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

The rho-variation and the oscillation of the heat and Poisson semigroups of the Laplacian and Hermite operators (i.e. Delta and -Delta + vertical bar x vertical bar(2)) are proved to be bounded from L p(R-n, w(x)dx) into itself (from L-1(R-n, w(x)dx) into weak-L-1(R-n, w(x)dx) in the case p = 1) for 1 <= p < infinity and w being a weight in the Muckenhoupt's A(p) class. In the case p = infinity it is proved that these operators do not map L-infinity into itself. Even more, they map L-infinity into BMO but the range of the image is strictly smaller that the range of a general singular integral operator.