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In this paper we obtain new characterizations of the uniformly convex andsmooth Banach spaces. These characterizations are established by usingLp-boundedness properties of Littlewood-Paley functions and area integralsassociated with heat semigroups and involving fractional derivatives. Ourresults apply for instance by considering the heat semigroups defined byHermite and Laguerre operators that do not satisfy the Markovian property.

Uniformly convex and smooth Banach spaces and Lp-boundedness properties of Littlewood-Paley and area functions associated with semigroups