Extension of Localisation Operators to Ultradistributional Symbols With Super-Exponential Growth

TL;DR

This paper extends the concept of localisation operators to include symbols with super-exponential growth, using ultradistributional frameworks, enabling analysis of operators with extremely fast-growing symbols.

Contribution

It introduces a method to define localisation operators with symbols exhibiting super-exponential growth within the ultradistribution setting, broadening the class of analyzable operators.

Findings

Symbols can have growth of the form exp(exp(l|x|^q)) in position space.

The extension applies to symbols as mappings between ultradistribution spaces.

Main results demonstrate the feasibility of analyzing highly rapidly growing symbols.

Abstract

In the Gelfand-Shilov setting, the localisation operator $A^{\varphi_1,\varphi_2}_a$ is equal to the Weyl operator whose symbol is the convolution of $a$ with the Wigner transform of the windows $\varphi_2$ and $\varphi_1$. We employ this fact, to extend the definition of localisation operators to symbols $a$ having very fast super-exponential growth by allowing them to be mappings from ${\mathcal D}^{\{M_p\}}(\mathbb R^d)$ into ${\mathcal D}'^{\{M_p\}}(\mathbb R^d)$, where $M_p$, $p\in\mathbb N$, is a non-quasi-analytic Gevrey type sequence. By choosing the windows $\varphi_1$ and $\varphi_2$ appropriately, our main results show that one can consider symbols with growth in position space of the form $\exp(\exp(l|\cdot|^q))$, $l,q>0$.