Bounds on the Phillips calculus of abstract first order differential operators

TL;DR

This paper establishes spectral multiplier bounds for abstract first order differential operators on $L^p$ spaces, using boundedness on $L^2$ and Sobolev embedding assumptions, extending Phillips calculus results.

Contribution

It introduces new spectral multiplier estimates for abstract differential operators assuming only $L^2$ boundedness, and proves an R-bounded H"ormander calculus for perturbed Hodge-Dirac operators.

Findings

Spectral multiplier bounds for specific group generators

R-bounded H"ormander calculus under Sobolev embedding

Square of perturbed Hodge-Dirac operator admits such calculus

Abstract

For an operator generating a group on $L^p$ spaces transference results give bounds on the Phillips functional calculus also known as spectral multiplier estimates. In this paper we consider specific group generators which are abstraction of first order differential operators and prove similar spectral multiplier estimates assuming only that the group is bounded on $L^2$ rather than $L^p$. We also prove an R-bounded H\"ormander calculus result by assuming an abstract Sobolev embedding property and show that the square of a perturbed Hodge-Dirac operator has such calculus.