Ultradistributions on $\mathbb R_{+}^{n}$. Solvability and hypoellipticity through series expansions of ultradistributions

TL;DR

This paper investigates ultradistributions on positive real spaces using Laguerre expansions, and solves differential equations involving ultradifferentiable operators, analyzing their solvability and hypoellipticity through series representations.

Contribution

It introduces a framework for analyzing ultradistributions via Laguerre expansions and provides solutions to complex differential equations, exploring solvability and hypoellipticity in ultradistribution spaces.

Findings

Laguerre expansions characterize ultradistribution spaces on $ plus^n$

Series solutions facilitate analysis of differential equations in ultradistribution spaces

Conditions for solvability and hypoellipticity are established

Abstract

In the first part we analyze space $\mathcal G^*(\mathbb R^{n}_+)$ and its dual through Laguerre expansions when these spaces correspond to a general sequence $\{M_p\}_{p\in\mathbb N_0}$, where $^*$ is a common notation for the Beurling and Roumieu cases of spaces. In the second part we are solving equation of the form $Lu=f,\; L=\sum_{j=1}^ka_jA_j^{h_j}+cE^{d}_y+bP(x,D_x),$ where $f$ belongs to the tensor product of ultradistribution spaces over compact manifolds without boundaries as well as ultradistribution spaces on $\mathbb R^n_+$ and $\mathbb R^m$; $A_j, j=1,...,k$, $E_y$ and $P(x,D_x)$ are operators whose eigenfunctions form orthonormal basis of corresponding $L^2-$space. The sequence space representation of solutions enable us to study the solvability and the hypoellipticity in the specified spaces of ultradistributions.