We derive the explicit fundamental solutions for a class of degenerate(or singular)one- parameter subelliptic differential operators on groups of Heisenberg(H)type.This extends the result of Kaplan for the sub-Laplaci...We derive the explicit fundamental solutions for a class of degenerate(or singular)one- parameter subelliptic differential operators on groups of Heisenberg(H)type.This extends the result of Kaplan for the sub-Laplacian on H-type groups,which in turn generalizes Folland's result on the Heisenberg group.As an application,we obtain a one-parameter representation formula for Sobolev functions of compact support on H-type groups.By choosing the parameter equal to the homogeneous dimension Q and using the Mose-Trudinger inequality for the convolutional type operator on stratified groups obtained in[18].we get the following theorem which gives the best constant for the Moser- Trudiuger inequality for Sobolev functions on H-type groups. Let G be any group of Heisenberg type whose Lie algebra is generated by m left invariant vector fields and with a q-dimensional center.Let Q=m+2q.Q'=Q-1/Q and Then. with A_Q as the sharp constant,where ▽G denotes the subelliptic gradient on G. This continues the research originated in our earlier study of the best constants in Moser-Trudinger inequalities and fundamental solutions for one-parameter subelliptic operators on the Heisenberg group [18].展开更多
For p ：〉 1, many improved or generalized results of the well-known Hardy＇s inequality have been established： In this paper, by means of the weight coefficient method, we establish the following Hardy type inequali...For p ：〉 1, many improved or generalized results of the well-known Hardy＇s inequality have been established： In this paper, by means of the weight coefficient method, we establish the following Hardy type inequality for p = -1：∑^n i=1（1/i∑^i j=1 aj）^-1〈∑^n i=1（1-π^2-9/3i）ai^-1,where ai 〉 0, i = 1,2,... ,n. For any fixed positive integer n 〉 2, we study the best constant Cn such that the inequality ∑^ni=1（1/i∑^ij=1aj）^-1≤cn∑^ni=1ai^-1holds. Moreover, by means ofthe Mathematica software, we givesome examples.展开更多
This paper proved the following three facts about the Lipschitz continuous property of Bernstein polynomials and Bezier nets defined on a triangle: suppose f(P) is a real valued function defined on a triangle T, (1) I...This paper proved the following three facts about the Lipschitz continuous property of Bernstein polynomials and Bezier nets defined on a triangle: suppose f(P) is a real valued function defined on a triangle T, (1) If f(P) satisfies Lipschitz continuous condition, i. e. f(P)∈Lip4α, then the corresponding Bernstein Bezier net fn∈LipAsecαψα, here ψ is the half of the largest angle of triangle T; (2) If Bernstein Bezier net fn∈ LipBα, then its elevation Bezier net Efn∈LipBα; and (3) If f(P)∈Lipαa, then the corresponding Bernstein polynomials Bn(f;P)∈LipAsecαψα, and the constant Asecαψ best in some sense.展开更多
The Hardy-Sobolev inequality with general weights is established, and it is shown that the constant is optimal. The two weights in this inequality are determined by a Bernoulli equation. In addition, the authors obtai...The Hardy-Sobolev inequality with general weights is established, and it is shown that the constant is optimal. The two weights in this inequality are determined by a Bernoulli equation. In addition, the authors obtain the Hardy-Sobolev inequality with general weights and remainder terms. By choosing special weights, it turns to be many versions of the Hardy-Sobolev inequality and the Caffarelli-Kohn-Nirenberg inequality with remainder terms in the literature.展开更多
In this paper, by introducing the norm ｜｜x｜｜ （x ∈ Rn）, a multiple Hardy- Hilbert＇s integral inequality with the best constant factor and it＇s equivalent form are given.
By using the weight function method,the matching parameters of the half discrete Hilbert type multiple integral inequality with a non-homogeneous kernel K(n,||x||ρ,m)=G(nλ1||x||ρmλ,2)are discussed,some equivalent ...By using the weight function method,the matching parameters of the half discrete Hilbert type multiple integral inequality with a non-homogeneous kernel K(n,||x||ρ,m)=G(nλ1||x||ρmλ,2)are discussed,some equivalent conditions of the optimal matching parameter are established,and the expression of the optimal constant factor is obtained.Finally,their applications in operator theory are considered.展开更多
It is well known that the doubly weighted Hardy-Littlewood-Sobolev inequality is as follows,｜∫Rn ∫Rn f（x）g（y）/｜x｜^α｜x-y｜^λ｜y｜^β dxdy｜≤ B（p,q,α,λ,β ,n）｜｜f｜｜Lp（Rn）｜｜g｜｜Lq（Rn）.The main ...It is well known that the doubly weighted Hardy-Littlewood-Sobolev inequality is as follows,｜∫Rn ∫Rn f（x）g（y）/｜x｜^α｜x-y｜^λ｜y｜^β dxdy｜≤ B（p,q,α,λ,β ,n）｜｜f｜｜Lp（Rn）｜｜g｜｜Lq（Rn）.The main purpose of this paper is to give the sharp constants B（p,q,,,,n）for the above inequality for three cases：（i）p=1 and q=1;（ii）p=1 and 1q 6∞,or 1p 6∞and q=1;（iii）1p,q∞and 1p＋1q=1.In addition,the explicit bounds can be obtained for the case 1p,q∞and 1/p＋1/q〉1.展开更多
We investigate a class of multilinear integral operators with the nonnegative kernels, and prove that the norms of the operators can be obtained by integral of the product of the kernel function and finitely many basi...We investigate a class of multilinear integral operators with the nonnegative kernels, and prove that the norms of the operators can be obtained by integral of the product of the kernel function and finitely many basic functions. Using the integral, we can easily calculate the sharp constants for the multilinear Hilbert inequality, the generalized Hardy-Littlewood-Sobolev inequality and the multilinear Hardy operator.展开更多
Let f be in the localized nonisotropic Sobolev space Wloc^1,p （H^n） on the n-dimensional Heisenberg group H^n = C^n ×R, where 1≤ p ≤ Q and Q = 2n ＋ 2 is the homogeneous dimension of H^n. Suppose that the sub...Let f be in the localized nonisotropic Sobolev space Wloc^1,p （H^n） on the n-dimensional Heisenberg group H^n = C^n ×R, where 1≤ p ≤ Q and Q = 2n ＋ 2 is the homogeneous dimension of H^n. Suppose that the subelliptic gradient is gloablly L^p integrable, i.e., fH^n ｜△H^n f｜^p du is finite. We prove a Poincaré inequality for f on the entire space H^n. Using this inequality we prove that the function f subtracting a certain constant is in the nonisotropic Sobolev space formed by the completion of C0^∞（H^n） under the norm of （∫H^n ｜f｜ Qp/Q-p）^Q-p/Qp ＋ （∫ H^n ｜△H^n f｜^p）^1/p. We will also prove that the best constants and extremals for such Poincaré inequalities on H^n are the same as those for Sobolev inequalities on H^n. Using the results of Jerison and Lee on the sharp constant and extremals for L^2 to L（2Q/Q-2） Sobolev inequality on the Heisenberg group, we thus arrive at the explicit best constant for the aforementioned Poincaré inequality on H^n when p=2. We also derive the lower bound of the best constants for local Poincaré inequalities over metric balls on the Heisenberg group H^n.展开更多
The purpose of the paper is to study sharp weak-type bounds for functions of bounded mean oscillation. Let 0 〈 p 〈 ∞ be a fixed number and let I be an interval contained in R. The author shows that for any φ ： I ...The purpose of the paper is to study sharp weak-type bounds for functions of bounded mean oscillation. Let 0 〈 p 〈 ∞ be a fixed number and let I be an interval contained in R. The author shows that for any φ ： I → R and any subset E I of positive measure, ｜I｜^-1/p/｜E｜1-1/p∫E｜φ -1/｜I｜∫Iφdy｜dx≤｜｜φ｜｜BMO（I）,0〈p≤2,｜I｜^-1/p/｜E｜1-1/p∫E｜φ -1/｜I｜∫Iφdy｜dx≤p/2^2/pe2/p-1｜｜φ｜｜BMO（I）p≥2. For each p, the constant on the right-hand side is the best possible. The proof rests on the explicit evaluation of the associated Bellman function. The result is a complement of the earlier works of Slavin, Vasyunin and Volberg concerning weak-type, L ^p and exponential bounds for the BMO class.展开更多
In this paper,the authors obtain the Dunkl analogy of classical L^(p)Hardy inequality for p>N+2γwith sharp constant((p-N-2γ)/p)^(p),where 2γis the degree of weight function associated with Dunkl operators,and L ...In this paper,the authors obtain the Dunkl analogy of classical L^(p)Hardy inequality for p>N+2γwith sharp constant((p-N-2γ)/p)^(p),where 2γis the degree of weight function associated with Dunkl operators,and L pHardy inequalities with distant function in some G-invariant domains.Moreover they prove two Hardy-Rellich type inequalities for Dunkl operators.展开更多
In this paper it is shown that a new Hilbert-type integral inequality can be established by introducing two parameters m（m ∈ N） and λ（λ 0）.And the constant factor expressed by the Bernoulli number and π is pro...In this paper it is shown that a new Hilbert-type integral inequality can be established by introducing two parameters m（m ∈ N） and λ（λ 0）.And the constant factor expressed by the Bernoulli number and π is proved to be the best possible.And then some important and especial results are enumerated.As applications,some equivalent forms are given.展开更多
A set of methods for interprocedural analysis is proposed. First, an ap-proach for interprocedural constant propagation is given. Then the concept of constant propagation is extended so as to meet the needs of data de...A set of methods for interprocedural analysis is proposed. First, an ap-proach for interprocedural constant propagation is given. Then the concept of constant propagation is extended so as to meet the needs of data dependence analysis. Besides certain constant, constant range can also be propagated. The related propagating rules are introduced, and an idea for computing Return function is given. This approach can solve almost all interprocedural constant propagation problems with non-recursive calls. Second, a muItiple-version par-allelizing technique is also proposed for alias problem. The work related to this paper has been implemented on a shared-memory parallel computer.展开更多
文摘We derive the explicit fundamental solutions for a class of degenerate(or singular)one- parameter subelliptic differential operators on groups of Heisenberg(H)type.This extends the result of Kaplan for the sub-Laplacian on H-type groups,which in turn generalizes Folland's result on the Heisenberg group.As an application,we obtain a one-parameter representation formula for Sobolev functions of compact support on H-type groups.By choosing the parameter equal to the homogeneous dimension Q and using the Mose-Trudinger inequality for the convolutional type operator on stratified groups obtained in[18].we get the following theorem which gives the best constant for the Moser- Trudiuger inequality for Sobolev functions on H-type groups. Let G be any group of Heisenberg type whose Lie algebra is generated by m left invariant vector fields and with a q-dimensional center.Let Q=m+2q.Q'=Q-1/Q and Then. with A_Q as the sharp constant,where ▽G denotes the subelliptic gradient on G. This continues the research originated in our earlier study of the best constants in Moser-Trudinger inequalities and fundamental solutions for one-parameter subelliptic operators on the Heisenberg group [18].
基金Foundation item： the National Natural Science Foundation of China （No. 10671136） the Natural Science Foundation of Sichuan Provincial Education Department （No. 2005A201）.
文摘For p ：〉 1, many improved or generalized results of the well-known Hardy＇s inequality have been established： In this paper, by means of the weight coefficient method, we establish the following Hardy type inequality for p = -1：∑^n i=1（1/i∑^i j=1 aj）^-1〈∑^n i=1（1-π^2-9/3i）ai^-1,where ai 〉 0, i = 1,2,... ,n. For any fixed positive integer n 〉 2, we study the best constant Cn such that the inequality ∑^ni=1（1/i∑^ij=1aj）^-1≤cn∑^ni=1ai^-1holds. Moreover, by means ofthe Mathematica software, we givesome examples.
基金Supported by NSF and SF of National Educational Committee
文摘This paper proved the following three facts about the Lipschitz continuous property of Bernstein polynomials and Bezier nets defined on a triangle: suppose f(P) is a real valued function defined on a triangle T, (1) If f(P) satisfies Lipschitz continuous condition, i. e. f(P)∈Lip4α, then the corresponding Bernstein Bezier net fn∈LipAsecαψα, here ψ is the half of the largest angle of triangle T; (2) If Bernstein Bezier net fn∈ LipBα, then its elevation Bezier net Efn∈LipBα; and (3) If f(P)∈Lipαa, then the corresponding Bernstein polynomials Bn(f;P)∈LipAsecαψα, and the constant Asecαψ best in some sense.
基金the National Natural Science Foundation of China(10771074,10726060)the Natural Science Foundation of Guangdong Province(04020077)
文摘The Hardy-Sobolev inequality with general weights is established, and it is shown that the constant is optimal. The two weights in this inequality are determined by a Bernoulli equation. In addition, the authors obtain the Hardy-Sobolev inequality with general weights and remainder terms. By choosing special weights, it turns to be many versions of the Hardy-Sobolev inequality and the Caffarelli-Kohn-Nirenberg inequality with remainder terms in the literature.
文摘In this paper, by introducing the norm ｜｜x｜｜ （x ∈ Rn）, a multiple Hardy- Hilbert＇s integral inequality with the best constant factor and it＇s equivalent form are given.
基金Supported by National Natural Science Foundation of China(Grant No.12071491)Guangzhou Science and Technology Plan Project(Grant No.202102080177).
文摘By using the weight function method,the matching parameters of the half discrete Hilbert type multiple integral inequality with a non-homogeneous kernel K(n,||x||ρ,m)=G(nλ1||x||ρmλ,2)are discussed,some equivalent conditions of the optimal matching parameter are established,and the expression of the optimal constant factor is obtained.Finally,their applications in operator theory are considered.
基金supported by National Natural Science Foundation of China(Grant Nos.11071250 and 11271162)
文摘It is well known that the doubly weighted Hardy-Littlewood-Sobolev inequality is as follows,｜∫Rn ∫Rn f（x）g（y）/｜x｜^α｜x-y｜^λ｜y｜^β dxdy｜≤ B（p,q,α,λ,β ,n）｜｜f｜｜Lp（Rn）｜｜g｜｜Lq（Rn）.The main purpose of this paper is to give the sharp constants B（p,q,,,,n）for the above inequality for three cases：（i）p=1 and q=1;（ii）p=1 and 1q 6∞,or 1p 6∞and q=1;（iii）1p,q∞and 1p＋1q=1.In addition,the explicit bounds can be obtained for the case 1p,q∞and 1/p＋1/q〉1.
基金supported by National Natural Science Foundation of China(Grant Nos.1147103911271162 and 11561062)
文摘We investigate a class of multilinear integral operators with the nonnegative kernels, and prove that the norms of the operators can be obtained by integral of the product of the kernel function and finitely many basic functions. Using the integral, we can easily calculate the sharp constants for the multilinear Hilbert inequality, the generalized Hardy-Littlewood-Sobolev inequality and the multilinear Hardy operator.
基金The first author is supported by Zhongdian grant of NSFCa global grant at Wayne State University and by NSF of USA
文摘Let f be in the localized nonisotropic Sobolev space Wloc^1,p （H^n） on the n-dimensional Heisenberg group H^n = C^n ×R, where 1≤ p ≤ Q and Q = 2n ＋ 2 is the homogeneous dimension of H^n. Suppose that the subelliptic gradient is gloablly L^p integrable, i.e., fH^n ｜△H^n f｜^p du is finite. We prove a Poincaré inequality for f on the entire space H^n. Using this inequality we prove that the function f subtracting a certain constant is in the nonisotropic Sobolev space formed by the completion of C0^∞（H^n） under the norm of （∫H^n ｜f｜ Qp/Q-p）^Q-p/Qp ＋ （∫ H^n ｜△H^n f｜^p）^1/p. We will also prove that the best constants and extremals for such Poincaré inequalities on H^n are the same as those for Sobolev inequalities on H^n. Using the results of Jerison and Lee on the sharp constant and extremals for L^2 to L（2Q/Q-2） Sobolev inequality on the Heisenberg group, we thus arrive at the explicit best constant for the aforementioned Poincaré inequality on H^n when p=2. We also derive the lower bound of the best constants for local Poincaré inequalities over metric balls on the Heisenberg group H^n.
基金supported by the NCN grant DEC-2012/05/B/ST1/00412
文摘The purpose of the paper is to study sharp weak-type bounds for functions of bounded mean oscillation. Let 0 〈 p 〈 ∞ be a fixed number and let I be an interval contained in R. The author shows that for any φ ： I → R and any subset E I of positive measure, ｜I｜^-1/p/｜E｜1-1/p∫E｜φ -1/｜I｜∫Iφdy｜dx≤｜｜φ｜｜BMO（I）,0〈p≤2,｜I｜^-1/p/｜E｜1-1/p∫E｜φ -1/｜I｜∫Iφdy｜dx≤p/2^2/pe2/p-1｜｜φ｜｜BMO（I）p≥2. For each p, the constant on the right-hand side is the best possible. The proof rests on the explicit evaluation of the associated Bellman function. The result is a complement of the earlier works of Slavin, Vasyunin and Volberg concerning weak-type, L ^p and exponential bounds for the BMO class.
基金supported by the National Natural Science Foundation of China(Nos.11771395,12071431)
文摘In this paper,the authors obtain the Dunkl analogy of classical L^(p)Hardy inequality for p>N+2γwith sharp constant((p-N-2γ)/p)^(p),where 2γis the degree of weight function associated with Dunkl operators,and L pHardy inequalities with distant function in some G-invariant domains.Moreover they prove two Hardy-Rellich type inequalities for Dunkl operators.
基金Supported by the Project of Scientific Research Fund of Hunan Provincial Education Department (GrantNo.09C789)
文摘In this paper it is shown that a new Hilbert-type integral inequality can be established by introducing two parameters m（m ∈ N） and λ（λ 0）.And the constant factor expressed by the Bernoulli number and π is proved to be the best possible.And then some important and especial results are enumerated.As applications,some equivalent forms are given.
文摘A set of methods for interprocedural analysis is proposed. First, an ap-proach for interprocedural constant propagation is given. Then the concept of constant propagation is extended so as to meet the needs of data dependence analysis. Besides certain constant, constant range can also be propagated. The related propagating rules are introduced, and an idea for computing Return function is given. This approach can solve almost all interprocedural constant propagation problems with non-recursive calls. Second, a muItiple-version par-allelizing technique is also proposed for alias problem. The work related to this paper has been implemented on a shared-memory parallel computer.
