Related papers: Sharp interpolation inequalities on the sphere : n…
We compute the optimal constant for a generalized Hardy-Sobolev inequality, and using the product of two symmetrizations we present an elementary proof of the symmetries of some optimal functions. This inequality was motivated by a…
We investigate the validity, as well as the failure, of Sobolev-type inequalities on Cartan-Hadamard manifolds under suitable bounds on the sectional and the Ricci curvatures. We prove that if the sectional curvatures are bounded from above…
It is shown by a counterexample that isocapacitary and isoperimetric constants of a multi-dimensional Euclidean domain starshaped with respect to a ball are not equivalent. Sharp integral inequalities involving the harmonic capacity which…
We present some new Sobolev inequalities on spheres corresponding to parabolic geometries of real rank-one semisimple Lie groups
The aim of this paper is to extend to the spaces L^2(R^d , (1+|v|)^2k dv) the spectral study led in L^2(R^d , exp(|v|^2/2)dv) by R. Ellis and M. Pinsky on the space inhomogeneous linearized Boltzmann operator for hard spheres. More…
Let $(E,\F,\mu)$ be a $\si$-finite measure space. For a non-negative symmetric measure $J(\d x, \d y):=J(x,y) \,\mu(\d x)\,\mu(\d y)$ on $E\times E,$ consider the quadratic form $$\E(f,f):= \frac{1}{2}\int_{E\times E} (f(x)-f(y))^2 \, J(\d…
In this paper we study sharp pointwise inequalities for maximal operators. In particular, we strengthen DeVore's inequality for the moduli of smoothness and a logarithmic variant of Bennett--DeVore--Sharpley's inequality for rearrangements.…
We establish sharp bilinear eigenfunction estimates for the Laplace-Beltrami operator on the standard three-sphere $\mathbb{S}^3$, eliminating the logarithmic loss that has persisted in the literature since the pioneering work of Burq,…
We prove limit equalities between the sharp constants in weighted Nikolskii-type inequalities for multivariate polynomials on an $m$-dimensional cube and ball and the corresponding constants for entire functions of exponential type.
We survey some interplays between spectral estimates of H\"ormander-type, degenerate Monge-Amp\`ere equations and geometric inequalities related to log-concavity such as Brunn-Minkowski, Santal\'o or Busemann inequalities.
In this paper, we consider the time change of the diffusion process on the 2-dimensional unit sphere generated by the Laplace-Beltrami operator, perturbed by means of a longitudinal vector field. First, this is done by addressing the…
Let $\Delta_0$ be the Laplace-Beltrami operator on the unit sphere $\mathbb{S}^{d-1}$ of $\mathbb{R}^d$. We show that the Hardy-Rellich inequality of the form $$ \int_{\mathbb{S}^{d-1}} \left | f (x)\right|^2 d\sigma(x) \leq c_d \min_{e\in…
The dual purpose of this article is to establish bilinear Poincare-type estimates associated to an approximation of the identity and to explore the connections between bilinear pseudo-differential operators and bilinear potential-type…
We present sharp interpolation theorems, including all limiting cases, for a class of quasilinear operators of joint weak type acting between Lorentz-Karamata spaces over $\sigma$-finite measure. This class contains many of the important…
Two sharp Chernoff type inequalities are obtained for star body in $\mathbb{R}^2$, one of which is an extension of the dual Chernoff-Ou-Pan inequality, and the other is the reverse Chernoff type inequality. Furthermore, we establish a…
In this paper, we obtain new upper bounds for the Lieb-Thirring inequality on the spheres of any dimension greater than $2$. As far as we have checked, our results improve previous results found in the literature for all dimensions greater…
The paper gives a detailed survey of recent results on elliptic problems in Hilbert spaces of generalized smoothness. The latter are the isotropic H\"ormander spaces $H^{s,\varphi}:=B_{2,\mu}$, with $\mu(\xi)=<\xi>^{s}\varphi(<\xi>)$ for…
In this paper, we study the dispersive properties of the wave equation associated with the shifted Laplace-Beltrami operator on real hyperbolic spaces, and deduce Strichartz estimates for a large family of admissible pairs. As an…
The purpose of this note is to investigate the concentration properties of spectral projectors on manifolds. This question has been intensively studied (by Logvinenko--Sereda, Nazarov, Jerison--Lebeau, Kovrizhkin,…
In this paper we are focusing on functional inequalities on compact simple edge spaces. More precisely we address the question whether the classical functional inequalities (Sobolev, Poincar\'e) hold in this setting, and as a by-product of…