The aim of this work is to give a Gårding inequality for pseudodifferential operators acting on functions in ${L}^{2}\left({\mathbb{R}}^{n}\right)$ supported in a closed regular region $F\subset {\mathbb{R}}^{n}$. A natural idea is to suppose that the symbol is non-negative in $F\times {\mathbb{R}}^{n}$. Assuming this, we show that this result is true for pseudo-differential operators of order one, when $F$ is the half-space, and under a supplementary weak hypothesis of degeneracy of the symbol on the boundary.

We consider the Fokker-Planck equation with a confining or anti-confining potential which behaves at infinity like a possibly high degree homogeneous function. Hypoellipticity techniques provide the well-posedness of the weak-Cauchy problem in both cases as well as instantaneous smoothing and exponential trend to equilibrium. Lower and upper bounds for the rate of convergence to equilibrium are obtained in terms of the lowest positive eigenvalue of the corresponding Witten laplacian, with detailed...

In this article we give a complete proof in one dimension of an a priori inequality involving pseudo-differential operators: if $a$ and $b$ are symbols in ${S}_{1,0}^{2}$ such that $\left|a\right|\le b$, then for all $\u03f5\>0$ we have the estimate $\parallel {a}^{w}u{\parallel}_{s}^{2}\le {C}_{\u03f5}(\parallel {b}^{w}u{\parallel}_{s}^{2}+\parallel u{\parallel}_{s+\u03f5}^{2})$ for all $u$ in the Schwartz space, where $\parallel \phantom{\rule{0.277778em}{0ex}}{\parallel}_{t}$ is the usual ${H}_{t}$ norm. We use microlocalization of levels I, II and III in the spirit of Fefferman’s SAK principle.

This note is an announcement of a forthcoming paper [] in collaboration with K. Pravda-Starov on global hypoelliptic estimates for Fokker-Planck and linear Landau-type operators. Linear Landau-type equations are a class of inhomogeneous kinetic equations with anisotropic diffusion whose study is motivated by the linearization of the Landau equation near the Maxwellian distribution. By introducing a microlocal method by multiplier which can be adapted to various hypoelliptic kinetic equations, we...

In this note we describe recent results on semiclassical random walk associated to a probability density which may also concentrate as the semiclassical parameter goes to zero. The main result gives a spectral asymptotics of the close to $1$ eigenvalues. This problem was studied in [] and relies on a general factorization result for pseudo-differential operators. In this note we just sketch the proof of this second theorem. At the end of the note, using the factorization, we give a new proof of the...

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