Assume that L is a non-negative self-adjoint operator on L^(2)(ℝ^(n))with its heat kernels satisfying the so-called Gaussian upper bound estimate and that X is a ball quasi-Banach function space onℝ^(n) satisfying som...Assume that L is a non-negative self-adjoint operator on L^(2)(ℝ^(n))with its heat kernels satisfying the so-called Gaussian upper bound estimate and that X is a ball quasi-Banach function space onℝ^(n) satisfying some mild assumptions.Let HX,L(ℝ^(n))be the Hardy space associated with both X and L,which is defined by the Lusin area function related to the semigroup generated by L.In this article,the authors establish various maximal function characterizations of the Hardy space HX,L(ℝ^(n))and then apply these characterizations to obtain the solvability of the related Cauchy problem.These results have a wide range of generality and,in particular,the specific spaces X to which these results can be applied include the weighted space,the variable space,the mixed-norm space,the Orlicz space,the Orlicz-slice space,and the Morrey space.Moreover,the obtained maximal function characterizations of the mixed-norm Hardy space,the Orlicz-slice Hardy space,and the Morrey-Hardy space associated with L are completely new.展开更多
This paper focuses on the continuity of the truncated Hardy-Littlewood maximal function.We first show that the truncated Hardy-Littlewood maximal function is lower semi-continuous.Then by investigating the behavior of...This paper focuses on the continuity of the truncated Hardy-Littlewood maximal function.We first show that the truncated Hardy-Littlewood maximal function is lower semi-continuous.Then by investigating the behavior of the truncated Hardy-Littlewood maximal function when the truncated parameterγchanges,we obtain an equivalent condition of the continuity of the truncated Hardy-Littlewood maximal function.展开更多
Let(X,d,μ)be a non-homogeneous metric measure space satisfying the geometrically doubling condition and the upper doubling condition.In this setting,the authors prove that the commutator M_(b)^(α)formed by b∈RBMO(...Let(X,d,μ)be a non-homogeneous metric measure space satisfying the geometrically doubling condition and the upper doubling condition.In this setting,the authors prove that the commutator M_(b)^(α)formed by b∈RBMO(μ)and the fractional maximal function M^((α))is bounded from Lebesgue spaces L^(p)(μ)into spaces L^(q)(μ),where 1/q=1/p-αforα∈(0,1)and p∈(1,1/α).Furthermore,the boundedness of the M_(b)^(α)on Orlicz spaces L^Φ(μ)is established.展开更多
Let 2≤p【∞ and let (f n) be a martingale. Using exponential bounds of the probabilities of the type P(|f n|】λ‖T(f n)‖ ∞) for some quasi-linear operators acting on martingales, we estimate upper bounds for t...Let 2≤p【∞ and let (f n) be a martingale. Using exponential bounds of the probabilities of the type P(|f n|】λ‖T(f n)‖ ∞) for some quasi-linear operators acting on martingales, we estimate upper bounds for the L p-norms of the maximal functions of martinglaes. Our result is the extension and improvements of the results obtained previously by HITCZENKO and ZENG .展开更多
In this paper, we get W 1,p(Rn)-boundedness for tangential maximal func- tion and nontangential maximal function , which improves J.Kinnunen, P.Lindqvist and Tananka’s results.
A new maximal function is introduced in the dual spaces of test function spaces on spaces of homogeneous type. Using this maximal function, we get new characterization of atomic H^p spaces.
For maximal multilinear Calderon-Zygmund singular integral operators, the sharp maximal function estimate and some weighted norm inequalities are obtained.
For a general normed vector space,a special optimal value function called a maximal time function is considered.This covers the farthest distance function as a special case,and has a close relationship with the smalle...For a general normed vector space,a special optimal value function called a maximal time function is considered.This covers the farthest distance function as a special case,and has a close relationship with the smallest enclosing ball problem.Some properties of the maximal time function are proven,including the convexity,the lower semicontinuity,and the exact characterizations of its subdifferential formulas.展开更多
Let X be an RD-space, i.e., a space of homogeneous type in the sense of Coifman and Weiss, which has the reverse doubling property. Assume that X has a “dimension” n. For α ∈ (0, ∞) denote by H α p (X), H d p (X...Let X be an RD-space, i.e., a space of homogeneous type in the sense of Coifman and Weiss, which has the reverse doubling property. Assume that X has a “dimension” n. For α ∈ (0, ∞) denote by H α p (X), H d p (X), and H *,p (X) the corresponding Hardy spaces on X defined by the nontangential maximal function, the dyadic maximal function and the grand maximal function, respectively. Using a new inhomogeneous Calderón reproducing formula, it is shown that all these Hardy spaces coincide with L p (X) when p ∈ (1,∞] and with each other when p ∈ (n/(n + 1), 1]. An atomic characterization for H ?,p (X) with p ∈ (n/(n + 1), 1] is also established; moreover, in the range p ∈ (n/(n + 1),1], it is proved that the space H *,p (X), the Hardy space H p (X) defined via the Littlewood-Paley function, and the atomic Hardy space of Coifman andWeiss coincide. Furthermore, it is proved that a sublinear operator T uniquely extends to a bounded sublinear operator from H p (X) to some quasi-Banach space B if and only if T maps all (p, q)-atoms when q ∈ (p, ∞)∩[1, ∞) or continuous (p, ∞)-atoms into uniformly bounded elements of B.展开更多
Let M be the multilinear maximal function and = （b1, . . . , bm） be a collection of locally integrable functions. Denote by M and [,M] the maximal commutator and the commutator of M with , respectively. In this pape...Let M be the multilinear maximal function and = （b1, . . . , bm） be a collection of locally integrable functions. Denote by M and [,M] the maximal commutator and the commutator of M with , respectively. In this paper, the multiple weighted strong and weak type estimates for operators M and [,M] are studied. Some characterizations of the class of functions bj are given, for which these operators satisfy some strong or weak type estimates.展开更多
In this paper,we systematically study several classes of maximal singular integrals and maximal functions with rough kernels in Fβ(S^n-1),a topic that relates to the Grafakos-Stefanov class.The boundedness and contin...In this paper,we systematically study several classes of maximal singular integrals and maximal functions with rough kernels in Fβ(S^n-1),a topic that relates to the Grafakos-Stefanov class.The boundedness and continuity of these operators on Triebel-Lizorkin spaces and Besov spaces are discussed.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
In the paper,we characterize a necessary and sufficient condition which ensures the continuities of the non-centered Hardy-Lit tiewood maximal function Mf and the centered Hardy-Lit tiewood maximal function Mcf on R^n...In the paper,we characterize a necessary and sufficient condition which ensures the continuities of the non-centered Hardy-Lit tiewood maximal function Mf and the centered Hardy-Lit tiewood maximal function Mcf on R^n.As two applications,we can easily deduce that Mcf and Mf are continuous if f is continuous,and Mf is continuous if f is of local bounded variation on R.展开更多
We obtain the characterizations of commutators of several versions of maximal functions on spaces of homogeneous type.In addition,with the aid of interpolation theory,we provide weighted version of the commutator theo...We obtain the characterizations of commutators of several versions of maximal functions on spaces of homogeneous type.In addition,with the aid of interpolation theory,we provide weighted version of the commutator theorems by establishing new characterizations of the weighted BMO space.Finally,a concrete example shows that the local version of commutators also has an independent interest.展开更多
Abstract In this paper, we study the Lp mapping properties of certain class of maximal oscillatory singular integral operators. We prove a general theorem for a class of maximal functions along surfaces. As a conseque...Abstract In this paper, we study the Lp mapping properties of certain class of maximal oscillatory singular integral operators. We prove a general theorem for a class of maximal functions along surfaces. As a consequence of such theorem, we establish the Lp boundedness of various maximal oscillatory singular integrals provided that their kernels belong to the natural space L log L（Sn-1）. Moreover, we highlight some additional results concerning operators with kernels in certain block spaces. The results in this paper substantially improve previously known results.展开更多
The author studies the L^p mapping properties of a class of maximal functions that are related to oscillatory singular integral operators. Lp estimates, as well as the corresponding weighted estimates of such maximal ...The author studies the L^p mapping properties of a class of maximal functions that are related to oscillatory singular integral operators. Lp estimates, as well as the corresponding weighted estimates of such maximal functions, are obtained. Moreover, several applications of our results are highlighted.展开更多
We give in this paper a necessary and sufficient condition of weighted weak and strong type norm inequalities for the vector-valued weighted maximal function.
In this paper, the authors point out that the methods used by Li(2004, 2005,2007) can be applied to study maximal functions on weighted harmonic AN groups.
In this paper,the authors study the multilinear commutators generated by a class of multilinear singular integral operators with generalized kernels and Lipschitz functions.By establishing the sharp maximal estimates,...In this paper,the authors study the multilinear commutators generated by a class of multilinear singular integral operators with generalized kernels and Lipschitz functions.By establishing the sharp maximal estimates,the boundedness of this kind of multilinear commutators on product of weighted Lebesgue spaces can be obtained.展开更多
In this article, we obtain the LP-boundedness of commutators of Lipschitz functions and singular integrals with non-smooth kernels on Euclidean spaces.
基金supported by the National Key Research and Development Program of China(2020YFA0712900)the National Natural Science Foundation of China(12371093,12071197,12122102 and 12071431)+2 种基金the Key Project of Gansu Provincial National Science Foundation(23JRRA1022)the Fundamental Research Funds for the Central Universities(2233300008 and lzujbky-2021-ey18)the Innovative Groups of Basic Research in Gansu Province(22JR5RA391).
文摘Assume that L is a non-negative self-adjoint operator on L^(2)(ℝ^(n))with its heat kernels satisfying the so-called Gaussian upper bound estimate and that X is a ball quasi-Banach function space onℝ^(n) satisfying some mild assumptions.Let HX,L(ℝ^(n))be the Hardy space associated with both X and L,which is defined by the Lusin area function related to the semigroup generated by L.In this article,the authors establish various maximal function characterizations of the Hardy space HX,L(ℝ^(n))and then apply these characterizations to obtain the solvability of the related Cauchy problem.These results have a wide range of generality and,in particular,the specific spaces X to which these results can be applied include the weighted space,the variable space,the mixed-norm space,the Orlicz space,the Orlicz-slice space,and the Morrey space.Moreover,the obtained maximal function characterizations of the mixed-norm Hardy space,the Orlicz-slice Hardy space,and the Morrey-Hardy space associated with L are completely new.
基金Supported by NSF of Zhejiang Province of China(LQ18A010002,LQ17A010002)。
文摘This paper focuses on the continuity of the truncated Hardy-Littlewood maximal function.We first show that the truncated Hardy-Littlewood maximal function is lower semi-continuous.Then by investigating the behavior of the truncated Hardy-Littlewood maximal function when the truncated parameterγchanges,we obtain an equivalent condition of the continuity of the truncated Hardy-Littlewood maximal function.
基金the Science Foundation for Youths of Gansu Province(Grant No.22JR5RA173)Master Foundation of Northwest Normal University(Grant No.2022KYZZ-S121).
文摘Let(X,d,μ)be a non-homogeneous metric measure space satisfying the geometrically doubling condition and the upper doubling condition.In this setting,the authors prove that the commutator M_(b)^(α)formed by b∈RBMO(μ)and the fractional maximal function M^((α))is bounded from Lebesgue spaces L^(p)(μ)into spaces L^(q)(μ),where 1/q=1/p-αforα∈(0,1)and p∈(1,1/α).Furthermore,the boundedness of the M_(b)^(α)on Orlicz spaces L^Φ(μ)is established.
文摘Let 2≤p【∞ and let (f n) be a martingale. Using exponential bounds of the probabilities of the type P(|f n|】λ‖T(f n)‖ ∞) for some quasi-linear operators acting on martingales, we estimate upper bounds for the L p-norms of the maximal functions of martinglaes. Our result is the extension and improvements of the results obtained previously by HITCZENKO and ZENG .
基金Supported by the key Academic Discipline of Zhejiang Province of China under Grant No.2005the Zhejiang Provincial Natural Science Foundation of China
文摘In this paper, we get W 1,p(Rn)-boundedness for tangential maximal func- tion and nontangential maximal function , which improves J.Kinnunen, P.Lindqvist and Tananka’s results.
文摘A new maximal function is introduced in the dual spaces of test function spaces on spaces of homogeneous type. Using this maximal function, we get new characterization of atomic H^p spaces.
基金Supported by the Natural Science Foundation of Hebei Province (08M001)the National Natural Science Foundation of China (10771049)
文摘For maximal multilinear Calderon-Zygmund singular integral operators, the sharp maximal function estimate and some weighted norm inequalities are obtained.
基金supported by the National Natural Science Foundation of China(11201324)the Fok Ying Tuny Education Foundation(141114)the Sichuan Technology Program(2022ZYD0011,2022NFSC1852).
文摘For a general normed vector space,a special optimal value function called a maximal time function is considered.This covers the farthest distance function as a special case,and has a close relationship with the smallest enclosing ball problem.Some properties of the maximal time function are proven,including the convexity,the lower semicontinuity,and the exact characterizations of its subdifferential formulas.
基金supported by the National Science Foundation of USA (Grant No. DMS 0400387)the University of Missouri Research Council (Grant No. URC-07-067)+1 种基金the National Science Foundation for Distinguished Young Scholars of China (Grant No. 10425106)the Program for New Century Excellent Talents in University of the Ministry of Education of China (Grant No. 04-0142)
文摘Let X be an RD-space, i.e., a space of homogeneous type in the sense of Coifman and Weiss, which has the reverse doubling property. Assume that X has a “dimension” n. For α ∈ (0, ∞) denote by H α p (X), H d p (X), and H *,p (X) the corresponding Hardy spaces on X defined by the nontangential maximal function, the dyadic maximal function and the grand maximal function, respectively. Using a new inhomogeneous Calderón reproducing formula, it is shown that all these Hardy spaces coincide with L p (X) when p ∈ (1,∞] and with each other when p ∈ (n/(n + 1), 1]. An atomic characterization for H ?,p (X) with p ∈ (n/(n + 1), 1] is also established; moreover, in the range p ∈ (n/(n + 1),1], it is proved that the space H *,p (X), the Hardy space H p (X) defined via the Littlewood-Paley function, and the atomic Hardy space of Coifman andWeiss coincide. Furthermore, it is proved that a sublinear operator T uniquely extends to a bounded sublinear operator from H p (X) to some quasi-Banach space B if and only if T maps all (p, q)-atoms when q ∈ (p, ∞)∩[1, ∞) or continuous (p, ∞)-atoms into uniformly bounded elements of B.
基金Supported by National Natural Science Foundation of China(Grant No.11271162)the Scientific Research Fund of Heilongjiang Provincial Education Department(Grant No.12531720)the Scientific Research Fund of Mudanjiang Normal University(Grant No.GY201305)
文摘Let M be the multilinear maximal function and = （b1, . . . , bm） be a collection of locally integrable functions. Denote by M and [,M] the maximal commutator and the commutator of M with , respectively. In this paper, the multiple weighted strong and weak type estimates for operators M and [,M] are studied. Some characterizations of the class of functions bj are given, for which these operators satisfy some strong or weak type estimates.
基金supported by National Natural Science Foundation of China(Grant No.11701333)Support Program for Outstanding Young Scientific and Technological Top-Notch Talents of College of Mathematics and Systems Science(Grant No.Sxy2016K01)+3 种基金supported by National Natural Science Foundation of China(Grant Nos.11471041 and 11671039)National Natural Science Foundation of China-Deutsche Forschungsgemeinschaft(Grant No.11761131002)supported by Grant-in-Aid for Scientific Research(C)(Grant No.15K04942)Japan Society for the Promotion of Science。
文摘In this paper,we systematically study several classes of maximal singular integrals and maximal functions with rough kernels in Fβ(S^n-1),a topic that relates to the Grafakos-Stefanov class.The boundedness and continuity of these operators on Triebel-Lizorkin spaces and Besov spaces are discussed.
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）.
基金This paper is supported by NSF of Zhejiang Province of China(Grant No.LQ18A010002 and No.LQ17A010002)in part by National Natural Foundation of China(Grant Nos.11871452 and 12001488).
文摘In the paper,we characterize a necessary and sufficient condition which ensures the continuities of the non-centered Hardy-Lit tiewood maximal function Mf and the centered Hardy-Lit tiewood maximal function Mcf on R^n.As two applications,we can easily deduce that Mcf and Mf are continuous if f is continuous,and Mf is continuous if f is of local bounded variation on R.
基金supported in part by the National Natural Science Foundation of China(Grant Nos.12171221,12071197)the Natural Science Foundation of Shandong Province(Grant Nos.ZR2019YQ04,2020KJI002).
文摘We obtain the characterizations of commutators of several versions of maximal functions on spaces of homogeneous type.In addition,with the aid of interpolation theory,we provide weighted version of the commutator theorems by establishing new characterizations of the weighted BMO space.Finally,a concrete example shows that the local version of commutators also has an independent interest.
文摘Abstract In this paper, we study the Lp mapping properties of certain class of maximal oscillatory singular integral operators. We prove a general theorem for a class of maximal functions along surfaces. As a consequence of such theorem, we establish the Lp boundedness of various maximal oscillatory singular integrals provided that their kernels belong to the natural space L log L（Sn-1）. Moreover, we highlight some additional results concerning operators with kernels in certain block spaces. The results in this paper substantially improve previously known results.
文摘The author studies the L^p mapping properties of a class of maximal functions that are related to oscillatory singular integral operators. Lp estimates, as well as the corresponding weighted estimates of such maximal functions, are obtained. Moreover, several applications of our results are highlighted.
文摘We give in this paper a necessary and sufficient condition of weighted weak and strong type norm inequalities for the vector-valued weighted maximal function.
文摘In this paper, the authors point out that the methods used by Li(2004, 2005,2007) can be applied to study maximal functions on weighted harmonic AN groups.
基金Supported by the National Natural Science Foundation of China(11671397,11571160,12071052)the Yue Qi Young Scholar of China University of Mining and Technology(Beijing)。
文摘In this paper,the authors study the multilinear commutators generated by a class of multilinear singular integral operators with generalized kernels and Lipschitz functions.By establishing the sharp maximal estimates,the boundedness of this kind of multilinear commutators on product of weighted Lebesgue spaces can be obtained.
文摘In this article, we obtain the LP-boundedness of commutators of Lipschitz functions and singular integrals with non-smooth kernels on Euclidean spaces.
