On the consequences of a Mihlin-Hörmander functional calculus: maximal and square function estimates

We prove that the existence of a Mihlin-Hormander functional calculus for an operator L implies the boundedness on \(L^p\) of both the maximal multiplier 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.