In the article we consider the fractional maximal operator Mα, 0 ≤α 〈 Q on any Carnot group G （i.e., nilpotent stratified Lie group） in the generalized Morrey spaces Mp,φ（G）, where Q is the homogeneous dimens...In the article we consider the fractional maximal operator Mα, 0 ≤α 〈 Q on any Carnot group G （i.e., nilpotent stratified Lie group） in the generalized Morrey spaces Mp,φ（G）, where Q is the homogeneous dimension of G. We find the conditions on the pair （φ1, φ2） which ensures the boundedness of the operator Ms from one generalized Morrey space Mp,φ1 （G） to another Mq,φ2 （G）, 1. 〈 p ≤q 〈 ∞. 1/p - 1/q = α/Q, and from the space M1,φ1 （G） to the weak space Wq,φ2 （G）, 1 〈 q 〈 ∞, 1 - 1/q = α/Q. Also find conditions on the φ which ensure the Adams type boundedness of the Ms from M α （G） from Mp,φ^1/p（G）to Mq,φ^1/q（G） for 1 〈p〈q〈∞ and fromM1,φ（G） toWMq,φ^1/q（G）for 1〈q〈∞. In the case b ∈ BMO（G） and 1 〈 p 〈 q 〈 ∞, find the sufficient conditions on the pair （φ1, φ2） which ensures the boundedness of the kth-order commutator operator Mb,α,k from Mp,φ1 （G） to Mq,φ2（G） with 1/p - 1/q = α/Q. Also find the sufficient conditions on the φ which ensures the boundedness of the operator Mb,α,k from Mp,φ^1/p（G） tom Mp,φ^1/p （G） for 1 〈p〈q〈∞. In all the cases the conditions for the boundedness of Mα are given it terms of supremaltype inequalities on （φ1, φ2） and φ , which do not assume any assumption on monotonicity of （φ1, φ2） and φ in r. As applications we consider the SchrSdinger operator -△G ＋ V on G, where the nonnegative potential V belongs to the reverse Holder class B∞（G）. The MB,φ1 - Mq,φ2 estimates for the operators V^γ（-△G ＋ V）^-β and V^γ△↓G（-△G ＋ V）^-β are obtained.展开更多
In this paper, we prove the boundedness of the fractional maximal operator, Hardy-Littlewood maximal operator and marcinkiewicz integrals associated with Schrodinger operator on Morrey spaces with variable exponent.
Let M α be the fractional maximal operators (0<α≤1) and (u,v) a pair of weight functions, u∈D ∞, σ=v~~~~^(-1/(p-1))∈A ∞. The boundedness of M α on some homogenous groups (G, ‖·‖, dx) and the cov...Let M α be the fractional maximal operators (0<α≤1) and (u,v) a pair of weight functions, u∈D ∞, σ=v~~~~^(-1/(p-1))∈A ∞. The boundedness of M α on some homogenous groups (G, ‖·‖, dx) and the covering Lemma of Calderon-Zygmund type are studied. Not only an adequate covering Lemma of Calderon-Zygmund type is shown, but also the boundedness of fractional maximal operators M α(0<α≤1) on some of homogeneous groups with respect to a given pair of weight functions (u,v) as above is proved. Moreover, a sufficient and necessary condition for M α∈B(u^qdx, v~~pdx), 0<α<1, 1<p<1α, and 1q=1p-α is also given. Finally, an application of the results is also obtained.展开更多
Let △ be full Laplacian on H-type group G. Then for every compact set D Ga local estimate of the Schrodinger maximal operator holds, that is,∫D^sup0〈t〈1｜e^it△f（x）｜^2dx≤｜｜f｜｜^2H^s,s〉1/2We also show that ...Let △ be full Laplacian on H-type group G. Then for every compact set D Ga local estimate of the Schrodinger maximal operator holds, that is,∫D^sup0〈t〈1｜e^it△f（x）｜^2dx≤｜｜f｜｜^2H^s,s〉1/2We also show that the above inequality fails when s 〈 1/4.展开更多
The aim of this paper is to establish several necessary and sufficient conditions in order that the weighted inequality ρ（M f 〉 λ）Φ（λ） ≤ C ∫_Ω~Ψ （C｜f｜）σdμ,λ 〉 0 or ρ（Mf〉λ） ≤ C∫-Ω~Φ（Cλ^...The aim of this paper is to establish several necessary and sufficient conditions in order that the weighted inequality ρ（M f 〉 λ）Φ（λ） ≤ C ∫_Ω~Ψ （C｜f｜）σdμ,λ 〉 0 or ρ（Mf〉λ） ≤ C∫-Ω~Φ（Cλ^-1 ｜f｜）σdμ,λ 〉0 holds for every uniformly integral martingale f=（f_n）, where M is the Doob＇s maximal operator, Φ, Ψ are both Φ-functions, and e, σ are weights.展开更多
We introduce a class of generalized Orlicz-type Auscher-Mourgoglou slice space,which is a special case of the Wiener amalgam.We prove versions of the Rubio de Francia extrapolation theorem in this space.As a consequen...We introduce a class of generalized Orlicz-type Auscher-Mourgoglou slice space,which is a special case of the Wiener amalgam.We prove versions of the Rubio de Francia extrapolation theorem in this space.As a consequence,we obtain the boundedness results for several classical operators,such as the Calderón-Zygmund operator,the Marcinkiewicz integrals,the Bochner-Riesz means and the Riesz potential,as well as variational inequalities for differential operators and singular integrals.As an application,we obtain global regularity estimates for solutions of non-divergence elliptic equations on generalized Orlicz-type slice spaces if the coefficient matrix is symmetric,uniformly elliptic and has a small(δ,R)-BMO norm for some positive numbers δ and R.展开更多
Let φ be a growth function, and let A ：= -（V- ia）. （V- ia）＋ V be a magnetic SchrSdinger operator on L2（Rn）, n≥ 2, where a ：= （a1, a2... an） ∈ r L1 loc（Rn） We establish the equivalent characteriza- L2 ...Let φ be a growth function, and let A ：= -（V- ia）. （V- ia）＋ V be a magnetic SchrSdinger operator on L2（Rn）, n≥ 2, where a ：= （a1, a2... an） ∈ r L1 loc（Rn） We establish the equivalent characteriza- L2 1oc（Rn, Rn） and 0 ≤ V ∈Lloc（Rn） tions of the Musielak-Orlicz-Hardy space HA,^（IRn）, defined by the Lusin area function associated with {e-t2A}t〉0, in terms of the Lusin area function associated with {e-t√A}t〉0, the radial maximal functions and the non- tangential maximal functions associated with {e-t2A}t〉o and {e-t√A}t〉0, respectively. The boundedness of the Riesz transforms LkA-U1/2, k ∈ {1, 2... n}, from HA,φ（Rn） to Lφ（Rn） is also presented, where Lk is the closure of δ/δxk iak in L2（Rn）. These results are new even when φ（x,t） ：= w（x）tp for all x ∈Rn and t∈ （0, ＋∞） with p ∈ （0, 1] and ω∈ A∞（Rn） （the class of Muckenhoupt weights on Rn）.展开更多
基金partially supported by the grant of Ahi Evran University Scientific Research Projects(FEN 4001.12.0018)partially supported by the grant of Ahi Evran University Scientific Research Projects(FEN 4001.12.0019)+1 种基金by the grant of Science Development Foundation under the President of the Republic of Azerbaijan project EIF-2010-1(1)-40/06-1partially supported by the Scientific and Technological Research Council of Turkey(TUBITAK Project No:110T695)
文摘In the article we consider the fractional maximal operator Mα, 0 ≤α 〈 Q on any Carnot group G （i.e., nilpotent stratified Lie group） in the generalized Morrey spaces Mp,φ（G）, where Q is the homogeneous dimension of G. We find the conditions on the pair （φ1, φ2） which ensures the boundedness of the operator Ms from one generalized Morrey space Mp,φ1 （G） to another Mq,φ2 （G）, 1. 〈 p ≤q 〈 ∞. 1/p - 1/q = α/Q, and from the space M1,φ1 （G） to the weak space Wq,φ2 （G）, 1 〈 q 〈 ∞, 1 - 1/q = α/Q. Also find conditions on the φ which ensure the Adams type boundedness of the Ms from M α （G） from Mp,φ^1/p（G）to Mq,φ^1/q（G） for 1 〈p〈q〈∞ and fromM1,φ（G） toWMq,φ^1/q（G）for 1〈q〈∞. In the case b ∈ BMO（G） and 1 〈 p 〈 q 〈 ∞, find the sufficient conditions on the pair （φ1, φ2） which ensures the boundedness of the kth-order commutator operator Mb,α,k from Mp,φ1 （G） to Mq,φ2（G） with 1/p - 1/q = α/Q. Also find the sufficient conditions on the φ which ensures the boundedness of the operator Mb,α,k from Mp,φ^1/p（G） tom Mp,φ^1/p （G） for 1 〈p〈q〈∞. In all the cases the conditions for the boundedness of Mα are given it terms of supremaltype inequalities on （φ1, φ2） and φ , which do not assume any assumption on monotonicity of （φ1, φ2） and φ in r. As applications we consider the SchrSdinger operator -△G ＋ V on G, where the nonnegative potential V belongs to the reverse Holder class B∞（G）. The MB,φ1 - Mq,φ2 estimates for the operators V^γ（-△G ＋ V）^-β and V^γ△↓G（-△G ＋ V）^-β are obtained.
基金supported by NSFC (No. 11201003)University NSR Project of Anhui Province (No. KJ2014A087)
文摘In this paper, we prove the boundedness of the fractional maximal operator, Hardy-Littlewood maximal operator and marcinkiewicz integrals associated with Schrodinger operator on Morrey spaces with variable exponent.
文摘Let M α be the fractional maximal operators (0<α≤1) and (u,v) a pair of weight functions, u∈D ∞, σ=v~~~~^(-1/(p-1))∈A ∞. The boundedness of M α on some homogenous groups (G, ‖·‖, dx) and the covering Lemma of Calderon-Zygmund type are studied. Not only an adequate covering Lemma of Calderon-Zygmund type is shown, but also the boundedness of fractional maximal operators M α(0<α≤1) on some of homogeneous groups with respect to a given pair of weight functions (u,v) as above is proved. Moreover, a sufficient and necessary condition for M α∈B(u^qdx, v~~pdx), 0<α<1, 1<p<1α, and 1q=1p-α is also given. Finally, an application of the results is also obtained.
基金supported by National Nature Science Foundation of China(11371036)
文摘Let △ be full Laplacian on H-type group G. Then for every compact set D Ga local estimate of the Schrodinger maximal operator holds, that is,∫D^sup0〈t〈1｜e^it△f（x）｜^2dx≤｜｜f｜｜^2H^s,s〉1/2We also show that the above inequality fails when s 〈 1/4.
基金Supported by the National Natural Science Foundation of China (1067114711071190)
文摘The aim of this paper is to establish several necessary and sufficient conditions in order that the weighted inequality ρ（M f 〉 λ）Φ（λ） ≤ C ∫_Ω~Ψ （C｜f｜）σdμ,λ 〉 0 or ρ（Mf〉λ） ≤ C∫-Ω~Φ（Cλ^-1 ｜f｜）σdμ,λ 〉0 holds for every uniformly integral martingale f=（f_n）, where M is the Doob＇s maximal operator, Φ, Ψ are both Φ-functions, and e, σ are weights.
基金supported by the National Natural Science Foundation of China(11726622)the Natural Science Foundation Projection of Chongqing,China(cstc2021jcyj-msxmX0705).
文摘We introduce a class of generalized Orlicz-type Auscher-Mourgoglou slice space,which is a special case of the Wiener amalgam.We prove versions of the Rubio de Francia extrapolation theorem in this space.As a consequence,we obtain the boundedness results for several classical operators,such as the Calderón-Zygmund operator,the Marcinkiewicz integrals,the Bochner-Riesz means and the Riesz potential,as well as variational inequalities for differential operators and singular integrals.As an application,we obtain global regularity estimates for solutions of non-divergence elliptic equations on generalized Orlicz-type slice spaces if the coefficient matrix is symmetric,uniformly elliptic and has a small(δ,R)-BMO norm for some positive numbers δ and R.
文摘Let φ be a growth function, and let A ：= -（V- ia）. （V- ia）＋ V be a magnetic SchrSdinger operator on L2（Rn）, n≥ 2, where a ：= （a1, a2... an） ∈ r L1 loc（Rn） We establish the equivalent characteriza- L2 1oc（Rn, Rn） and 0 ≤ V ∈Lloc（Rn） tions of the Musielak-Orlicz-Hardy space HA,^（IRn）, defined by the Lusin area function associated with {e-t2A}t〉0, in terms of the Lusin area function associated with {e-t√A}t〉0, the radial maximal functions and the non- tangential maximal functions associated with {e-t2A}t〉o and {e-t√A}t〉0, respectively. The boundedness of the Riesz transforms LkA-U1/2, k ∈ {1, 2... n}, from HA,φ（Rn） to Lφ（Rn） is also presented, where Lk is the closure of δ/δxk iak in L2（Rn）. These results are new even when φ（x,t） ：= w（x）tp for all x ∈Rn and t∈ （0, ＋∞） with p ∈ （0, 1] and ω∈ A∞（Rn） （the class of Muckenhoupt weights on Rn）.
