Let μΩ,b^m be the higher order commutator generated by Marcinkiewicz integral μΩ and a BMO（R^n） function b（x）. In this paper, we will study the continuity of μΩ and μΩ,b^m on homogeneous Morrey-Herz spaces.
A weak type endpoint estimate for the maximal multilinear singular integral operator TAf（x）=supε〉0｜∫｜x-y｜〉εΩ（x-y）/｜x-y｜^n＋1（A（x）-A（y）-VA（y）（x-y））f（y）dy｜ is established, where Ω is homogen...A weak type endpoint estimate for the maximal multilinear singular integral operator TAf（x）=supε〉0｜∫｜x-y｜〉εΩ（x-y）/｜x-y｜^n＋1（A（x）-A（y）-VA（y）（x-y））f（y）dy｜ is established, where Ω is homogeneous of degree zero, integrable on the unit sphere and has vanishing moment of order one, and A has derivatives of order one in BMO（R^n）. A regularity condition on Ω which implies an LlogL type estimate of TA is given.展开更多
We establish the theory of Orlicz-Hardy spaces generated by a wide class of functions.The class will be wider than the class of all the N-functions.In particular,we consider the non-smooth atomic decomposition.The rel...We establish the theory of Orlicz-Hardy spaces generated by a wide class of functions.The class will be wider than the class of all the N-functions.In particular,we consider the non-smooth atomic decomposition.The relation between Orlicz-Hardy spaces and their duals is also studied.As an application,duality of Hardy spaces with variable exponents is revisited.This work is different from earlier works about Orlicz-Hardy spaces H（Rn）in that the class of admissible functions is largely widened.We can deal with,for example,Ф（r）≡（rp1（log（e＋1/r））q1,0〈r≤1,r^p2 （log（e＋r））q2,r〉1,with p1,p2∈（0,∞）and q1,q2∈（.∞,∞）,where we shall establish the boundedness of the Riesz transforms on H（Rn）.In particular,is neither convex nor concave when 0〈p1〈1〈p2〈∞,0〈p21〈p1〈∞or p1=p2=1 and q1,q20.If（r）≡r（log（e＋r））q,then H（Rn）=H（logH）q（Rn）.We shall also establish the boundedness of the fractional integral operators I of order∈（0,∞）.For example,I is shown to be bounded from H（logH）1^1-α/n（Rn）to Ln/（n-α）（log L）（Rn）for 0〈α〈n.展开更多
The boundedness of multilinear Calderdn-Zygmund operators and their commutators with bounded mean oscillation （BMO） functions in variable exponent Morrey spaces are obtained.
In this paper,we first introduce the weak Lipschitz spaces WLip_(q,α),1<q<∞,0<α<1 which are the analog of weak Lebesgue spaces L^(q,∞)in the setting of Lipschitz space.We obtain the equivalence between...In this paper,we first introduce the weak Lipschitz spaces WLip_(q,α),1<q<∞,0<α<1 which are the analog of weak Lebesgue spaces L^(q,∞)in the setting of Lipschitz space.We obtain the equivalence between the norm ||·||_(Lipα)and ||·||_(()WLip_(q,α)).As an application,we show that the commutator M_(β)~b is bounded from L~p to L^(q,∞) for some p ∈(1,∞) and 1/p-1/q=(α+β)/n if and only if b is in Lip_(α).We also introduce the weak central bounded mean oscillation space WCBMO_(q,α) and give a characterization of WCBMO_(q,α) via the boundedness of the commutators of Hardy type operators.展开更多
基金Supported by NSF of China (10371087)NSF of Anhui Province(07021019)Education Committee of Anhui Province(KJ2007A009)
文摘Let μΩ,b^m be the higher order commutator generated by Marcinkiewicz integral μΩ and a BMO（R^n） function b（x）. In this paper, we will study the continuity of μΩ and μΩ,b^m on homogeneous Morrey-Herz spaces.
文摘A weak type endpoint estimate for the maximal multilinear singular integral operator TAf（x）=supε〉0｜∫｜x-y｜〉εΩ（x-y）/｜x-y｜^n＋1（A（x）-A（y）-VA（y）（x-y））f（y）dy｜ is established, where Ω is homogeneous of degree zero, integrable on the unit sphere and has vanishing moment of order one, and A has derivatives of order one in BMO（R^n）. A regularity condition on Ω which implies an LlogL type estimate of TA is given.
基金supported by Grant-in-Aid for Scientific Research (C) of Japan Society for the Promotion of Science (Grant No. 24540159)Grant-in-Aid for Young Scientists (B) of Japan Society for the Promotion of Science (Grant No. 24540085)
文摘We establish the theory of Orlicz-Hardy spaces generated by a wide class of functions.The class will be wider than the class of all the N-functions.In particular,we consider the non-smooth atomic decomposition.The relation between Orlicz-Hardy spaces and their duals is also studied.As an application,duality of Hardy spaces with variable exponents is revisited.This work is different from earlier works about Orlicz-Hardy spaces H（Rn）in that the class of admissible functions is largely widened.We can deal with,for example,Ф（r）≡（rp1（log（e＋1/r））q1,0〈r≤1,r^p2 （log（e＋r））q2,r〉1,with p1,p2∈（0,∞）and q1,q2∈（.∞,∞）,where we shall establish the boundedness of the Riesz transforms on H（Rn）.In particular,is neither convex nor concave when 0〈p1〈1〈p2〈∞,0〈p21〈p1〈∞or p1=p2=1 and q1,q20.If（r）≡r（log（e＋r））q,then H（Rn）=H（logH）q（Rn）.We shall also establish the boundedness of the fractional integral operators I of order∈（0,∞）.For example,I is shown to be bounded from H（logH）1^1-α/n（Rn）to Ln/（n-α）（log L）（Rn）for 0〈α〈n.
基金The first author was supported by the TianYuan Special Funds of the National Natural Science Foundation of China （Grant No. 11426221） and the High Level Introduction of Talent Research Start-up Fund by Central South University of Forestory and Technology （Grant No. 1040212） the second author was supported by the National Natural Science Foundation of China （Grant No. 11361020）.
文摘The boundedness of multilinear Calderdn-Zygmund operators and their commutators with bounded mean oscillation （BMO） functions in variable exponent Morrey spaces are obtained.
基金Supported by the National Natural Science Foundation of China (Grant No.11871452)the Natural Science Foundation of Henan Province (Grant No.202300410338)the Nanhu Scholar Program for Young Scholars of Xinyang Normal University。
文摘In this paper,we first introduce the weak Lipschitz spaces WLip_(q,α),1<q<∞,0<α<1 which are the analog of weak Lebesgue spaces L^(q,∞)in the setting of Lipschitz space.We obtain the equivalence between the norm ||·||_(Lipα)and ||·||_(()WLip_(q,α)).As an application,we show that the commutator M_(β)~b is bounded from L~p to L^(q,∞) for some p ∈(1,∞) and 1/p-1/q=(α+β)/n if and only if b is in Lip_(α).We also introduce the weak central bounded mean oscillation space WCBMO_(q,α) and give a characterization of WCBMO_(q,α) via the boundedness of the commutators of Hardy type operators.
