Hardy–Adams Inequalities on ℍ2 × ℝ n-2

Let ℍ 2 {\mathbb{H}^{2}} be the hyperbolic space of dimension 2. Denote by M n = ℍ 2 × ℝ n - 2 {M^{n}=\mathbb{H}^{2}\times\mathbb{R}^{n-2}} the product manifold of ℍ 2 {\mathbb{H}^{2}} and ℝ n - 2 ( n ≥ 3 ) {\mathbb{R}^{n-2}(n\geq 3)} . In this paper we establish some sharp Hardy–Adams inequalities on M n {M^{n}} , though M n {M^{n}} is not with strictly negative sectional curvature. We also show that the sharp constant in the Poincaré–Sobolev inequality on M n {M^{n}} coincides with the best Sobolev constant, which is of independent interest.

Energy asymptotics in the three-dimensional Brezis–Nirenberg problem

For a bounded open set $$\Omega \subset {\mathbb {R}}^3$$ Ω ⊂ R 3 we consider the minimization problem $$\begin{aligned} S(a+\epsilon V) = \inf _{0\not \equiv u\in H^1_0(\Omega )} \frac{\int _\Omega (|\nabla u|^2+ (a+\epsilon V) |u|^2)\,dx}{(\int _\Omega u^6\,dx)^{1/3}} \end{aligned}$$ S ( a + ϵ V ) = inf 0 ≢ u ∈ H 0 1 ( Ω ) ∫ Ω ( | ∇ u | 2 + ( a + ϵ V ) | u | 2 ) d x ( ∫ Ω u 6 d x ) 1 / 3 involving the critical Sobolev exponent. The function a is assumed to be critical in the sense of Hebey and Vaugon. Under certain assumptions on a and V we compute the asymptotics of $$S(a+\epsilon V)-S$$ S ( a + ϵ V ) - S as $$\epsilon \rightarrow 0+$$ ϵ → 0 + , where S is the Sobolev constant. (Almost) minimizers concentrate at a point in the zero set of the Robin function corresponding to a and we determine the location of the concentration point within that set. We also show that our assumptions are almost necessary to have $$S(a+\epsilon V)<S$$ S ( a + ϵ V ) < S for all sufficiently small $$\epsilon >0$$ ϵ > 0 .

Logarithmic Sobolev inequalities for Moebius measures on spheres

In this paper, using the method in [1], i.e., reduce Moebius measures {\mu_{x}^{n}} indexed by {|x|<1} on spheres {S^{n-1}} ( {n\geq 3} ) to one-dimensional diffusions on {[0,\pi]} , we obtain that the optimal Poincaré constant is not greater than {\frac{2}{n-2}} and the optimal logarithmic Sobolev constant denoted by {C_{\rm LS}(\mu_{x}^{n})} behaves like {\frac{1}{n}\log(1+\frac{1}{1{-}|x|})} . As a consequence, we claim that logarithmic Sobolev inequalities are strictly stronger than {L^{2}} -transportation-information inequalities.

Calabi flow on toric varieties with bounded Sobolev constant, I

Let (X, P) be a toric variety. In this note, we show that the C0-norm of the Calabi flow φ(t) on X is uniformly bounded in [0, T) if the Sobolev constant of φ(t) is uniformly bounded in [0, T). We also show that if (X, P) is uniform K-stable, then the modified Calabi flow converges exponentially fast to an extremal Kähler metric if the Ricci curvature and the Sobolev constant are uniformly bounded. At last, we discuss an extension of our results to a quasi-proper Kähler manifold.