1、Chin.Quart.J.of Math.2023,38(2):184195Multilinear Calder on-Zygmund Operators with Kernels ofDini Type and Commutators in Variable ExponentCentral Morrey SpacesMA Teng(Department of Mathematics,Xinjiang Institute of Engineering,Xinjiang 830011,China)Abstract:In this paper,we obtain the boundedness o
2、f multilinear Calder on-Zygmundoperators with kernels of Dini type and commutators with variable exponent-centralBMO functions in variable exponent central Morrey spaces.Keywords:Multilinear Calder on-Zygmund operators;Variable exponent;Central Morreyspaces;BMO2000 MR Subject Classification:42B35,47
3、B38,47B47CLC number:O174.2Document code:AArticle ID:1002-0462(2023)02-0184-12DOI:10.13371/ki.chin.q.j.m.2023.02.0061.IntroductionIn the last three decades,the interest for variable exponent spaces has been increasing yearby year due to their applications in electrorheological fluid and differential
4、equation.Also,variable exponent spaces are generalized from the standard function spaces.And variable indexfunctions are widely used in the study on Laplace equations of non-standard growth,whichare derived from problems in nonlinear elasticity.There are many great results in this area,see 6,10,16.V
5、ariable Lebesgue spaces become one of the most important function spaces due tothe fundamental paper 12 by Kov a cik and R akosn k.Recently,Cruz-Uribe,Fiorenza,Martelland Perez 4,5 proved that many classical operators in harmonic analysis,such as maximaloperators,singular integrals,commutators and f
6、ractional integrals were bounded on the variableLebesgue space.In 2015,Mizuta,Ohno and Shimomura introduced the non-homogeneouscentral Morrey spaces of variable exponent in 15.Recently,Wang et al.introduced the centralBMO spaces with variable exponent and gave the boundedness of some operators in 17
7、.Fu andwang 7 defined the-central BMO spaces with variable exponent and studied the boundednessof the singular integral operator with rough kernel and its commutator on central Morrey spaceswith variable exponent.Received date:2021-12-07Biographies:MA Teng(1995-),male,native of Urumqi,Xinjiang,lectu
8、rer of Xinjiang Institute of Engineering,engages in harmonic analysis.Corresponding author MA Teng:.184No.2MA Teng:Multilinear Calder on-Zygmund Operators in185The classic Calder on-Zygmund operators were first introduced by Coifman and Meyer 1,2.Calder on-Zygmund operators with kernels satisfying t
9、he standard estimates were then furtherinvestigated by many authors in the last few decades.Recently,there are a number of studiesconcerning singular integrals which possess rough associated kernels so that they do not belongto the standard Calder on-Zygmund operators classes.In particular,Yabuta 19
10、 introduced theCalder on-Zygmund operators with kernels of Dini type.And on the weightedLpspaces,thecorresponding boundedness was obtained when the weight functions belonged to Muckenhouptclass.Moreover,the multilinear Calder on-Zygmund theory was first studied by Coifman andMeyer in 1,2.This theory
11、 was further investigated by many authors in the last few decades,see 8,9,11,13.In particular,Lu and Zhang 14 eastablished the multiple-weight norminequalities for multilinear Calder on-Zygmund operators with kernels of Dini type and theircommutators.Inspired by the above work,we will study the boun
12、dedness of the multilinear Calder on-Zygmund operators with kernels of Dini type and their commutators on central Morrey spaceswith variable exponent.We now give the definition of the multilinear Calder on-Zygmund operators with kernels oftype(t).Throughout this paper,we always assume that(t):0,)0,)
13、is a nondecreasingfunction with 0(1)0,we say that Dini(a)if|Dini(a)=Z10a(t)tdt.Obviously,Dini(a1)Dini(a2)if 0a1a2.Xj=0(2j)Z10(t)tdtif Dini(1).Definition 1.1.A locally integrable functionK(x,y1,ym)defined away from the diagonalx=y1=ymin(Rn)m+1is called a kernel of type(t),if there is a positive const
14、antAsuchthat|K(x,y1,ym)?A?Pmi=1|xyi|?mn,?K(x,y1,ym)K(z,y1,ym)?A?Pmi=1|xyi|?mn?|zx|Pmi=1|xyi|?,whenever|zx|12max1im|xyi|.?K(x,y1,yj,ym)K(x,y1,y0j,ym)?A?Pmi=1|xyi|?mn?|yy0j|Pmi=1|xyi|?,186CHINESE QUARTERLY JOURNAL OF MATHEMATICSVol.38whenever|yjy0j|12max1im|xyi|.We say that T is a Multilinear Calder o
15、n-Zygmund operators with kernels of type(t),T(f1,fm)(x)=Z(Rn)mK(x,y1,ym)f1(y1)fm(ym)dy1dym,when x not belong toTmi=1suppfiand each fiCc.Letbbe a measurable locally integrable function andTbe a linear operator.Then thecommutator b,T is defined by b,Tf=bTf T(bf).In 3,R.Coifman,R.Rochberg and G.Weiss p
16、roved that the commutator b,T is bounded onLp(Rn)ifbBMO(Rn),1p0,ZRn?|f(x)|?p(x)dx0:ZRn?|f(x)|?p(x)dx1?.These spaces are referred as variableLpspaces,since they generalized the standardLpspaces,Lp()(Rn)is isometrically isomorphic to Lp(Rn)if p(x)=p is a constant.Denote byP(Rn)the set of measurable functionsp(x)such thatp1 andp+01|B(x,r)|ZB(x,r)|f(y)|dy.The setB(Rn)consists ofp()P(Rn)satisfying the condition thatMis bounded onLp()(Rn)5.No.2MA Teng:Multilinear Calder on-Zygmund Operators in187Lemma