Continuity and Positivity Problems in Pseudo-differential Calculus

Abstract: The paper deals with various positivity and continuity questions arising in the Weyl calculus of pseudo-differential operators. Let W be a symplectic vector space. In the first part of the paper we discuss positivity and continuity properties in spaces s_p(W) of distributions, whose Weyl quantizations are in the Schatten-von Neumann class of the order p> 1. In particular we prove some Young related inequalities for ordinary products and convolution products of elements in s_p(W) and L p(W). An important ingredient in the proofs is the use of twisted convolutions. In the second part of the paper we develop a decomposition technique in the space s₁(W) to treat some lower bound problems in the Weyl calculus. As an application we prove a result which is related to Hörmander's improvement of Melin's inequality. We also give a survey of some entropy inequalities that are useful when studying lower bounds for pseudo-differential operators after these have been represented in the form of Toeplitz operators.