In this paper we extend Hardy-Littlewood-Sobolev inequalities on compact Riemannian manifolds for dimension $n e 2$. As one application, we solve a generalized Yamabe problem on locally conforamlly flat manifolds via a new designed energy functional
and a new variational approach. Even for the classic Yamabe problem on locally conformally flat manifolds, our approach provides a new and relatively simpler solution.
In this paper we formulate new curvature functions on $mathbb{S}^n$ via integral operators. For certain even orders, these curvature functions are equivalent to the classic curvature functions defined via differential operators, but not for all even
orders. Existence result for antipodally symmetric prescribed curvature functions on $mathbb{S}^n$ is obtained. As a corollary, the existence of a conformal metric for an antipodally symmetric prescribed $Q-$curvature functions on $mathbb{S}^3$ is proved. Curvature function on general compact manifold as well as the conformal covariance property for the corresponding integral operator are also addressed, and a general Yamabe type problem is proposed.
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 c
ase 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.
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.
We exam various geometric active contour methods for radar image segmentation. Due to special properties of radar images, we propose our new model based on modified Chan-Vese functional. Our method is efficient in separating non-meteorological noises from meteorological images.
We establish an analog Hardy inequality with sharp constant involving exponential weight function. The special case of this inequality (for n=2) leads to a direct proof of Onofri inequality on S^2.
In this paper we shall give an analytic proof of the fact that the Liouville energy on a topological two sphere is bounded from below. Our proof does not rely on the uniformization theorem and the Onofri inequality, thus it is essentially needed in t
he alternative proof of the uniformization theorem via the Calabi flow. Such an analytic approach also sheds light on how to obtain the boundedness for E_1 energy in the study of general Kahler manifolds.
In this paper we continue our studies of the one dimensional conformal metric flows, which were introduced in [8]. In this part we mainly focus on evolution equations involving fourth order derivatives. The global existence and exponential convergenc
e of metrics for the 1-Q and 4-Q flows are obtained.
