Beurling-Pollard type theorems

We establish a version of the Beurling-Pollard theorem for operator synthesis and apply it to derive some results on linear operator equations and to prove a Beurling-Pollard type theorem for Varopoulos tensor algebras. Additionally we establish a Beurling-Pollard theorem for weighted Fourier algebras and use it to obtain ascent estimates for operators that are functions of generalized scalar operators.