The ρ-variation and the oscillation of the heat and Poisson semigroups of the Laplacian and Hermite operators (i.e. and − + |x| 2 ) are proved to be bounded fromin 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 B M O but the range of the image is strictly smaller that the range of a general singular integral operator.
The weak type 1 for the Mehler maximal function is studied via a precise estimate for the ‘maximal kernel’. This, in turn, allows the geometry involved in this setting to be described.
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