Une classe d'opérateurs pseudo-différentiels partiellement hypoelliptique-analytiques
The aim of this work is to give a Gårding inequality for pseudodifferential operators acting on functions in supported in a closed regular region . A natural idea is to suppose that the symbol is non-negative in . Assuming this, we show that this result is true for pseudo-differential operators of order one, when is the half-space, and under a supplementary weak hypothesis of degeneracy of the symbol on the boundary.
Let WF⁎ be the wave front set with respect to , quasi analyticity or analyticity, and let K be the kernel of a positive operator from to ’. We prove that if ξ ≠ 0 and (x,x,ξ,-ξ) ∉ WF⁎(K), then (x,y,ξ,-η) ∉ WF⁎(K) and (y,x,η,-ξ) ∉ WF⁎(K) for any y,η. We apply this property to positive elements with respect to the weighted convolution , where is appropriate, and prove that if for every and (0,ξ) ∉ WF⁎(u), then (x,ξ) ∉ WF⁎(u) for any x.
We study boundedness properties of commutators of general linear operators with real-valued BMO functions on weighted spaces. We then derive applications to particular important operators, such as Calderón-Zygmund type operators, pseudo-differential operators, multipliers, rough singular integrals and maximal type operators.
We study the Weyl asymptotics of the distribution of eigenvalues of non-self-adjoint (pseudo)differential operators with small random multiplicative perturbations in arbitrary dimension. We were led to quite essential improvements of many of the probabilistic aspects.
We discuss continuity properties of the Weyl product when acting on classical modulation spaces. In particular, we prove that is an algebra under the Weyl product when p ∈ [1,∞] and 1 ≤ q ≤ min(p,p’).
This paper begins with a broad survey of the state of the art in matters of solvability for differential and pseudo-differential equations. Then we proceed with a Hilbertian lemma which we use to prove a new solvability result.
We define a class of pseudodifferential operators with symbols a(x,ξ) without any regularity assumptions in the x variable and explore their boundedness properties. The results are applied to obtain estimates for certain maximal operators associated with oscillatory singular integrals.