Convolution operators on Banach lattices with shift-invariant norms

TL;DR

This paper investigates convolution operators on Banach lattices with shift-invariant norms, establishing bounds on their norms using Fourier transforms and total variation of measures, extending previous lattice theory results.

Contribution

It generalizes Banach lattice classes and applies Laplace transform methods to derive bounds on convolution operator norms in these spaces.

Findings

Lower bound of convolution operator norm by Fourier-Stieltjes transform's L_infinity norm

Upper bound of convolution operator norm by measure's total variation

Extension of bounds to Banach lattices of locally integrable functions

Abstract

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 [6]. 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_\infty 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.