Abstract
We present a class of global pseudo-differential operators of infinite order which are intrinsically related to the spaces of tempered ultradistributions as well as the symbolic calculus these operators enjoy. We also give the notion of hypoellipticity in this setting and consider the complex powers of non-negative hypoelliptic pseudo-differential operators. Finally, we give the construction of the heat parametrix and present an application of these results to semigroups whose infinitesimal generators are square roots of non-negative operators having hypoelliptic Weyl symbols.
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