Resumen de Global pseudodifferential operators in spaces of ultradifferentiable functions
Alberto Asensio López
In this thesis we study pseudodifferential operators, which are integral operators of the form $$ f\mapsto \int_{\mathbb{R}^d} \big( \int_{\mathbb{R}^d}e^{i(x-y)\cdot\xi} a(x,y,\xi) f(y) dy\big) d\xi, $$ in the global class of ultradifferentiable functions of Beurling type $\mathcal{S}_\omega(\mathbb{R}^d)$ as introduced by Björck, when the weight function $\omega$ is given in the sense of Braun, Meise, and Taylor.
We develop a symbolic calculus for these operators, treating also the change of quantization, the existence of pseudodifferential parametrices and applications to global wave front sets.
The thesis consists of four chapters. In Chapter 1 we introduce global symbols and amplitudes and show that the corresponding pseudodifferential operators are well defined and continuous in $\mathcal{S}_\omega(\mathbb{R}^d)$. These results are extended in Chapter 2 for arbitrary quantizations, which leads to the study of the transpose of any quantization of a pseudodifferential operator, and the composition of two different quantizations of pseudodifferential operators. In Chapter 3 we develop the method of the parametrix, providing sufficient conditions for the existence of left parametrices of a pseudodifferential operator, which motivates in Chapter 4 the definition of a new global wave front set for ultradistributions in $\mathcal{S}'_\omega(\mathbb{R}^d)$ given in terms of Weyl quantizations. Then, we compare this wave front set with the Gabor wave front set defined by the STFT and give applications to the regularity of Weyl quantizations.
