Fractional square functions and potential spaces

Carlos Segovia and Richard Wheeden defined fractional square functions involving fractional derivatives. They obtained characterizations of potential spaces via square functions. Our aim in this paper is to reconsider the ideas of Segovia and Wheeden under the light of the semigroups of operators. We develop a quite general theory of fractional square functions associated to certain classes of operators. We present some examples of differential operators where our theory applies. We recover in a more compact way the results of Segovia and Wheeden and we obtain new characterizations of the potential spaces associated to the harmonic oscillator and Ornstein–Uhlenbeck operators.