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In this paper we revisit the theory of one-parameter semigroups of linearoperators on Banach spaces in order to prove quantitative bounds for boundedholomorphic semigroups. Subsequently, relying on these bounds we obtain newquantitative versions of two recent results of Xu related to the vector-valuedLittlewood--Paley--Stein theory for symmetric diffusion semigroups.

Quantitative estimates for bounded holomorphic semigroups