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.
This is an open access article distributed under the Creative Commons Attribution License
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited