Geometric Properties of some Banach Algebras related to the Fourier algebra on Locally Compact Groups

Let $A_p(G)$ denote the Figa-Talamanca-Herz Banach Algebra of the locally compact group}} $G$, thus $A_2(G)$ {\it{is the Fourier Algebra of $G$. If $G$ is commutative then $A_2(G)=L^1(\widehat{G}{)}^{\wedge }$. Let $A_p^r(G)=A_p\cap L^r(G)$ with norm $\mathrm{\parallel }u{\mathrm{\parallel }}_{A_p^r}=\mathrm{\parallel }u{\mathrm{\parallel }}_{A_p}+\mathrm{\parallel }u{\mathrm{\parallel }}_{L^r}$.We investigate for which $p$, $r$, and $G$ do the Banach algebras $A_p^r(G)$ {\it{have the Banach space geometric properties: The Radon-Nikodym Property (RNP), the Schur Property (SP) or the Dunford-Pettis Property (DPP). The results are new even if $G=R$ (the real line) or $G=Z$ (the additive integers).