Weighted resolvent estimates for the spatially periodic Stokes equations

Abstract

We consider spatially periodic Laplace and Stokes resolvent problems in the whole space and show corresponding weighted resolvent estimates with weights in the Muckenhoupt class. A main tool is the use of Fourier techniques on the Schwartz-Bruhat space and on the tempered distributions together with a weighted transference principle a la Andersen and Mohanty (Proc Amer Math Soc 137(5):1689–1697, 2009) and a splitting of the function spaces into a mean-value free part and a nonperiodic part of functions defined on a lower-dimensional whole space, which then enables us to make use of a weighted Mikhlin multiplier theorem.