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

By R. Crescimbeni, R. Macías, T. Menárguez, José L. Torrea, B. Viviani. 2009

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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 Ap 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.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/57265bdfc99e8103f6ee746ed1aec7e67eb21d2b