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.
H^\infty functional calculus and square function estimates for Ritt operators
A sharp equivalence between $H^\infty$ functional calculus and square function estimates
An L^1-estimate for certain spectral multipliers associated with the Ornstein--Uhlenbeck operator
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