A method for obtaining time-periodic Lp estimates

We introduce a method for showing a priori  LpLp estimates for time-periodic, linear, partial differential equations set in a variety of domains such as the whole space, the half space and bounded domains. The method is generic and can be applied to a wide range of problems. We demonstrate it on the heat equation. The main idea is to replace the time axis with a torus in order to reformulate the problem on a locally compact abelian group and to employ Fourier analysis on this group. As a by-product, maximal LpLp regularity for the corresponding initial-value problem follows without   the notion of RR-boundedness. Moreover, we introduce the concept of a time-periodic fundamental solution.