On the Riesz Transforms for Gaussian Measures

Let B be an n × n positive-definite symmetric matrix, and LB the second order partial differential operator in Rn defined by LBu = 12Δ − Bx · ∇u. The operator LB is self-adjoint with respect to the Gaussian probability measure γBn(x)dx, where γBn(x) = Cn, B exp(−Bx · x). In this paper a class of Riesz′s transforms naturally associated with LB is studied. It is shown that these transformations are bounded in the spaces LpγBn(Rn), p > 1, with a constant independent of the dimension an depending only on p and the number of different eigenvalues of the matrix B. The proof of this result is analytic and uses appropriate square-functions defined in terms of semigroups of operators related to LB and the Littlewood-Paley-Stein theory. The result contains as a particular case some inequalities proved by Meyer and Gundy using probabilistic methods.