Boundedness of Parametrized Littlewood-Paley Operators with Non-doubling Measures on Morrey Spaces?

LI Tie,JIANG Yin-sheng,ZHOU Jiang

(College of Mathematics and System Sciences,Xinjiang University,Urumqi,Xinjiang 830046,China)

Abstract: Letμbe a non-negative Radon measure on Rdwhich only satis fies the following growth condition that there exists a positive constant C such thatμ(B(x,r))≤Crnfor all x∈Rd,r>0 and some fixed n∈(0,d].We will prove that for suitable indexes ρ and λ the parametrized functionand Mρare bounded on(k,μ)spaces.

Keywords:Non-doubling measures, Morrey space, Parametrized Littlewood-Paley Operators, Parametrized Marcinkiewiczintegral

0 Introduction

Letμbe a nonnegative Radon measure on Rdwhich only satis fies the following growth condition that for allx∈Rdand allr>0,

whereC0andnare positive constants andn∈(0,d],andB(x,r)is the open ball centered atxand having radiusr.Such a measureμmay be non-doubling.We recall thatμis said to be a doubling measure,if there is a positive constantCsuch that for anyx∈supp(μ)andr>0,μ(B(x,2r))≤Cμ(B(x,r)),and that the doubling condition is a key assumption in the classical theory of harmonic analysis.In recent years,many classical results concerning the theory of Calder′on-Zygmund operators and function spaces have been proved still valid if the Lebesgue measure is substituted by a measureμas in(1),for example,see[1,2].

LetKbeμ-locally integrable function on Rd×Rd{(x,y):x=y}.Assume that there exists a positive constantCsuch that for anyx,y∈Rdwithx,y,

and for anyx,y,y0∈Rd,

The parametrized Marcinkiewicz integral Mρ(f)associated to the above kernelKand the measureμas in(1)is de fined by

where ρ ∈ (0,∞).For anyx∈ Rd,the parametrized area integralandfunctionare de fined respectively by

with ? homogeneous of degree zero and ω ∈Lipα(Sd?1)for some α ∈ (0,1],thenKsatis fies(2)and(3).Under these condtions,Mρin(4)is just the parametrized Marcinkiewicz integral introduced by Ho¨rmander in[3],andandas in(5)and(6),respectively,are the parametrized area integral and the parametrizedfunction considered by Sakamoto and Yabuta in[4].When ρ =1,the operatorMρas(4)is just the Marcinkiewicz integral with non-doubling measures in[5],where the boundedness of such operator in Lebesgue spaces and Hardy spaces was established under the assumption that Mρis bounded onL2(μ).There are some results about the boundedness of M,Mρandwith non-doubling Measures onLp(μ)spaces.In 2007,the boundedness of Marcinkiewicz integral M with non-doubling measures was established onLp(μ)spaces by Hu,Lin and Yang in[5].In 2008,Lin and Meng in[6]proved that the boundedness of parametrized Marcinkiewicz integral Mρandfunctionwith non-doubling measures onLp(μ)spaces.In 2010,the boundedness of Marcinkiewicz integral M with non-doubling measures was established on Morrey spacesby Zhang and Li in[7].

Throughout this paper,we always assume that the parametrized Marcinkiewicz integral with non-doubling measures Mρas in(4)is bounded onL2(μ).The main purpose of this paper is to establish that for suitable indexes ρ and λ the parametrizedfunctionand Mρare bounded on(k,μ)spaces.

To state our results,we still recall some notations and de finitions.The Morrey spaces were introduced by Sawano and Tanaka in[8].

De finition 1Letk>1 and 1≤q≤p<∞.We recall the Morrey spaceMpq(k,μ)as

It is easy to observe thatLp(μ)=(k,μ),and Ho¨lder’s inequality tells usfor all 1≤q2≤q1≤p,then we haveLp=(k,μ)?(k,μ)?k,μ).The space,μ)is a Banach space with its norm(7)and the parameterk>1 appearing in the de finition does not a ffect it.

Our main theorems are formulated as follows.

Theorem 1LetKbe aμ-locally integrable function on Rd×Rd{(x,y):x=y}satisfying(2)and(3),and Mρbe as in(4)with ρ∈).Then,fork>1 and 1

To state our next theorems,we need a stronger condition for the kernelK,for some fixed σ>2,

Theorem 2LetKbe a μ-locally integrable function on Rd×Rd{(x,y):x=y}satisfying(2)and(8),andbe as in(6)with ρ ∈).Then fork>1,λ>2 and 1

Note that for any ρ∈(0,∞),λ∈(1,∞),andx∈Rd,

The above inequality together with Theorem 2 deduce that the following theorem.

Theorem 3LetKbe a μ-locally integrable function on Rd×Rd{(x,y):x=y}satisfying(2)and(8),andbe as in(6)with ρ ∈).Then,fork>1 and 1

For a cubeQ?Rd,we mean a closed cube whose sides parallel to the coordinate axes and we denote its side length byl(Q)and its center byxQ.In what follows,Cdenotes a positive constant that is independent of main parameters involved but whose value may differ from line to line.For a μ-measurable setE,χEdenotes its characteristic function.For anyp∈[1,∞],we denote byp0its conjugate index,

1 Proof of main result

Before proving our theorems we introduce two lemmas which will be used on later.

Lemma 1[6]LetKbe aμ-locally integrable function on Rd×Rd{(x,y):x=y}satisfying(2)and(3),and Mρbe as in(4)with ρ ∈).Then,for anyp∈ (1,∞),Mρis bounded onLp(μ).That is,

Lemma 2[6]LetKbe a μ-locally integrable function on Rd×Rd{(x,y):x=y}satisfying(2),(8)andbe as in(6)with ρ ∈).Then,for anyp∈(1,∞),is bounded onLp(μ).That is,

Proof of Theorem 1For anyf∈(μ),and fixedQ,we setf1(x)=fχ2Q(x)andf2(x)=f(x)?f1(x).Then

From Lemma 1,we can deduce that

As forI2,note that for anyx∈Q,y∈(2Q)c,|x?y|～|xQ?y|.Applying the Minkowiski inequality and(2)we obtain that

Thus

The estimates forI1andI2give thatand then the proof of Theorem 1 is completed.

Proof of Theorem 2We still adapt the notation in the proof of Theorem 1 and obtain that

From Lemma 2,it follows that

As forM2,note that for anyx∈Q,z∈(2Q)c,|x?z|～|xQ?z|,and by(2),we have that

Applying the Minkowski inequality and condition(2),it leads to that for anyz∈Rd2Qandx∈Q,

As forF2(z),notice that for anyx,y,z∈Rdsatisfying|y?x|

To estimateF3(z),we first have that for anyx,y,z∈Rdsatisfying 2|y?z|≤|x?z|,|x?z|≤2|y?x|,then|y?x|≤3|x?z|/2.Choose 0< δandx∈Q,we have

Combining the estimates forF1(z),F2(z)andF3(z),we obtain that for anyz∈Rd2Q,