Convolution Operators on Banach Lattices with Shift-Invariant Norms

Let G be a locally compact abelian group and let mu be a complex valued regular Borel measure on G. In this paper we consider a generalisation of a class of Banach lattices introduced in Johansson (Syst Control Lett 57:105-111, 2008). We use Laplace transform methods to show that the norm of a convolution operator with symbol mu on such a space is bounded below by the L(infinity) norm of the Fourier-Stieltjes transform of mu. We also show that for any Banach lattice of locally integrable functions on G with a shift-invariant norm, the norm of a convolution operator with symbol mu is bounded above by the total variation of mu.