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.
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Partially supported by Ministerio de Educación y Ciencia (Spain), Grant MTM2005-08350-C03-01.
R. Crescimbeni was partially supported by Fundación Carolina, Ministerio de Educación de la República Argentina and Universidad Nacional del Comahue. R. A. Macías and B. Viviani were partially supported by Facultad de Ingeniera Química-UNL.
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Crescimbeni, R., Macías, R.A., Menárguez, T. et al. The ρ-variation as an operator between maximal operators and singular integrals. J. Evol. Equ. 9, 81–102 (2009). https://doi.org/10.1007/s00028-009-0003-0
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DOI: https://doi.org/10.1007/s00028-009-0003-0