Semimartingales and geometric inequalities on manifolds

The purpose of this paper is twofold. First, we use resultson Jacobi fields to study the stochastic differential equations(SDEs) for exp_Xt( exp_Xt^-1(Y_t)) with specially constructed coupledsemimartingales X and Y on a complete, simply connected Riemannianmanifold M with constant sectional curvature. Secondly, we applythese SDEs to obtain an analogue for M of a result of Borellconcerning an inequality relating the solutions of the parabolicequation / t = 1/2 – h, with Dirichlet boundary condition,on three convex sets in Euclidean space. From the latter, therefollows an inequality involving the first eigenvalues of theLaplacian on those convex sets with the Dirichlet boundary condition,analogous to an inequality in Euclidean space which is equivalentto the Brunn–Minkowski inequality of these eigenvaluesobtained by Brascamp and Lieb.     

Semimartingales and geometric inequalities on locally symmetric manifolds

We generalise, to complete, connected and locally symmetric Riemannian manifolds, the construction of coupled semimartingales X and Y given in Le and Barden (J Lond Math Soc 75:522–544, 2007). When such a manifold has non-negative curvature, this makes it possible for the stochastic anti-development of the corresponding semimartingale ${\rm exp}_{X_t} \big(\alpha\,{\rm exp}^{-1}_{X_t}(Y_t)\big)$ to be a time-changed Brownian motion with drift when X and Y are. As an application, we use the latter result to strengthen, and extend to locally symmetric spaces, the results of Le and Barden (J Lond Math Soc 75:522–544, 2007) concerning an inequality involving the solutions of the parabolic equation $\frac{\partial\psi} {\partial t} = \frac{1}{2}\Delta\psi - h\,\psi$ with Dirichlet boundary condition and an inequality involving the first eigenvalues of the Laplacian, both on three related convex sets.     

Sharp constants in weighted trace inequalities on Riemannian manifolds

In this paper, we establish some sharp weighted trace inequalities ${W^{1,2}(\rho^{1-2 \sigma}, M) \hookrightarrow L^{\frac{2n}{n-2 \sigma}}(\partial M)}$ on n + 1 dimensional compact smooth manifolds with smooth boundaries, where ρ is a defining function of M and ${\sigma \in (0,1)}$ . This is stimulated by some recent work on fractional (conformal) Laplacians and related problems in conformal geometry, and also motivated by a conjecture of Aubin.     

Sharp Sobolev trace inequalities on Riemannian manifolds with boundaries

In this paper, we establish some sharp Sobolev trace inequalities on n-dimensional, compact Riemannian manifolds with smooth boundaries. More specifically, let q = 2(n - 1)/(n - 2), 1/S = inf {∫ |∇u|² : ∇u ∈ L²(R₊ⁿ), ∫ |u|^q = 1}. We establish for any Riemannian manifold with a smooth boundary, denoted as (M, g), that there exists some constant A = A(M, g) > 0, (∫_dM|u|^q ds_g)^2/q < or = to S ∫_M |∇_gu|² dv_g + A ∫_dMu² ds_g, for all u ∈ H¹ (M). The inequality is sharp in the sense that the inequality is false when S is replaced by any smaller number. © 1997 John Wiley & Sons, Inc.     

Sharp Sobolev-Poincaré inequalities on compact Riemannian manifolds

Given a smooth compact Riemannian -manifold, , we return in this article to the study of the sharp Sobolev-Poincaré type inequality

where is the critical Sobolev exponent, and is the sharp Euclidean Sobolev constant. Druet, Hebey and Vaugon proved that is true if , that is true if and the sectional curvature of is a nonpositive constant, or the Cartan-Hadamard conjecture in dimension is true and the sectional curvature of is nonpositive, but that is false if and the scalar curvature of is positive somewhere. When is true, we define as the smallest in . The saturated form of reads as

We assume in this article that , and complete the study by Druet, Hebey and Vaugon of the sharp Sobolev-Poincaré inequality . We prove that is true, and that possesses extremal functions when the scalar curvature of is negative. A fairly complete answer to the question of the validity of under the assumption that the scalar curvature is not necessarily negative, but only nonpositive, is also given.

     

Sharp inequalities for geometric maximal operators associated with general measures

We determine the best constants in the weak-type (p, p) and L ^p estimates for geometric maximal operator on (?, µ). It is also shown that in higher dimensions such inequalities fail to hold.     

Some geometric inequalities for varifolds on Riemannian manifolds based on monotonicity identities

Annals of Global Analysis and Geometry - Using Rauch’s comparison theorem, we prove several monotonicity inequalities for Riemannian submanifolds. Our main result is a general Li–Yau...     

Sharp inequalities via truncation

We show that Sobolev-Poincaré and Trudinger inequalities improve to inequalities on Lorentz-type scales provided they are stable under truncations.     

Gagliardo-Nirenberg inequalities on manifolds

We prove Gagliardo-Nirenberg inequalities on some classes of manifolds, Lie groups and graphs.     

Sharp inequalities for convolution-type operators

Let μ be a probability measure on [− a, a], a > 0, and let x₀ε[− a, a], f ε Cⁿ([−2a, 2a]), n 0 even. Using moment methods we derive best upper bounds to ¦∫_−a^a ([f(x₀ + y) + f(x₀ − y)]/2) μ(dy) − f(x₀)¦, leading to sharp inequalities that are attainable and involve the second modulus of continuity of f⁽ⁿ⁾ or an upper bound of it.     

New geometric aspects of Moser–Trudinger inequalities on Riemannian manifolds: the non-compact case

In the first part of the paper we investigate some geometric features of Moser–Trudinger inequalities on complete non-compact Riemannian manifolds. By exploring rearrangement arguments, isoperimetric estimates, and gluing local uniform estimates via Gromov's covering lemma, we provide a Coulhon, Saloff-Coste and Varopoulos type characterization concerning the validity of Moser–Trudinger inequalities on complete non-compact n-dimensional Riemannian manifolds $(n\ge 2)$ with Ricci curvature bounded from below. Some sharp consequences are also presented both for non-negatively and non-positively curved Riemannian manifolds, respectively. In the second part, by combining variational arguments and a Lions type symmetrization-compactness principle, we guarantee the existence of a non-zero isometry-invariant solution for an elliptic problem involving the n-Laplace–Beltrami operator and a critical nonlinearity on n-dimensional homogeneous Hadamard manifolds. Our results complement in several directions those of Y. Yang [J. Funct. Anal., 2012].     

Analytic and geometric isoperimetric inequalities

We prove uncountably many new analytic and geometric isoperimetric inequalities associated with the solutions of second order ordinary differential equations.     

Sharp inequalities related to Gosper's formula

The purpose of this Note is to construct a new type of Stirling series, which extends the Gosper's formula for big factorials. New sharp inequalities for the gamma and digamma functions are established. Finally, numerical computations which demonstrate the superiority of our new series over the classical Stirling's series are given.     

Sharp Sobolev inequalities with interior norms

In this paper, we establish some new sharp Sobolev inequalities on any smooth bounded domain . Let and S be the sharp constants corresponding to the Sobolev embedding and trace inequalities respectively. If , there exist constants , such that and If , for any , there exist constants such that and Received March 15, 1997 / Accepted January 30, 1998     

Sharp constants in hardy type inequalities

We prove new weighted Hardy type inequalities with sharp constants and describe their applications to inequalities in multidimensional domains.     

Sharp Sobolev inequalities of second order

Let (M, g) be a smooth compact Riemannian manifold of dimension n≥5, and ₂ ² (M) be the Sobolev space consisting of functions in L²(M) whose derivatives up to the order two are also in L²(M). Thanks to the Sobolev embedding theorem, there exist positive constants A and B such that for any U ∈ H ₂ ² (M),