On the consequences of a Mihlin-H\"ormander functional calculus: maximal and square function estimates

TL;DR

This paper demonstrates that a Mihlin-H"ormander functional calculus for an operator ensures the boundedness of associated maximal and square function operators on L^p spaces, broadening the understanding of spectral multiplier effects.

Contribution

It establishes new boundedness results for maximal and square functions derived from spectral multipliers under Mihlin-H"ormander conditions.

Findings

Maximal operators are bounded on L^p.

Square functions are bounded on L^p.

Results apply to multipliers with finite smoothness and integral conditions.

Abstract

We prove that the existence of a Mihlin-H\"ormander functional calculus for an operator $L$ implies the boundedness on $L^p$ of both the maximal operators and the continuous square functions build on spectral multipliers of $L.$ The considered multiplier functions are finitely smooth and satisfy an integral condition at infinity. In particular multipliers of compact support are admitted.