A novel representation is developed as a measure for multilinear fractional embedding. Corresponding extensions are given for the Bourgain-Brezis-Mironescu theorem and Pitt＇s inequal- ity. New results are obtained fo...A novel representation is developed as a measure for multilinear fractional embedding. Corresponding extensions are given for the Bourgain-Brezis-Mironescu theorem and Pitt＇s inequal- ity. New results are obtained for diagonal trace restriction on submanifolds as an application of the Hardy-Littlewood-Sobolev inequality. Smoothing estimates are used to provide new structural un- derstanding for density functional theory, the Coulomb interaction energy and quantum mechanics of phase space. Intriguing connections are drawn that illustrate interplay among classical inequalities in Fourier analysis.展开更多
Using product and convolution theorems on Lorentz spaces, we characterize the sufficient and necessary conditions which ensure the validity of the doubly weighted Hardy-Littlewood-Sobolev inequality. It should be poin...Using product and convolution theorems on Lorentz spaces, we characterize the sufficient and necessary conditions which ensure the validity of the doubly weighted Hardy-Littlewood-Sobolev inequality. It should be pointed out that we con- sider whole ranges of p and q, i.e., 0 〈 p ≤∞ and 0 〈 q ≤∞.展开更多
In this paper, we prove that the supremum sup{ ∫B∫B|u（y）|p（|y|）|u（x）|p（|x|）/|x-y|μdxdy : u ∈ H0,rad1（B）, ||?||uL2（B）= 1}is attained, where B denotes the unit ball in RN（N ≥3）, μ ∈（0, N）, p（r） ...In this paper, we prove that the supremum sup{ ∫B∫B|u（y）|p（|y|）|u（x）|p（|x|）/|x-y|μdxdy : u ∈ H0,rad1（B）, ||?||uL2（B）= 1}is attained, where B denotes the unit ball in RN（N ≥3）, μ ∈（0, N）, p（r） = 2μ*+ rt, t ∈（0, min{N/2-μ/4, N-2}） and 2μ*=（2N-μ）/（N-2） is the critical exponent for the Hardy-Littlewood-Sobolev inequality.展开更多
We obtain the optimal integrability for positive solutions of the Euler-Lagrange system of the weighted Hardy-Littlewood-Sobolev inequality in R^n ：{u（x）=1/｜x｜^α｜∫R^n v（y）^q｜y｜^β｜x-y｜^λdy,v（x）=1/｜x...We obtain the optimal integrability for positive solutions of the Euler-Lagrange system of the weighted Hardy-Littlewood-Sobolev inequality in R^n ：{u（x）=1/｜x｜^α｜∫R^n v（y）^q｜y｜^β｜x-y｜^λdy,v（x）=1/｜x｜^β∫R^n u（y）^p｜y｜^α｜x-y｜^λdy.C. Jin and C. Li [Calc. Var. Partial Differential Equations, 2006, 26： 447-457] developed some very interesting method for regularity lifting and obtained the optimal integrability for p, q 〉 1. Here, based on some new observations, we overcome the difficulty there, and derive the optimal integrability for the case of p, q ≥1 and pq ≠1. This integrability plays a key role in estimating the asymptotic behavior of positive solutions when ｜x｜ →0 and when ｜x｜→∞.展开更多
The purpose of this paper is five-fold. First, we employ the harmonic analysis techniques to establish the following Hardy–Littlewood–Sobolev inequality with the fractional Poisson kernel on the upper half space ■ ...The purpose of this paper is five-fold. First, we employ the harmonic analysis techniques to establish the following Hardy–Littlewood–Sobolev inequality with the fractional Poisson kernel on the upper half space ■ where f ∈ L^p(?R_+~n), g ∈ Lq(R_+~n) and p, q'∈(1, +∞), 2 ≤α < n satisfying (n-1)/np+1/q'+(2-α)/n= 1.Second, we utilize the technique combining the rearrangement inequality and Lorentz interpolation to show the attainability of best constant C_(n,α,p,q'). Third, we apply the regularity lifting method to obtain the smoothness of extremal functions of the above inequality under weaker assumptions. Furthermore,in light of the Pohozaev identity, we establish the sufficient and necessary condition for the existence of positive solutions to the integral system of the Euler–Lagrange equations associated with the extremals of the fractional Poisson kernel. Finally, by using the method of moving plane in integral forms, we prove that extremals of the Hardy–Littlewood–Sobolev inequality with the fractional Poisson kernel must be radially symmetric and decreasing about some point ξ_0 ∈ ?R_+~n. Our results proved in this paper play a crucial role in establishing the Stein–Weiss inequalities with the Poisson kernel in our subsequent paper.展开更多
In this paper, we are concerned with properties of positive solutions of the following Euler-Lagrange system associated with the weighted Hardy-Littlewood-Sobolev inequality in discrete form{uj =∑ k ∈Zn vk^q/（1 ＋ ...In this paper, we are concerned with properties of positive solutions of the following Euler-Lagrange system associated with the weighted Hardy-Littlewood-Sobolev inequality in discrete form{uj =∑ k ∈Zn vk^q/（1 ＋ ｜j｜）^α（1 ＋ ｜k- j｜）^λ（1 ＋ ｜k｜）^β,（0.1）vj =∑ k ∈Zn uk^p/（1 ＋ ｜j｜）^β（1 ＋ ｜k- j｜）^λ（1 ＋ ｜k｜）^α,where u, v 〉 0, 1 〈 p, q 〈 ∞, 0 〈 λ 〈 n, 0 ≤α ＋ β≤ n- λ,1/p＋1〈λ＋α/n and 1/p＋1＋1/q＋1≤λ＋α＋β/n：=λ^-/n. We first show that positive solutions of（0.1） have the optimal summation interval under assumptions that u ∈ l^p＋1（Z^n） and v ∈ l^q＋1（Z^n）. Then we show that problem（0.1） has no positive solution if 0 〈λˉ pq ≤ 1 or pq 〉 1 and max{（n-λ^-）（q＋1）/pq-1,（n-λ^-）（p＋1）/pq-1} ≥λ^-.展开更多
文摘A novel representation is developed as a measure for multilinear fractional embedding. Corresponding extensions are given for the Bourgain-Brezis-Mironescu theorem and Pitt＇s inequal- ity. New results are obtained for diagonal trace restriction on submanifolds as an application of the Hardy-Littlewood-Sobolev inequality. Smoothing estimates are used to provide new structural un- derstanding for density functional theory, the Coulomb interaction energy and quantum mechanics of phase space. Intriguing connections are drawn that illustrate interplay among classical inequalities in Fourier analysis.
基金supported in part by National Natural Foundation of China (Grant Nos. 11071250 and 11271162)
文摘Using product and convolution theorems on Lorentz spaces, we characterize the sufficient and necessary conditions which ensure the validity of the doubly weighted Hardy-Littlewood-Sobolev inequality. It should be pointed out that we con- sider whole ranges of p and q, i.e., 0 〈 p ≤∞ and 0 〈 q ≤∞.
基金supported by National Natural Science Foundation of China(Grant Nos.11831009 and 11571130)
文摘In this paper, we prove that the supremum sup{ ∫B∫B|u（y）|p（|y|）|u（x）|p（|x|）/|x-y|μdxdy : u ∈ H0,rad1（B）, ||?||uL2（B）= 1}is attained, where B denotes the unit ball in RN（N ≥3）, μ ∈（0, N）, p（r） = 2μ*+ rt, t ∈（0, min{N/2-μ/4, N-2}） and 2μ*=（2N-μ）/（N-2） is the critical exponent for the Hardy-Littlewood-Sobolev inequality.
基金Acknowledgements The authors wish to express their appreciations to the anonymous referees. Their suggestions have greatly improved this paper. They are also grateful to Prof. Congming Li for many fruitful discussion. The first author was supported by the National Natural Science Foundation of China （Grant No. 11171158）, the Natural Science Foundation of Jiangsu （No. BK2012846）, and the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry.
文摘We obtain the optimal integrability for positive solutions of the Euler-Lagrange system of the weighted Hardy-Littlewood-Sobolev inequality in R^n ：{u（x）=1/｜x｜^α｜∫R^n v（y）^q｜y｜^β｜x-y｜^λdy,v（x）=1/｜x｜^β∫R^n u（y）^p｜y｜^α｜x-y｜^λdy.C. Jin and C. Li [Calc. Var. Partial Differential Equations, 2006, 26： 447-457] developed some very interesting method for regularity lifting and obtained the optimal integrability for p, q 〉 1. Here, based on some new observations, we overcome the difficulty there, and derive the optimal integrability for the case of p, q ≥1 and pq ≠1. This integrability plays a key role in estimating the asymptotic behavior of positive solutions when ｜x｜ →0 and when ｜x｜→∞.
基金partly supported by a US NSF granta Simons Collaboration grant from the Simons Foundation
文摘The purpose of this paper is five-fold. First, we employ the harmonic analysis techniques to establish the following Hardy–Littlewood–Sobolev inequality with the fractional Poisson kernel on the upper half space ■ where f ∈ L^p(?R_+~n), g ∈ Lq(R_+~n) and p, q'∈(1, +∞), 2 ≤α < n satisfying (n-1)/np+1/q'+(2-α)/n= 1.Second, we utilize the technique combining the rearrangement inequality and Lorentz interpolation to show the attainability of best constant C_(n,α,p,q'). Third, we apply the regularity lifting method to obtain the smoothness of extremal functions of the above inequality under weaker assumptions. Furthermore,in light of the Pohozaev identity, we establish the sufficient and necessary condition for the existence of positive solutions to the integral system of the Euler–Lagrange equations associated with the extremals of the fractional Poisson kernel. Finally, by using the method of moving plane in integral forms, we prove that extremals of the Hardy–Littlewood–Sobolev inequality with the fractional Poisson kernel must be radially symmetric and decreasing about some point ξ_0 ∈ ?R_+~n. Our results proved in this paper play a crucial role in establishing the Stein–Weiss inequalities with the Poisson kernel in our subsequent paper.
基金supported by NNSF of China(11261023,11326092),NNSF of China(11271170)Startup Foundation for Doctors of Jiangxi Normal University+1 种基金GAN PO 555 Program of JiangxiNNSF of Jiangxi(20122BAB201008)
文摘In this paper, we are concerned with properties of positive solutions of the following Euler-Lagrange system associated with the weighted Hardy-Littlewood-Sobolev inequality in discrete form{uj =∑ k ∈Zn vk^q/（1 ＋ ｜j｜）^α（1 ＋ ｜k- j｜）^λ（1 ＋ ｜k｜）^β,（0.1）vj =∑ k ∈Zn uk^p/（1 ＋ ｜j｜）^β（1 ＋ ｜k- j｜）^λ（1 ＋ ｜k｜）^α,where u, v 〉 0, 1 〈 p, q 〈 ∞, 0 〈 λ 〈 n, 0 ≤α ＋ β≤ n- λ,1/p＋1〈λ＋α/n and 1/p＋1＋1/q＋1≤λ＋α＋β/n：=λ^-/n. We first show that positive solutions of（0.1） have the optimal summation interval under assumptions that u ∈ l^p＋1（Z^n） and v ∈ l^q＋1（Z^n）. Then we show that problem（0.1） has no positive solution if 0 〈λˉ pq ≤ 1 or pq 〉 1 and max{（n-λ^-）（q＋1）/pq-1,（n-λ^-）（p＋1）/pq-1} ≥λ^-.
