The fractional Hardy inequality with a remainder term

We calculate the regional fractional Laplacian on some power function on an interval. As an application, we prove Hardy inequality with an extra term for the fractional Laplacian on the interval with the optimal constant. As a result, we obtain the fractional Hardy inequality with best constant and an extra lower-order term for general domains, following the method developed by M. Loss and C. Sloane [arXiv:0907.3054v1 [math.AP]]Comment: Major change

Hardy inequalities and non-explosion results for semigroups

We prove non-explosion results for Schr\"odinger perturbations of symmetric transition densities and Hardy inequalities for their quadratic forms by using explicit supermedian functions of their semigroups.Comment: 21 pages, updated reference

Exponential rate of convergence independent from the dimension in a mean-field system of particles

International audienceThis article deals with a mean-field model. We consider a large number of particles interacting through their empirical law. We know that there is a unique invariant probability for this diffusion. We look at functional inequalities. In particular, we briefly show that the diffusion satisfies a Poincaré inequality. Then, we establish a so-called WJ-inequality, which is independent from the number of particles

Comparability and regularity estimates for symmetric nonlocal Dirichlet forms

The aim of this work is to study comparability of nonlocal Dirichlet forms. We provide sufficient conditions on the kernel for local and global comparability. As an application we prove a-priori estimates in H\"{o}lder spaces for solutions to integrodifferential equations. These solutions are defined with the help of symmetric nonlocal Dirichlet forms.Comment: 17 pages, 1 figur