On the trace theorem to Volterra-type equations with local or non-local derivatives

TL;DR

This paper develops trace and extension theorems for evolution equations with local or non-local derivatives in weighted vector-valued $L_p$ spaces, using generalized interpolation, stochastic processes, and Hardy's inequality, applicable to time-fractional equations.

Contribution

It introduces a generalized real interpolation method and derives new trace and extension theorems for equations with local and non-local derivatives in weighted spaces.

Findings

Established trace and extension theorems for evolution equations with derivatives

Applied results to time-fractional equations with broad temporal weights

Utilized stochastic process theory and Hardy's inequality in proofs

Abstract

This paper considers traces at the initial time for solutions of evolution equations with local or non-local derivatives in vector-valued $L_p$ spaces with $A_p$ weight. To achieve this, we begin by introducing a generalized real interpolation method. Within the framework of generalized interpolation theory, we make use of stochastic process theory and two-weight Hardy's inequality to derive our trace and extension theorems. Our results encompass findings applicable to time-fractional equations with broad temporal weight functions.