Consider a pseudo-differential operator T_(a)f(x)=∫_(R^(n))e^(ix,ζ)a(x,ζ)f(ζ)dζwhere the symbol a is in the rough Hormander class L^(∞)S_(ρ)^(m)with m∈R andρ∈[0,1].In this note,when 1≤p≤2,if n(ρ-1)/p and ...Consider a pseudo-differential operator T_(a)f(x)=∫_(R^(n))e^(ix,ζ)a(x,ζ)f(ζ)dζwhere the symbol a is in the rough Hormander class L^(∞)S_(ρ)^(m)with m∈R andρ∈[0,1].In this note,when 1≤p≤2,if n(ρ-1)/p and a∈L^(∞)S_(ρ)^(m),then for any f∈S(R^(n))and x∈R^(n),we prove that M(T_(a)f)(x)≤C(M(|f|^(p))(x))^(1/p) where M is the Hardy-Littlewood maximal operator.Our theorem improves the known results and the bound on m is sharp,in the sense that n(ρ-1)/p can not be replaced by a larger constant.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
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.
In this article, the authors introduce two operators-geometrical maximal operator Mo and the closely related limiting operator M0^＊, then they give sufficient conditions under which the equality M0=MM0^＊ holds, and ...In this article, the authors introduce two operators-geometrical maximal operator Mo and the closely related limiting operator M0^＊, then they give sufficient conditions under which the equality M0=MM0^＊ holds, and characterize the equivalent relations between the weak or strong type weighted inequality and the property of W∞-weight or W∞^＊-weight for the geometrical maximal operator in the case of two-weight condition. What should be mentioned is that the new operator-the geometrical minimal operator is equal to the limitation of the minimal operator sequence, and the results for the minimal operator proved in [12] makes the proof elegant and evident.展开更多
Let M f be the Kakeya maximal function in d-dimensional Euclidean space, with same base consisting of cylinders of eccentricity N. The inequality shoum for a base satisfying a direction condition, where ?and c are con...Let M f be the Kakeya maximal function in d-dimensional Euclidean space, with same base consisting of cylinders of eccentricity N. The inequality shoum for a base satisfying a direction condition, where ?and c are constants depending only on d.展开更多
In this note the authors give the weighted Lp-boundedness for a class of maximal singular integral operators with rough kernel. The result in this note is an improvement and extension of the result obtained by Chen a...In this note the authors give the weighted Lp-boundedness for a class of maximal singular integral operators with rough kernel. The result in this note is an improvement and extension of the result obtained by Chen and Lin in 1990.展开更多
With Ω ∈ L (log + L)(S n-1 ) and suitable h ∈ L γ (R 1)(1<γ≤2),the weak type (1,1) of the square function g(f)(x) =k｜ψ k*f｜ 2 12(x) and the maximal operator M ψ(f)(x) = sup ...With Ω ∈ L (log + L)(S n-1 ) and suitable h ∈ L γ (R 1)(1<γ≤2),the weak type (1,1) of the square function g(f)(x) =k｜ψ k*f｜ 2 12(x) and the maximal operator M ψ(f)(x) = sup k｜ψ k｜*｜f｜(x) where ψ(x)=｜x｜ -n Ω(x)h(｜x｜),ψ k(x)=ψ 2 k (x), are studied in this paper.As a corollary,the weak bounds of M Ω(f) proved by Christ in 1988 are given and the previous weak type results for M ψ(f)(x) are improved.In addition,the weighted weak type (1,1) estimates of the Littlewood Paley function g ψ(f) with power weights is also proved.展开更多
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 0<β<1 andΩbe a proper open and non-empty subset of R^(n).In this paper,the object of our investigation is the multilinear local maximal operator Mβ,defined by M_(β)((f))(x)=sup_(Q(∈)xQ∈Fβ)Π_(i=1)^m1/...Let 0<β<1 andΩbe a proper open and non-empty subset of R^(n).In this paper,the object of our investigation is the multilinear local maximal operator Mβ,defined by M_(β)((f))(x)=sup_(Q(∈)xQ∈Fβ)Π_(i=1)^m1/|Q|∫_(Q)|f_(i)(y_(i))|dy_(i),where F_(β)={Q(x,l):x∈Ω,l<βd(x,Ω^(c))},Q=Q(x,l)is denoted as a cube with sides parallel to the axes,and x and l denote its center and half its side length.Two-weight characterizations for the multilinear local maximal operator M_(β)are obtained.A formulation of the Carleson embedding theorem in the multilinear setting is proved.展开更多
Let Ф be a Young function and MФ be the maximal operator associated with in the space of homogeneous type. In this paper, the composition of the maximal operators of type MФ is considered, and the result establishe...Let Ф be a Young function and MФ be the maximal operator associated with in the space of homogeneous type. In this paper, the composition of the maximal operators of type MФ is considered, and the result established by Carrozza and Passarelli Di Napoli is generalized to the space of homogeneous type.展开更多
It provides the boundary proof of the maximal operator on Lipschitz space. i.e.,if f ∈Lip α(R n) (0<α<1) and inf x∈R nM(f)(x) <∞, then almost every x∈R n has M(f)(x) <∞ and exists a constant C indep...It provides the boundary proof of the maximal operator on Lipschitz space. i.e.,if f ∈Lip α(R n) (0<α<1) and inf x∈R nM(f)(x) <∞, then almost every x∈R n has M(f)(x) <∞ and exists a constant C independent of f and x ,such that ‖M(f)‖ ∧ α ≤C‖f‖ ∧ α .展开更多
In this paper we discuss the weak type(IP,I)boundedness of a class of maximal operators T and themaximal strong,mean boundedness of a family of the operators {T on the atomic IP spaces on compaet Lie groups.Also,we ob...In this paper we discuss the weak type(IP,I)boundedness of a class of maximal operators T and themaximal strong,mean boundedness of a family of the operators {T on the atomic IP spaces on compaet Lie groups.Also,we obtain the correspoding convergent rosults.展开更多
For the plane curves Γ,the maximal operator associated to it is defined by Mf(x)=sup|∫f(x-Γ(t))(r^(-1)t)r^(-1)dt| where is a Schwartz function.For a certain class of curves in R^2,M is shown to bounded on (H(R^2)...For the plane curves Γ,the maximal operator associated to it is defined by Mf(x)=sup|∫f(x-Γ(t))(r^(-1)t)r^(-1)dt| where is a Schwartz function.For a certain class of curves in R^2,M is shown to bounded on (H(R^2),Weak L^1(R^2).This extends the theorem of Stein & Wainger and the theo- rem of Weinberg.展开更多
We consider a first order periodic system in R^(N),involving a time dependent maximal monotone operator which need not have a full domain and a multivalued perturbation.We prove the existence theorems for both the con...We consider a first order periodic system in R^(N),involving a time dependent maximal monotone operator which need not have a full domain and a multivalued perturbation.We prove the existence theorems for both the convex and nonconvex problems.We also show the existence of extremal periodic solutions and provide a strong relaxation theorem.Finally,we provide an application to nonlinear periodic control systems.展开更多
In this paper,the authors show that the maximal operators of the multilinear Calderón-Zygmund singular integrals are bounded from a product of weighted Hardy spaces into a weighted Lebesgue spaces,which essential...In this paper,the authors show that the maximal operators of the multilinear Calderón-Zygmund singular integrals are bounded from a product of weighted Hardy spaces into a weighted Lebesgue spaces,which essentially extend and improve the previous known results obtained by Grafakos and Kalton(2001)and Li,Xue and Yabuta(2011).The corresponding estimates on variable Hardy spaces are also established.展开更多
In this paper,we establish the strong and weak boundedness of the multilinear maximal operator in the setting of the Choquet integral with respect to theαdimensional Hausdorff content.Our results cover Orobitg and Ve...In this paper,we establish the strong and weak boundedness of the multilinear maximal operator in the setting of the Choquet integral with respect to theαdimensional Hausdorff content.Our results cover Orobitg and Verdera’s results in[8].展开更多
Let p is an element of (n/(n + 1), 1]. The authors investigate the (H-b(p)(R-n), L-p(R-n))-type and (H-b(p,infinity)(R-n), L-p,L-infinity(R-n))-type continuities for the maximal operators associated with the commutato...Let p is an element of (n/(n + 1), 1]. The authors investigate the (H-b(p)(R-n), L-p(R-n))-type and (H-b(p,infinity)(R-n), L-p,L-infinity(R-n))-type continuities for the maximal operators associated with the commutators of Bochner-Riesz operators with BMO(R-n) functions, where H-b(p)(R-n) and H-b(p,infinity)(R-n) are, respectively, the variants of the standard Hardy spaces and the standard weak Hardy spaces.展开更多
基金Supported by the National Natural Science Foundation of China(11871436,12071437)。
文摘Consider a pseudo-differential operator T_(a)f(x)=∫_(R^(n))e^(ix,ζ)a(x,ζ)f(ζ)dζwhere the symbol a is in the rough Hormander class L^(∞)S_(ρ)^(m)with m∈R andρ∈[0,1].In this note,when 1≤p≤2,if n(ρ-1)/p and a∈L^(∞)S_(ρ)^(m),then for any f∈S(R^(n))and x∈R^(n),we prove that M(T_(a)f)(x)≤C(M(|f|^(p))(x))^(1/p) where M is the Hardy-Littlewood maximal operator.Our theorem improves the known results and the bound on m is sharp,in the sense that n(ρ-1)/p can not be replaced by a larger constant.
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 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.
基金supported by the NSF of China and the aid financial plan for the backbone of the young teachers of university of Henan
文摘In this article, the authors introduce two operators-geometrical maximal operator Mo and the closely related limiting operator M0^＊, then they give sufficient conditions under which the equality M0=MM0^＊ holds, and characterize the equivalent relations between the weak or strong type weighted inequality and the property of W∞-weight or W∞^＊-weight for the geometrical maximal operator in the case of two-weight condition. What should be mentioned is that the new operator-the geometrical minimal operator is equal to the limitation of the minimal operator sequence, and the results for the minimal operator proved in [12] makes the proof elegant and evident.
基金The author is partly supported by the Grants-in-Aid for Scientific Reseach,The Ministry of Educa-ion,Science and Culture,Japan.
文摘Let M f be the Kakeya maximal function in d-dimensional Euclidean space, with same base consisting of cylinders of eccentricity N. The inequality shoum for a base satisfying a direction condition, where ?and c are constants depending only on d.
文摘In this note the authors give the weighted Lp-boundedness for a class of maximal singular integral operators with rough kernel. The result in this note is an improvement and extension of the result obtained by Chen and Lin in 1990.
文摘With Ω ∈ L (log + L)(S n-1 ) and suitable h ∈ L γ (R 1)(1<γ≤2),the weak type (1,1) of the square function g(f)(x) =k｜ψ k*f｜ 2 12(x) and the maximal operator M ψ(f)(x) = sup k｜ψ k｜*｜f｜(x) where ψ(x)=｜x｜ -n Ω(x)h(｜x｜),ψ k(x)=ψ 2 k (x), are studied in this paper.As a corollary,the weak bounds of M Ω(f) proved by Christ in 1988 are given and the previous weak type results for M ψ(f)(x) are improved.In addition,the weighted weak type (1,1) estimates of the Littlewood Paley function g ψ(f) with power weights is also proved.
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 partly by the Natural Science Foundation from the Education Department of Anhui Province(KJ2017A847)The second author was supported by NSFC(11671039,11871101)NSFC-DFG(11761131002).
文摘Let 0<β<1 andΩbe a proper open and non-empty subset of R^(n).In this paper,the object of our investigation is the multilinear local maximal operator Mβ,defined by M_(β)((f))(x)=sup_(Q(∈)xQ∈Fβ)Π_(i=1)^m1/|Q|∫_(Q)|f_(i)(y_(i))|dy_(i),where F_(β)={Q(x,l):x∈Ω,l<βd(x,Ω^(c))},Q=Q(x,l)is denoted as a cube with sides parallel to the axes,and x and l denote its center and half its side length.Two-weight characterizations for the multilinear local maximal operator M_(β)are obtained.A formulation of the Carleson embedding theorem in the multilinear setting is proved.
基金Supported by the NSF from the Education Department of Henan Province(2007110006)
文摘Let Ф be a Young function and MФ be the maximal operator associated with in the space of homogeneous type. In this paper, the composition of the maximal operators of type MФ is considered, and the result established by Carrozza and Passarelli Di Napoli is generalized to the space of homogeneous type.
文摘It provides the boundary proof of the maximal operator on Lipschitz space. i.e.,if f ∈Lip α(R n) (0<α<1) and inf x∈R nM(f)(x) <∞, then almost every x∈R n has M(f)(x) <∞ and exists a constant C independent of f and x ,such that ‖M(f)‖ ∧ α ≤C‖f‖ ∧ α .
文摘In this paper we discuss the weak type(IP,I)boundedness of a class of maximal operators T and themaximal strong,mean boundedness of a family of the operators {T on the atomic IP spaces on compaet Lie groups.Also,we obtain the correspoding convergent rosults.
文摘For the plane curves Γ,the maximal operator associated to it is defined by Mf(x)=sup|∫f(x-Γ(t))(r^(-1)t)r^(-1)dt| where is a Schwartz function.For a certain class of curves in R^2,M is shown to bounded on (H(R^2),Weak L^1(R^2).This extends the theorem of Stein & Wainger and the theo- rem of Weinberg.
基金supported by the NSFC(12071413)the Guangxi Natural Sci-ence Foundation(2023GXNSFAA026085)the European Union's Horizon 2020 Research and Innovation Programme under the Marie Sklodowska-Curie grant agreement No.823731 CONMECH。
文摘We consider a first order periodic system in R^(N),involving a time dependent maximal monotone operator which need not have a full domain and a multivalued perturbation.We prove the existence theorems for both the convex and nonconvex problems.We also show the existence of extremal periodic solutions and provide a strong relaxation theorem.Finally,we provide an application to nonlinear periodic control systems.
基金supported by the National Natural Science Foundation of China(Nos.11871101,12171399)NSFC-DFG(No.11761131002)+3 种基金the Natural Science Foundation of Fujian Province(No.2021J05188)the Scientific Research Project of The Education Department of Fujian Province(No.JAT200331)the President’s fund of Minnan Normal University(No.KJ2020020)the Institute of Meteorological Big Data-Digital Fujian,Fujian Key Laboratory of Data Science and Statistics and Fujian Key Laboratory of Granular Computing and Applications(Minnan Normal University)。
文摘In this paper,the authors show that the maximal operators of the multilinear Calderón-Zygmund singular integrals are bounded from a product of weighted Hardy spaces into a weighted Lebesgue spaces,which essentially extend and improve the previous known results obtained by Grafakos and Kalton(2001)and Li,Xue and Yabuta(2011).The corresponding estimates on variable Hardy spaces are also established.
基金supported by the National Natural Science Foundation of China (Grant Nos.11871452 and 12071473)Beijing Information Science and Technology University Foundation (Grant No.2025031).
文摘In this paper,we establish the strong and weak boundedness of the multilinear maximal operator in the setting of the Choquet integral with respect to theαdimensional Hausdorff content.Our results cover Orobitg and Verdera’s results in[8].
基金Tang Lin and Yang Dachun are supported in part by the NNSF and the SEDF of China.
文摘Let p is an element of (n/(n + 1), 1]. The authors investigate the (H-b(p)(R-n), L-p(R-n))-type and (H-b(p,infinity)(R-n), L-p,L-infinity(R-n))-type continuities for the maximal operators associated with the commutators of Bochner-Riesz operators with BMO(R-n) functions, where H-b(p)(R-n) and H-b(p,infinity)(R-n) are, respectively, the variants of the standard Hardy spaces and the standard weak Hardy spaces.
