| 1. |
- Toft, Joachim
(author)
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An Embedding Result for some General Symbol Classes in the Weyl Calculus
- 1999
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Conference paper (peer-reviewed)abstract
- The paper deals with inclusion relations between s(p) and H-s(p). Here s(p) is the set of all a is an element ofJ such that the Weyl operator a(w) (x, D) is a Schatten-von Neumann operator on L-2 to the order p is an element of [1, infinity], and H-s(p) is the Sobolev space of distributions with s derivatives in L-p. At the same time we compute the trace norm for a(w) (x, D), when a is an arbitrary Gauss function.
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| 2. |
- Toft, Joachim
(author)
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Continuity and Positivity Problems in Pseudo-differential Calculus
- 1996
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Doctoral thesis (other academic/artistic)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 sp(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 sp(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.
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