Convolution operators on weighted spaces of continuous functions and supremal convolution

Abstract

The convolution of two weighted balls of measures is proved to be contained in a third weighted ball if and only if the supremal convolution of the corresponding two weights is less than or equal to the third weight. Here supremal convolution is introduced as a type of convolution in which integration is replaced with supremum formation. Invoking duality the equivalence implies a characterization of equicontinuity of weight-bounded sets of convolution operators having weighted spaces of continuous functions as domain and range. The overall result is a constructive method to define weighted spaces on which a given set of convolution operators acts as an equicontinuous family of endomorphisms. The result is applied to linear combinations of fractional Weyl integrals and derivatives with orders and coefficients from a given bounded set.

Fractional glassy relaxation and convolution modules of distributions

“…It has been shown recently by the authors [19], that the space E can be endowed with a weighted topology in a natural way such that one obtains bicontinous operators. By means of Theorem 10, Eq.…”

“…Given the need for concretely characterized extended domains in applications a first step was taken in [18,19] where fractional Weyl integrals have been interpreted as convolutions of Radon measures with continuous functions. Locally convex topologies generated by weighted supremum norms were identified such that a given set of convolution operators acts as an equicontinuous family of endomorphisms on certain weighted spaces of continuous functions [19].…”

“…Given the need for concretely characterized extended domains in applications a first step was taken in [18,19] where fractional Weyl integrals have been interpreted as convolutions of Radon measures with continuous functions. Locally convex topologies generated by weighted supremum norms were identified such that a given set of convolution operators acts as an equicontinuous family of endomorphisms on certain weighted spaces of continuous functions [19]. Our objective in this work is to combine the benefits of our approach [18,19] with those of Marchaud's [26], with those of fractional powers [2,12,15,17,20,47], with those of translation invariant distribution spaces [22,41], and with the algebraic benefits of operational calculus [13,30,48].…”

“…Abstract operational approaches [13,30,48] or fractional powers (with appropriate domain restrictions) [2,12,15,17,20,47] automatically guarantee invertibility and the validity of algebraic relations between operators. Merging these benefits with the advantage of "maximal" domains for certain sets of integral operators on weighted spaces in [18,19] has therefore been the objective of this work.…”

On extremal domains and codomains for convolution of distributions and fractional calculus

“…Formation of K -translation shells equals supremal convolution with the indicator function 1 K . Supremal convolution arises in the context of convolution operators of measures on weighted spaces of continuous functions [30]. By virtue of [30,Prop.…”