Wang Hongbin

(Department of Mathematical and Physical Science, Zibo Normal College, Zibo 255130, China)

Triebel-Lizorkin Spaces and Besov Spaces Associated with Different Homogeneities and Boundedness of Composition Operators

Wang Hongbin

(Department of Mathematical and Physical Science, Zibo Normal College, Zibo 255130, China)

In this paper, the author introduces new Triebel-Lizorkin spaces and Besov spaces associated with different homogeneities and proves that the composition of two Calderón-Zygmund singular integral operators with different homogeneities is bounded on these new Triebel-Lizorkin spaces and Besov spaces.

Triebel-Lizorkin spaces; Besov spaces; Calderón-Zygmund operators; composition; discrete Calderón’s identity

1 Introduction and statement of main results

Forx=(x′,xn)∈n-1×and δ>0, we consider two kinds of homogeneities onN

δ°e(x′,xn)=(δx′,δxn),

δ°h(x′,xn)=(δx′,δ2xn).

The first are the classical isotropic dilations occurring in the classical Calder n-Zygmund singular integrals, while the second are non-isotropic and related to the heat equations (also Heisenberg groups). Lete(ζ) be a function onnhomogeneous of degree 0 in the isotropic sense and smooth away from the origin. Similarly, suppose thatH(ζ)is a function onnhomogeneous of degree 0 in the non-isotropic sense, and also smooth away from the origin. Then it is well-known that the Fourier multipliersT1defined by(ζ)=e(ζ)(ζ)andT2givenby(ζ)=h(ζ)(ζ) are both bounded onLpfor 1

To state more precisely our main results, we first recall some notions and notations. Forx=(x′,xm)∈m-1×we denote |x|e=(|x′|2+|xm|2)and |x|h=(|x′|2+|xm|.Wealsousenotationsj∧k=min{j,k}andj∨k=max{j,k}.LetΨ(1)∈s(m)with

(1.1)

(1.3)

(1.5)

where the convergence of series inL2(m) andS′/P(m) (the space of tempered distributions modulo polynomials) follows from the results in the classical case. See [1] for more details.

Now, we give the definition of Triebel-Lizorkin spaces and Besov spaces associated with different homogeneities.

Definition 1.1 Let 0

where

where

Thesingularintegralsconsideredinthispaperaredefinedby

Definition 1.2 A locally integrable functionK1onRn{0} is said to be a Calderón- Zygmund kernel associated with the isotropic homogeneity if

(1.6)

for all |α|≥0 and

(1.7)

for all 0

We say that an operatorT1is a Calderón-Zygmund singular integral operator associated with the isotropic homogeneity ifT1(f)(x)=p.v.(K1*f)(x), whereK1satisfies conditions of (1.6) and (1.7).

(1.8)

forall|α|≥0,β≥0and

and

(1.9)

forall0

WesaythatanoperatorT2isaCalderón-Zygmundsingularintegraloperatorassociatedwiththenon-isotropichomogeneityifT2(f)(x)=p.v.(K2*f)(x),whereK2satisfiesconditionsof(1.8)and(1.9).

Ourmainresultsarethefollowing

Theorem 1.1 LetT1andT2be Calder n-Zygmund singular integral operators with isotropic and non-isotropic homogeneity, respectively. Then for 0

2 Some lemmas

Intheproofofourmainresult,wewillusethefollowinglemmas.

×Ψj,k(x′-2-(j∧k)l′,xm-2-(j∧2k)lm),

where the series converges inL2(m)，S∞(m)，andS′/P(m).

Lemma 2.2 (Almost orthogonality estimates[2]) Suppose thatΨj,kandφj′,k′satisfy the same conditions in (1.1)-(1.4). Then for any given integersLandM, there exists a constantC=C(L,M)>0 such that

DenotebyMsthestrongmaximaloperatordefinedby

wherethesupremumistakenoverallopenrectanglesinmthatcontainthepointx.

SimilartotheproofofLemma3.2in[2],wehavethefollowingestimateofthediscreteversionofthemaximalfunction.

Lemma 2.3 LetI,I′be dyadic cubes inm-1andJ,J′be dyadic intervals inwith the side lengthsl(I)=2-(j∧k),l(I′)=2-(j′∧k′)andl(J)=2-(J∧2k),l(J′)=22(j′∧2k′), and the left lower corners ofI,I′and the left end points ofJ,J′ are 2-(j∧k)l′,2-(j′∧k′)l″,2-(j∧2k)lmand,respectively.Thenforanyu′,v′∈I,um,vm∈J,any0

whereC1=C2(m-1)(1/δ-1)(j′∧k′-j∧k)+2(1/δ-1)(j′∧2k′-j∧2k)+and (a-b)+=max{a-b,0}.

and

whereMisafixedlargepositiveintegerdependingonp,qands.Wealsoletφ(2)∈S(m)withsuppφ(2)?B(0,1),

and

ThediscreteCalderón-typeidentityisgivenbythefollowing

Lemma 2.4 Letφ(1)andφ(2)satisfy conditions from (2.1) to (2.4) and Let 0

×(φj,k*h)(2-(j∧k)-Nl′,2-(j∧2k)-Nlm),

wheretheseriesconvergesinL2(m),I are dyadic cubes inm-1andJare dyadic intervals inwith side-lengthl(I)=2-(j∧k)-Nandl(J)=2-(j∧2k)-N, and the left lower corners of I and the left end points of J are 2-(j∧k)-Nl′ and 2-(j∧2k)-Nlm, respectively. Moreover,

and

Proof By taking the Fourier transform, we have that forf∈L2(m),

Applying Coifman's decomposition of the identity operator, we obtain

×(φj,k*f)(2-(j∧k)-Nl′, 2-(j∧2k)-Nlm)+RN(f)(x′,xm)

∶=TN(f)(x′,xm)+RN(f)(x′,xm),

where

wheretheseriesIare dyadic cubes inm-1andJare dyadic intervals inwith side-lengthl(I)=2-(j∧k)-Nandl(J)=2-(j∧2k)-N, and the left lower corners ofIand the left end points ofJare 2-(j∧k)-Nl′ and 2-(j∧2k)-Nlm, respectively.

Similar to the proof of Theorem 4.1 in [2], we only need to prove

whereCistheconstantindependentoffandN.

Thisproves(2.8)andhenceLemma2.4follows.

Similarly,toshowTheorem1.2,weneed

Lemma 2.5 Let φ(1)and φ(2)satisfy conditions from (2.1) to (2.4) and let 0

×(φj,k*h)(2-(j∧k)-Nl′,2-(j∧2k)-Nlm),

wheretheseriesconvergesinL2,Iare dyadic cubes inm-1andJare dyadic intervals inwith side-lengthl(I)=2-(j∧k)-Nandl(J)2-(j∧2k)-N, and the left lower corners ofIandtheleftendpointsofJare 2-(j∧k)-Nl′ and 2-(j∧2k)-Nlm, respectively. Moreover,

and

whereCis the constant independent offandN.

3 Proof of Theorem 1.1

Theorem 3.1 Let 0

Proof Letf∈S′/P(m). We denotexI=2-(j∧k)l′ ,xJ=2-(j∧2k)lm,xI′=2-(j′∧k′)l″ andxJ′=.DiscreteCaldern'sidentityonS′/P(m)andthealmostorthogonalityestimatesyieldthatfor<δ

whereinthelastinequalityweusethefactsthat(j′∧k′-j∧k)+≤|j-j′|(k-k′|,(j′∧2k′-j∧2k)+≤|j-j′|+2|k-k′|andwehavechosenL>max{m(1/δ-1)+|s1|,(m+1)(1/δ-1)+|s2|}sothat

Applying Fefferman-Stein's vector-valued strong maximal inequality onLp/δ(lq/δ)yields

Bysymmetry,wegettheconverseinequality.HencetheproofofTheorem3.1iscomplete.

AsaconsequenceofTheorem3.1, L2(m)∩(m)isdensein(m).Indeed,wehavethefollowing

Corollary 3.1 Let 0

E={(j,k,l′,lm)∶|j|≤N,|k|≤N,|l′|≤N,|lm|≤N}

and

×Ψj,k(x′-2-(j∧k)l′,xm-2-(j∧2k)lm),

whereΨj,kare the same as Lemma 2.1.

SinceΨj,k∈S∞(m), we obviously havefN∈S∞(m). Repeating the same proof as in Theorem 3.1, we can conclude that≤C.ToseethatfNtendstofinm),bythediscreteCaldern'sidentityinS′/P(m)inLemma2.1,

×Ψj,k(x′-2-(j∧k)l′,xm-2-(j∧2k)lm),

wheretheseriesconvergesinS′/P(m).

Therefore,

Arguing as in the proof of Theorem 3.1, we derive

Repeating the same proof as in Theorem 3.1, we have

Corollary 3.2 Let 0

whereφj,k,φj′,k′,handNarethesameasinLemma2.4.

SimilartotheproofofTheorem1.7in[2],wehave

4 Proof of Theorem 1.2

Theorem 4.1 Let 0

Proof Letf∈S′/P(m). We denotexI=2-(j∧k)l′,xJ=2-(j∧2k)lm,xI′=2-(j′∧k′)l″ andxJ′=. Discrete Calderón's identity onS′/P(m) and the almost orthogonality estimates yield that for<δ

|Ψj,k*f(xI,xJ)|

When1≤q<∞,applyingH?lderinequalityandwhen0

where in the last inequality we use the facts that (j′∧k′-j∧k)+≤|j-j′|+|k-k′|,

(j′∧2k′-j∧2k)+≤|j-j′|+2|k-k′|,and we have chosenL>max{m(1/δ-1)+|s1|,(m+1)(1/δ-1)+|s2|} so that

TheproofofTheorem4.1iscomplete.

AsaconsequenceofTheorem4.1, L2(m)∩(m)isdensein(m). Indeed we have the following

Corollary 4.1 Let 0

E1={(j,k)∶|j|≤N,|k|≤N}，E2={(l′,lm)∶|l′|≤N,|lm|≤N}，

and

×Ψj,k(x′-2-(j∧k)l′,xm-2-(j∧2k)lm)

whereΨj,kare the same as Lemma 2.1.

SinceΨj,k∈S∞(m), we obviously havefN∈S∞(m). Repeating the same proof as in Theorem 4.1, we conclude that≤C.ToseethatfNtendstofinm),bythediscreteCaldern'sidentityinS′/P(m)inLemma2.1,

×Ψj,k(x′-2-(j∧k)l′,xm-2-(j∧2k)lm)

wheretheseriesconvergesinS′/P(m).

RepeatingthesameproofofTheorem4.1,wehave

Corollary 4.2 Let 0

whereφj,k*φj′,k′handNarethesameasinLemma2.5.

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(責(zé)任編輯：胡安波)

2014-06-21

王洪彬(1981-)，男，山東淄博人，博士，淄博師范高等專科學(xué)校數(shù)理系教師，主要從事調(diào)和分析方向研究。

O174.2

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(2014)04-0049-10

注：本文為淄博師范高等?？茖W(xué)校校級課題“變指標(biāo)Herz型空間中算子的有界性”[13xk023]的階段性研究成果。