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.
In this paper we establish a T 1 criterion for the boundedness of Hermite-Calderón -Zygmund operators on the B M O H (R n ) space naturally associated to the Hermite operator H . We apply this criterion in a systematic way to prove the boundedness on B M O H (R n ) of certain harmonic analysis operators related to H (Riesz transforms, maximal operators, Littlewood-Paley g-functions and variation operators).
Abstract. In this paper we establish L p -boundedness properties for Laplace type transform spectral multipliers associated with the Schrödinger operator L = −∆ + V . We obtain for this type of multipliers pointwise representation as principal value integral operators. We also characterize the UMD Banach spaces in terms of the L p -boundedness of the imaginary powers L iγ , γ ∈ R, of L.
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