Lp regularity for convolution operator equations in Banach spaces

TL;DR

This paper applies advanced Fourier multiplier theorems to derive regularity and lower bound estimates for elliptic and parabolic convolution operator equations in Banach spaces, relevant to heat conduction in materials with memory.

Contribution

It introduces new operator-valued Fourier multiplier techniques to analyze regularity and estimates for convolution operator equations in Banach spaces, including elliptic and parabolic types.

Findings

Established lower bound estimates for elliptic integro-differential equations.

Analyzed separability properties of parabolic convolution equations in heat conduction.

Discussed optimal regularity conditions for elliptic and parabolic equations.

Abstract

Here we utilize operator--valued Lq-Lp Fourier multiplier theorems to establish lower bound estimates for large class of elliptic integro-differential equations in Rd. Moreover, we investigate separability properties of parabolic convolution operator equations that arise in heat conduction problems in materials with fading memory. Finally, we give some remarks on optimal regularity of elliptic differential equations and Cauchy problem for parabolic equations.