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

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.