The ρ-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 ${L^p(\mathbb{R}^n,\,w(x){\rm d}x)}$ into itself (from ${L^1(\mathbb{R}^n,\,w(x){\rm d}x)}$ into weak-${L^1(\mathbb{R}^n,\,w(x){\rm d}x)}$ in the case p = 1) for 1 ≤ p < ∞ and w being a weight in the Muckenhoupt’s A _p 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.