On the inverse of the sum of two sectorial operators

TL;DR

This paper investigates the inverse of the sum of two sectorial operators on Banach spaces, establishing conditions for maximal regularity in abstract parabolic and hyperbolic problems, with implications for $L^p$-regularity.

Contribution

It introduces a boundedness condition for a specific operator $H^$-calculus that ensures maximal regularity for solutions of abstract linear operator equations.

Findings

Boundedness of a special $H^$-calculus operator implies maximal regularity.

Provides $L^p$-regularity conditions for abstract parabolic problems.

Establishes sufficient conditions for solutions to abstract hyperbolic problems.

Abstract

We study an abstract linear operator equation on a Banach space by using the inverse of the sum of two sectorial operators. We prove that the boundedness of a special type of operator valued $H^\infty$-calculus is sufficient for maximal regularity of the solution. We apply the result to the abstract parabolic problem, to give a maximal $L^{p}$-regularity condition. We also study the abstract hyperbolic problem and give a sufficient condition for the existence of solution.