Non-spherical Harish-Chandra Fourier transforms on real reductive groups

The Harish-Chandra Fourier transform, $f\mapsto \mathcal{H}f,$ is a linear topological algebra isomorphism of the spherical (Schwartz) convolution algebra ${\mathcal{C}}^p(G\mathrm{/}\mathrm{/}K)$ (where $K$ is a maximal compact subgroup of any arbitrarily chosen group $G$ in the Harish-Chandra class and $0<p\le 2$) onto the (Schwartz) multiplication algebra $\bar{\mathcal{Z}}({\mathfrak{F}}^{ϵ})$ (of $\mathfrak{w}-$invariant members of $\mathcal{Z}({\mathfrak{F}}^{ϵ}),$ with $ϵ=(2\mathrm{/}p)-1$). This is the well-known Trombi-Varadarajan theorem for spherical functions on the real reductive group, $G\mathrm{.}$ Even though ${\mathcal{C}}^p(G\mathrm{/}\mathrm{/}K)$ is a closed subalgebra of ${\mathcal{C}}^p(G),$ a similar theorem cannot however be proved for the full Schwartz convolution algebra ${\mathcal{C}}^p(G)$ except; for ${\mathcal{C}}^p(G\mathrm{/}K)$ (whose method is essentially that of Trombi-Varadarajan, as shown by M. Eguchi); for few specific examples of groups (notably $G=SL(2,\mathbb{R})$) and; for some notable values of $p$ (with restrictions on $G$ and/or on members of ${\mathcal{C}}^p(G)$). In this paper, we construct an appropriate image of the Harish-Chandra Fourier transform for the full Schwartz convolution algebra ${\mathcal{C}}^p(G),$ without any restriction on any of $G,p$ and members of ${\mathcal{C}}^p(G)\mathrm{.}$ Our proof, that the Harish-Chandra Fourier transform, $f\mapsto \mathcal{H}f,$ is a linear topological algebra isomorphism on ${\mathcal{C}}^p(G),$ equally shows that its image ${\mathcal{C}}^p(\widehat{G})$ can be nicely decomposed, that the full invariant harmonic analysis is available and implies that the definition of the Harish-Chandra Fourier transform may now be extended to include all $p-$tempered distributions on $G$ and to the zero-Schwartz spaces