Spazi ultradifferenziabili a due pesi ed applicazioni

This thesis is concerned with ultradifferentiable classes, that are intermediate scales of spaces between real anlytic and smooth functions, and contain as particular cases Gevrey and Gelfand-Shilov spaces. In the first part we treat Gevrey spaces and ultradifferentiable classes, defined in the style of Braun-Meise-Taylor, both in the Beurling and in the Roumieu case. Then we analize the space of ultradifferentiable rapidly decreasing functions of Beurling type, defined as in Björck. For these classes we study basic properties, we analyze their topology, the action of the Fourier transform and the corresponding spaces of ultradistributions. As in the classical case, the space of the ultradifferentiable functions with compact support and the space of the ultradifferentiable functions can be considered; they are studied in their Beurling and Roumieu versions. More precisely, what we have done is to suppose a different control on the function and its Fourier transform through two different weight functions. This change stems from the fact that there exist function spaces, for example the space of Gelfand-Shilov functions (of Gevrey type), for which there is a different control of this type. We study the properties of this new space, taht is a good functional setting for the action of time-frequency transformations like the Gabor transform and the Wigner-like transform. In the last 60 years, the classes of ultradifferentiable functions and their duals have been intensively studied for very different purposes and have become the right setting to study many different problems in analysis as well as in the field of PDEs. Therefore, in the last chapter of the thesis we give an application of the spaces we have defined to PDEs. In fact, the mapping properties of the operator Wig allow us to give a regularity result on linear partial differential equations with polynomial coefficients.