The Figa–Talamanca–Herz–Lebesgue Banach algebras $A_p^r(G)=A_p(G)\cap L^r(G)$
Functional analytic properties of the Banach Algebras $A_p^r(G)$ such as Arens regularity, amenability, weak sequential completeness, sets of synthesis and strict containment for fixed $p$ and increasing $r$, are investigated.
It is found that $A_p^r(G)$, $1\leq r<\infty$, behave very differently to the Figa–Talamanca–Herz algebras $A_p(G)$.(Published Online April 26 2006)
(Received March 14 2005)