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Register a new userSharp Hardy-Littlewood-Sobolev inequality on the upper half space
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There are at least two directions concerning the extension of classical sharp Hardy-Littlewood-Sobolev inequality: (1) Extending the sharp inequality on general manifolds; (2) Extending it for the negative exponent $lambda=n-alpha$ (that is for the case of $alpha>n$). In this paper we confirm the possibility for the extension along the first direction by establishing the sharp Hardy-Littlewood-Sobolev inequality on the upper half space (which is conformally equivalent to a ball). The existences of extremal functions are obtained; And for certain range of the exponent, we classify all extremal functions via the method of moving sphere.
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In this paper we establish the reversed sharp Hardy-Littlewood-Sobolev (HLS for short) inequality on the upper half space and obtain a new HLS type integral inequality on the upper half space (extending an inequality found by Hang, Wang and Yan in cite{HWY2008}) by introducing a uniform approach. The extremal functions are classified via the method of moving spheres, and the best constants are computed. The new approach can also be applied to obtain the classical HLS inequality and other similar inequalities.
In this paper, we prove the following reversed Hardy-Littlewood-Sobolev inequality with extended kernel begin{equation*} int_{mathbb{R}_+^n}int_{partialmathbb{R}^n_+} frac{x_n^beta}{|x-y|^{n-alpha}}f(y)g(x) dydxgeq C_{n,alpha,beta,p}|f|_{L^{p}(partialmathbb{R}_+^n)} |g|_{L^{q}(mathbb{R}_+^n)} end{equation*} for any nonnegative functions $fin L^{p}(partialmathbb{R}_+^n)$ and $gin L^{q}(mathbb{R}_+^n)$, where $ngeq2$, $p, qin (0,1)$, $alpha>n$, $0leqbeta<frac{alpha-n}{n-1}$, $p>frac{n-1}{alpha-1-(n-1)beta}$ such that $frac{n-1}{n}frac{1}{p}+frac{1}{q}-frac{alpha+beta-1}{n}=1$. We prove the existence of extremal functions for the above inequality. Moreover, in the conformal invariant case, we classify all the extremal functions and hence derive the best constant via a variant method of moving spheres, which can be carried out emph{without lifting the regularity of Lebesgue measurable solutions}. Finally, we derive the sufficient and necessary conditions for existence of positive solutions to the Euler-Lagrange equations by using Pohozaev identities. Our results are inspired by Hang, Wang and Yan cite{HWY}, Dou, Guo and Zhu cite{DGZ} for $alpha<n$ and $beta=1$, and Gluck cite{Gl} for $alpha<n$ and $betageq0$.
This paper is devoted to a new family of reverse Hardy-Littlewood-Sobolev inequalities which involve a power law kernel with positive exponent. We investigate the range of the admissible parameters and characterize the optimal functions. A striking open question is the possibility of concentration which is analyzed and related with nonlinear diffusion equations involving mean field drifts.
The classical sharp Hardy-Littlewood-Sobolev inequality states that, for $1<p, t<infty$ and $0<lambda=n-alpha <n$ with $ 1/p +1 /t+ lambda /n=2$, there is a best constant $N(n,lambda,p)>0$, such that $$ |int_{mathbb{R}^n} int_{mathbb{R}^n} f(x)|x-y|^{-lambda} g(y) dx dy|le N(n,lambda,p)||f||_{L^p(mathbb{R}^n)}||g||_{L^t(mathbb{R}^n)} $$ holds for all $fin L^p(mathbb{R}^n), gin L^t(mathbb{R}^n).$ The sharp form is due to Lieb, who proved the existence of the extremal functions to the inequality with sharp constant, and computed the best constant in the case of $p=t$ (or one of them is 2). Except that the case for $pin ((n-1)/n, n/alpha)$ (thus $alpha$ may be greater than $n$) was considered by Stein and Weiss in 1960, there is no other result for $alpha>n$. In this paper, we prove that the reversed Hardy-Littlewood-Sobolev inequality for $0<p, t<1$, $lambda<0$ holds for all nonnegative $fin L^p(mathbb{R}^n), gin L^t(mathbb{R}^n).$ For $p=t$, the existence of extremal functions is proved, all extremal functions are classified via the method of moving sphere, and the best constant is computed.
In this paper, we investigate a stochastic Hardy-Littlewood-Sobolev inequality. Due to the stochastic nature of the inequality, the relation between the exponents of intgrability is modified. This modification can be understood as a regularization by noise phenomenon. As a direct application, we derive Strichartz estimates for the white noise dispersion which enables us to address a conjecture from [3].
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