.yCu/
Z
Ik.u/
f .x/d .x/D0:
Therefore, Z
InIk.u/
sup
n;a2N
Q
tn˛af d D Z
InIk.u/
sup
n2k;a2N
Q
tn˛af d : Recall thatBD jnj. Then
A˛naTn˛a D
2B 1
X
jD0
A˛2aB 1
Cn.B/ jDj
D
2B 1
X
jD0
A˛na 1
.B/CjD2B j
By Lemma1we have A˛naTn˛a
DD2B
2B 1
X
jD0
A˛na 1
.B/Cj !2B 1
2B 1
X
jD0
A˛na 1
.B/CjDj: It is easy to have that A1˛a
n D2B.´/P2B 1 jD0 A˛na 1
.B/Cj D0, for any´2InIk. This holds becauseD2B.´/D0forBD jnj kand´2InIk. By the help of the Abel transform we get:
ˇ ˇ ˇ ˇ ˇ ˇ
2B 1
X
jD0
A˛na 1
.B/CjDj
ˇ ˇ ˇ ˇ ˇ ˇ
D ˇ ˇ ˇ ˇ ˇ ˇ
2B 1
X
jD0
.A˛na 1
.B/Cj A˛na 1
.B/CjC1/
j
X
iD0
DiCA˛na 1
.B/C2B 2B 1
X
iD0
Di
ˇ ˇ ˇ ˇ ˇ ˇ
D ˇ ˇ ˇ ˇ ˇ ˇ
.1 ˛a/
2B 1
X
jD0
A˛na 1
.B/Cj
jC1
n.B/CjC1Kj1CA˛na 12BK21B 1
ˇ ˇ ˇ ˇ ˇ ˇ
D ˇ ˇ ˇ ˇ ˇ
.1 ˛a/
2k 1
X
jD0
A˛na 1
.B/Cj
jC1
n.B/CjC1Kj1C.1 ˛a/
2B 1
X
jD2k
A˛na 1
.B/Cj
jC1 n.B/CjC1Kj1 CA˛na 12BK21B 1
ˇ ˇ ˇ ˇ ˇ
.1 ˛a/
2k 1
X
jD0
A˛na 1
.B/Cj
jC1 n.B/CjC1
ˇ ˇ ˇKj1
ˇ ˇ ˇ
C.1 ˛a/
2B 1
X
jD2k
A˛na 1
.B/Cj
jC1 n.B/CjC1
ˇ ˇ ˇKj1
ˇ ˇ
ˇCA˛na 12Bˇ ˇK21B 1
ˇ ˇ
DWICIICIII:
These equalities above immediately proves inequalityˇ ˇTn˛aˇ
ˇ QTn˛a.
Since for any j < 2k we have that the Fej´er kernel Kj1.yCx/ depends (with respect tox) only on coordinatesx0; : : : ; xk 1, thenR
Ik.u/f .x/jKj1.yCx/jd .x/D jKj1.yCu/jR
Ik.u/f .x/d .x/D0givesR
Ik.u/f .x/I.yCx/d .x/D0.
On the other hand, IID.1 ˛a/
2B 1
X
jD2k
A˛na 1
.B/Cj
jC1
n.B/CjC1jKj1j sup
j2k
jKj1j.1 ˛a/
n
X
jD0
Aj˛a 1DA˛na.1 ˛a/ sup
j2k
jKj1j: This by Lemma 3 in [6] gives
Z
InIk
sup
n2k;a2N
1 A˛na
IId Z
InIk
sup
j2k
jKj1jd C:
The situation withIII is similar. Namely, A˛na 1n
A˛na D ˛an
.˛aCn/ ˛a< 1:
So, just as in the case ofII we apply Lemma 3 in [6]
Z
InIk
sup
n2k;a2N
1 A˛na
IIId Z
InIk
sup
n2k;a2N
jK21jnj 1jd C:
Therefore, substituting´DxCy2InIk (sincex2Ik.u/andy2InIk.u/) Z
InIk.u/
sup
n2k;a2N
Q tn˛af d
D Z
InIk.u/
sup
n2k;a2N
ˇ ˇ ˇ ˇ Z
Ik.u/
f .x/TQn˛a.yCx/d .x/
ˇ ˇ ˇ ˇ
d .y/
Z
InIk.u/
Z
Ik.u/jf .x/j sup
n2k;a2N
1
A˛na .II.yCx/CIII.yCx// d .x/
D Z
Ik.u/jf .x/j Z
InIk
sup
n2k;a2N
1
A˛na .II.´/CIII.´// d .´/d .x/
C Z
Ik.u/jf .x/jd .x/:
This completes the proof of Lemma2.
A straightforward corollary of this lemma is:
Corollary 1. Let1 > ˛a> 0. Then we havekTn˛ak1 k QTn˛ak1C,ktn˛afk1; kQtn˛afk1Ckfk1andktn˛agk1;kQtn˛agk1Ckgk1for all natural numbersa; n, whereC is some absolute constant andf 2L1; g2L1. That is, operatorstQn˛a; tn˛a
is of type.L1; L1/and.L1; L1/(uniformly inn).
Proof. The proof is a straightforward consequence of Lemma2and an easy es- timation below. LetBD jnj. Then
A˛naTQn˛a
1 kD2Bk1 2B 1
X
jD0
A˛na 1
.B/Cj
C.1 ˛a/
2B 1
X
jD0
A˛na 1
.B/Cj
jC1
n.B/CjC1kKj1k1CA˛na 12BkK21B 1k1: Then bykD2Bk1D1;kKj1k1C we complete the proof of Corollary1.
Lemma 3. Letn; N be any natural numbers and0 < ˛ < 1. Then we have A˛n
A˛N 2 nC1
N ˛
: Proof. By definition we have
A˛n A˛N D
1 ˛
nC1C˛
1 ˛
NC˛
1 ˛
nC2
1 ˛
NC1
: It is well-known that
1 ˛
i.nC1/C1
1 ˛
.iC1/.nC1/
1 ˛
.iC1/.nC1/
nC1
e 1iC˛1 for everyn2N. This gives
1 ˛
nC2
1 ˛
NC1
e 1˛PbnNC1c
iD2 1 i
e 1˛loge j N
nC1
k 1Cc
2 e 1˛loge
N nC1
D2 nC1
N ˛
;
where c 0:5772 is the Euler-Mascheroni constant. This completes the proof of
Lemma3.
Recall that the two variable functionP .n; ˛/DP1
iD0ni2i ˛forn2N; ˛2Rand K2Rdetermines the set of natural numbers
N˛;K D
n2NW P .n; ˛n/ n˛n K
:
LetnD2hsC C2h0, wherehs> > h00are integers. That is,jnj Dhs. Let n.j /WD2hj C C2h0. This meansnDn.s/. Define the following kernel function and operators
KQn˛nWD QTn˛.s/n C
s
X
lD0
A˛nn.l 1/
A˛nn.s/ D2hlCA˛nn.l 1/
A˛nn.s/ TQn˛ln1
!
and
Q
n˛nf WDf QKn˛n; Q˛f WD sup
n2N˛;K
jf QKn˛j:
In the sequel we prove that maximal operatorQ˛f is quasi-local. This is the very base of the proof of the main results of this paper. That is, Theorem1and Theorem 2.
Lemma 4. Let1 > ˛n> 0,f 2L1.I /such that suppf Ik.u/,R
Ik.u/f d D0 for some dyadic intervalIk.u/. Then we have
Z
InIk.u/Q˛f d CKkfk1; where constantCKcan depend only onK.
Proof. Recall thatnD2hsC C2h0, wherehs> > h00are integers. That is,jnj Dhs. Letn.j /WD2hjC C2h0. This meansnDn.s/. Use also the notation
Q Kn˛.s/n
D QTn˛.s/n C
s
X
lD0
A˛nn.l 1/
A˛nn.s/ D2hl CA˛nn.l 1/
A˛nn.s/ TQn˛ln1
!
DWG1CG2CG3:
Sincen.l 1/< 2h.l 1/C1, then by Lemma3we have A˛nn.l 1/
A˛nn.s/ 2 n.l 1/C1 n.s/
!˛n
22˛n.hl 1C1/
2˛nhs C2hl 1˛n n˛n : That is, by the above written we also have
Z
InIk.u/
sup
n2N
ˇ ˇ ˇ ˇ Z
Ik.u/
f .x/G2.yCx/d .x/
ˇ ˇ ˇ ˇ
d .y/
Z
InIk.u/
sup
n2N s
X
lD0
2hl 1˛n n˛n
ˇ ˇ ˇ ˇ Z
Ik.u/
f .x/D2h.yCx/d .x/
ˇ ˇ ˇ ˇ
d .y/D0
sincef D2hD0forhkbecause of theAk measurability ofD2h andR
f D0.
Besides, forh > k D2h.yCx/D0(yCx…Ik).
As a result of these estimations above and by the help of Lemma2, that is the quasi-locality of operatortQ˛Dsupn;a2NjQtn˛ajwe conclude
Z
InIk.u/
sup
n2N
ˇ ˇ ˇ ˇ Z
Ik.u/
f .x/.G1.yCx/CG3.yCx//d .x/
ˇ ˇ ˇ ˇ
d .y/
CK Z
InIk.u/
sup
n;a2N
ˇ ˇ ˇ ˇ Z
Ik.u/
f .x/TQn˛a.yCx/d .x/
ˇ ˇ ˇ ˇ
d .y/
CKkfk1:
This completes the proof of Lemma4.
Lemma 5. The operatorQ˛is of type.L1; L1/ .Q˛f WDsupn2N˛;Kˇ ˇQn˛nfˇ
ˇ/:
Proof. By the help of the method of Lemma4and by Corollary1we have
KQn˛n 1D
KQn˛.s/n
1
TQn˛.s/n
1C
s
X
lD0
A˛nn.l 1/
A˛nn.s/ kD2hlk1CA˛nn.l 1/
A˛nn.s/ k QTn˛ln1k1
!
CCC
s
X
lD0
A˛nn.l 1/
A˛nn.s/ CK
becausen2N˛;K. HenceQ˛ is of type.L1; L1/(with constantCK). This com-
pletes the proof of Lemma5.
Now, we can prove the main tool in order to have Theorem1for operator˛f WD supn2N˛;Kˇ
ˇn˛nfˇ ˇ:
Lemma 6. The operatorsQ˛and˛are of weak type.L1; L1/.
Proof. First, we prove Lemma6for operatorQ˛. We apply the Calderon-Zygmund decomposition lemma [9]. That is, letf 2L1andkfk1< ı. Then there is a decom- position:
f Df0C
1
X
jD1
fj
such that kf0k1C ı ;kf0k1Ckfk1andIj DIkj.uj/are disjoint dyadic in- tervals for which
suppfj Ij ; Z
Ij
fjd D0 ; jFj Ckf1k ı
.uj 2I ; kj 2N ; j 2P /, whereF D [1iD1Ij. By the-sublinearity of the max- imal operator with an appropriate constantCK we have
.Q˛f > 2CKı/.Q˛f0> CKı/C.Q˛.
1
X
iD1
fi/ > CKı/WDICII :
Since by Lemma5k Q˛f0k1CKkf0k1CKıthen we haveID0. So, .Q˛.
1
X
iD1
fi/ > CKı/ jFj C.FN\ f Q˛.
1
X
iD1
fi/ > CKıg/
CKkfk1 ı CCK
ı
1
X
iD1
Z
InIjQ˛fjd DWCKkfk1 ı CCK
ı
1
X
iD1
IIIj;
where
IIIj WD Z
InIjQ˛fjd
Z
InIkj.uj/
sup
n2N˛;K
ˇ ˇ ˇ ˇ ˇ Z
Ikj.uj/
fj.x/KQn˛n.yCx/d .x/
ˇ ˇ ˇ ˇ ˇ
d .y/:
The forthcoming estimation ofIIIj is given by the help Lemma4 IIIj CKkfjk1:
That is, operatorQ˛is of weak type.L1; L1/. Next, we prove the estimation jKn˛nj QKn˛n: (1) To prove (1) recall again thatnD2hsC C2h0, wherehs> > h00are integers.
SincenD2hsCn.s 1/, then we have
2hsCn.s 1/
X
jD2hs
A˛nn.s 11/
C2hs jDj D
n.s 1/
X
kD0
A˛nn.s 11/ kD2hsCk
DD2hs
n.s 1/
X
kD0
A˛nn.s 11/ kC!2hs
n.s 1/
X
kD0
A˛nn.s 11/ kDk DD2hsA˛nn.s 1/C!2hsA˛nn.s 1/Kn˛.sn 1/:
So, by the help of the equalities above we get Kn˛.s/n DTn˛.s/n CA˛nn.s 1/
A˛nn.s/
D2hs C!2hsKn˛.sn 1/
:
Apply this last formula recursively and Lemma 2. (Note that n. 1/ D0; T0˛n D K0˛nD0; A˛0nD1.)
jKn˛nj D jKn˛.s/n j jTn˛.s/nj C
s
X
lD0
0
@
s
Y
jDl
A˛nn.j 1/
A˛nn.j / D2hl C
s
Y
jDl
A˛nn.j 1/
A˛nn.j / jTn˛ln1j 1 A
D jTn˛.s/nj C
s
X
lD0
A˛nn.l 1/
A˛nn.s/ D2hl CA˛nn.l 1/
A˛nn.s/ jTn˛ln1j
!
QKn˛.s/n D QKn˛n:
This completes the proof of inequality (1). This inequality gives that the operator˛ is also of weak type.L1; L1/since
.˛f > 2CKı/.Q˛jfj> 2CKı/CKkjfjk1
ı DCKkfk1 ı :
This completes the proof of Lemma6.
Proof of Theorem1. LetP 2Pbe a polynomial whereP .x/DP2k 1
iD0 ci!i. Then for all natural number n2k, n2N˛;K we have that SnP P. Consequently, the statementn˛nP !P holds everywhere (of course not only for restrictedn2 N˛;K). Now, let; ı > 0; f 2L1. LetP2Pbe a polynomial such thatkf Pk1< ı.
Then . lim
n2N˛;K
jn˛nf fj> / . lim
n2N˛;K
jn˛n.f P /j>
3/C. lim
n2N˛;K
jn˛nP Pj>
3/ C. lim
n2N˛;K
jP fj>
3/ . sup
n2N˛;K
jn˛n.f P /j>
3/C0C3
kP fk1CKkP fk13 CK
ı because˛is of weak type.L1; L1/(with any fixedK > 0). This holds for allı > 0.
That is, for an arbitrary > 0we have . lim
n2N˛;K
jn˛nf fj> /D0 and consequently we also have
. lim
n2N˛;K
jn˛nf fj> 0/D0:
This finally gives
n2limN˛;K
jn˛nf fj D0 a:e;
n˛nf !f a:e: .n2N˛;K/:
This completes the proof of Theorem1.
Proof of Theorem2. Inequality (1), Lemma5and Lemma6by the interpolation theorem of Marcinkiewicz [9] give that the operator˛ is of type.Lp; Lp/for all 1 < p 1. In the sequel we prove that operatorQ˛f Dsupn2N˛;Kjf QKn˛jis of type.H; L/.
Letabe an atom (a¤1can be supposed ), suppaIk.x/ ;kak12kfor some k2N andx2I. Then,n < 2k,n2N˛;K impliesa QKn˛D0becauseKQn˛ isAk measurable forn < 2k andR
Ik.x/a.t /d .t /D0. That is, Q
˛aD sup
N˛;K3n2k
j Qn˛nfj:
By the help Lemma4we have Z
InIk.x/Q˛a d D Z
InIk.x/
sup
N˛;K3n2k
ˇ ˇ ˇ ˇ Z
Ik.x/
a.y/KQn˛n.´Cy/d .y/
ˇ ˇ ˇ ˇ
d .´/
CK Z
Ik.x/ja.y/jd .y/
CKkak1
CK:
Since the operatorQ˛is of type.L2; L2/(i.ek Q˛fk2CKkfk2for allf 2L2.I /), then we have
k Q˛ak1D Z
InIk.x/Q˛aC Z
Ik.x/Q˛a CKC jIk.x/j12k Q˛ak2
CKCCK2 2kkak2 CKCCK2 2k2k2 CK:
That isk Q˛ak1CKand consequently the-sublinearity ofQ˛gives k Q˛fk1
1
X
iD0
jijk Q˛aik1
CK 1
X
iD0
jij CKkfkH for allP1
iD0iai 2H. That is, the operatorQ˛is of type.H; L/. This by inequality (1) and then byk˛fk1 k Q˛jfjk1CKkjfjkH completes the proof of Theorem
2.
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Authors’ addresses
Anas Ahmad Abu Joudeh
Institute of Mathematics, University of Debrecen, H-4010 Debrecen, Pf. 12, Hungary
E-mail address:anas.abujoudeh@mailbox.unideb.edu.hu, mr anas judeh@yahoo.com
Gy¨orgy G´at
Institute of Mathematics, University of Debrecen, H-4010 Debrecen, Pf. 12, Hungary E-mail address:gat.gyorgy@science.unideb.hu