$b$-coercive convolution equations in weighted function spaces and applications

Abstract

We study the $b$-separability properties of elliptic convolution operators in weighted Besov spaces and establish sharp estimates for the resolvents of the convolution operators. As a result, it is shown that these operators are positive and, in addition, play the role of negative generators of analytic semigroups. Moreover, the maximal $b$-regularity properties of the Cauchy problem for a parabolic convolution equation are established. Finally, these results are applied to obtain the maximal regularity properties for anisotropic integro-differential equations and the system of infinitely many convolution equations.