Fundamental Results for Pseudo-Differential Operators of Type 1, 1
Department of Mathematical Sciences, Aalborg University, Fredrik Bajers Vej 7G, DK-9220 Aalborg Øst, Denmark
Academic Editor: Palle E.T. Jorgensen
Received: 4 March 2016 / Revised: 4 May 2016 / Accepted: 6 May 2016 / Published: 19 May 2016
This paper develops some deeper consequences of an extended definition, proposed previously by the author, of pseudo-differential operators that are of type
in Hörmander’s sense. Thus, it contributes to the long-standing problem of creating a systematic theory of such operators. It is shown that type
-operators are defined and continuous on the full space of temperate distributions, if they fulfil Hörmander’s twisted diagonal condition, or more generally if they belong to the self-adjoint subclass; and that they are always defined on the temperate smooth functions. As a main tool the paradifferential decomposition is derived for type
-operators, and to confirm a natural hypothesis the symmetric term is shown to cause the domain restrictions; whereas the other terms are shown to define nice type
-operators fulfilling the twisted diagonal condition. The decomposition is analysed in the type
-context by combining the Spectral Support Rule and the factorisation inequality, which gives pointwise estimates of pseudo-differential operators in terms of maximal functions.
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MDPI and ACS Style
Johnsen, J. Fundamental Results for Pseudo-Differential Operators of Type 1, 1. Axioms 2016, 5, 13.
Johnsen J. Fundamental Results for Pseudo-Differential Operators of Type 1, 1. Axioms. 2016; 5(2):13.
Johnsen, Jon. 2016. "Fundamental Results for Pseudo-Differential Operators of Type 1, 1." Axioms 5, no. 2: 13.
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