Related papers: A sharp weighted anisotropic Poincar\'e inequality…

We compute the best constants in some second order dilation invariant inequalities, with weights being powers of the distance from the origin.

We study weighted Sobolev inequalities on open convex cones endowed with $\alpha$-homogeneous weights satisfying a certain concavity condition. We establish a so-called reduction principle for these inequalities and characterize optimal…

In this paper, we prove a version of weighted inequalities of exponential type for fractional integrals with sharp constants in any domain of finite measure in $\mathbb{R}^{n}$. Using this we prove a sharp singular Adams inequality in high…

The aim of this paper is to establish various functional inequalities for the convolution of a compactly supported measure and a standard Gaussian distribution on Rd. We especially focus on getting good dependence of the constants on the…

We use "generalized" version of total variation, coarea formulas, isoperimetric inequalities to obtain sharp estimates for solutions (and for their gradients) to anisotropic elliptic equations with a lower order term, comparing them with…

Motivated by a long-standing conjecture of Polya and Szeg\"o about the Newtonian capacity of convex bodies, we discuss the role of concavity inequalities in shape optimization, and we provide several counterexamples to the…

In this paper we provide a Bonnesen-style inequality which gives a lower bound for the isoperimetric deficit corresponding to a closed convex curve in terms of some geometrical invariants of this curve. Moreover we give a geometrical…

In this paper, we study weighted fractional Sobolev-Poincar\'e inequalities for irregular domains. The weights considered here are distances to the boundary to certain powers, and the domains are the so-called $s$-John domains and…

We establish a Shearer-type inequality for the Poincar\'e constant, showing that the Poincar\'e constant corresponding to the convolution of a collection of measures can be nontrivially controlled by the Poincar\'e constants corresponding…

In this paper, we show that the convex domains of the hyperbolic space which are almost extremal for the Faber-Krahn or the Payne-Polya-Weinberger inequalities are close to geodesic balls. Our proof is also valid in other space forms and…

In this article, we consider a class of incompressible stochastic third-grade fluids (non-Newtonian fluids) equations on two- as well as three-dimensional Poincar\'e domains $\mathcal{O}$ (which may be bounded or unbounded). Our aims are to…

We prove fractional Sobolev-Poincar\'e inequalities in unbounded John domains and we characterize fractional Hardy inequalities there.

We obtain new proofs with improved constants of the Khintchine-type inequality with matrix coefficients in two cases. The first case is the Pisier and Lust-Piquard noncommutative Khintchine inequality for $p=1$, where we obtain the sharp…

We prove a sharp upper bound for the first Dirichlet eigenvalue of a class of nonlinear elliptic operators which includes the p-Laplace and the pseudo-p-Laplace operators. Moreover, we prove a stability result by means of a suitable…

The positive semidefinite rank of a convex body $C$ is the size of its smallest positive semidefinite formulation. We show that the positive semidefinite rank of any convex body $C$ is at least $\sqrt{\log d}$ where $d$ is the smallest…

We prove a counterpart of the log-convex density conjecture in the hyperbolic plane.

We prove a sharp global $W^{2,\,p}$ estimate for potentials of optimal transport maps that take a certain class of non-convex planar domains to convex ones.

For a bounded convex domain \Omega in R^N we prove refined Hardy inequalities that involve the Hardy potential corresponding to the distance to the boundary of \Omega, the volume of $\Omega$, as well as a finite number of sharp logarithmic…

We give an optimal upper bound of the degree of quasi-smooth hypersurfaces which are invariant by a one-dimensional holomorphic foliation on a compact toric orbifold, i.e. on a complete simplicial toric variety. This bound depends only on…

In this paper, we establish a sharp remainder formula for the Poincar\'e inequality for Baouendi-Grushin vector fields in the setting of $L^{p}$ for complex-valued functions. In special cases, we recover previously known results.…