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

Abstract

In the first part we analyze space ${\mathcal{G}}^{\ast }({\mathbb{R}}_{+}^n)$ and its dual through Laguerre expansions when these spaces correspond to a general sequence $\{M_p{\}}_{p\in {\mathbb{N}}_0}$, where ${}^{\ast }$ is a common notation for the Beurling and Roumieu cases of spaces. In the second part we are solving equation of the form $Lu=f,L={\sum }_{j=1}^k{a}_j{A}_j^{h_j}+c{E}_y^d+bP(x,D_x),$ 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.