A Rigorous Analysis of Integro-Differential Operators with Non-Singular Kernels

Abstract

Integro-differential operators with non-singular kernels have been much discussed among fractional calculus researchers. We present a mathematical study to clearly establish the rigorous foundations of this topic. By considering function spaces and mapping results, we show that operators with non-singular kernels can be defined on larger function spaces than operators with singular kernels, as differentiability conditions can be removed. We also discover an analogue of the Sonine invertibility condition, giving two-sided inversion relations between operators with non-singular kernels that are not possible for operators with singular kernels.