The rho-variation as an operator between maximal operators and singular integrals

Abstract

The ρ-variation and the oscillation of the heat and Poisson semigroups of the Laplacian and Hermite operators (i.e. Δ and −Δ + |x|2) are proved to be bounded from Lp(Rn w(x)dx) into itself (fromL1(Rn w(x)dx) into weak-L1(Rn w(x)dx) in the case p = 1) for 1 ≤ p < ∞ and w being a weight in the Muckenhoupt?s Ap class. In the case p = ∞ it is proved that these operators do not map L∞ into itself. Even more, they map L∞ into BMO but the range of the image is strictly smaller that the range of a general singular integral operator.