On a class of translation-invariant spaces of quasianalytic ultradistributions

A class of translation-invariant Banach spaces of quasianalytic ultradistributions is introduced and studied. They are Banach modules over a Beurling algebra. Based on this class of Banach spaces, we define corresponding test function spaces ${\mathcal{D}}_E^{\ast }$ and their strong duals ${\mathcal{D}}_{E_{\ast }^{\mathrm{\prime }}}^{\mathrm{\prime }\ast }$ of quasianalytic type, and study convolution and multiplicative products on ${\mathcal{D}}_{E_{\ast }^{\mathrm{\prime }}}^{\mathrm{\prime }\ast }$. These new spaces generalize previous works about translation-invariant spaces of tempered (non-quasianalytic ultra-) distributions; in particular, our new considerations apply to the settings of Fourier hyperfunctions and ultrahyperfunctions. New weighted ${\mathcal{D}}_{L_{\eta }^p}^{\mathrm{\prime }\ast }$ spaces of quasianalytic ultradistributions are analyzed.