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

In the first part we analyze space \({\mathcal {G}}^*({\mathbb {R}}^{n}_+)\) ${\mathcal{G}}^{\ast }({\mathbb{R}}_{+}^n)$ and its dual through Laguerre expansions when these spaces correspond to a general sequence \(\{M_p\}_{p\in {\mathbb {N}}_0}\) ${\{M_p\}}_{p\in {\mathbb{N}}_0}$ , where * is a common notation for the Beurling and Roumieu cases of spaces. In the second part we are solving equation of the form \(Lu=f,\; L=\sum _{j=1}^ka_jA_j^{h_j}+cE^{d}_y+bP(x,D_x),\) $Lu=f,L={\sum }_{j=1}^k{a}_j{A}_j^{h_j}+c{E}_y^d+bP(x,D_x),$ where f belongs to the tensor product of ultradistribution spaces over compact manifolds without boundaries as well as ultradistribution spaces on \({\mathbb {R}}^n_+\) ${\mathbb{R}}_{+}^n$ and \({\mathbb {R}}^m\) ${\mathbb{R}}^m$ ; \(A_j, j=1,...,k\) $A_j,j=1,...,k$ , \(E_y\) $E_y$ and \(P(x,D_x)\) $P(x,D_x)$ are operators whose eigenfunctions form orthonormal basis of corresponding \(L^2-\) $L^2-$ space. The sequence space representation of solutions enable us to study the solvability and the hypoellipticity in the specified spaces of ultradistributions.