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Schwartz kernels on the Heisenberg group

Alessandro Veneruso (2003)

Bollettino dell'Unione Matematica Italiana

Let H n be the Heisenberg group of dimension 2 n + 1 . Let L 1 , , L n be the partial sub-Laplacians on H n and T the central element of the Lie algebra of H n . We prove that the kernel of the operator m L 1 , , L n , - i T is in the Schwartz space S H n if m S R n + 1 . We prove also that the kernel of the operator h L 1 , , L n is in S H n if h S R n and that the kernel of the operator g L , - i T is in S H n if g S R 2 . Here L = L 1 + + L n is the Kohn-Laplacian on H n .

Self-affine measures that are L p -improving

Kathryn E. Hare (2015)

Colloquium Mathematicae

A measure is called L p -improving if it acts by convolution as a bounded operator from L q to L² for some q < 2. Interesting examples include Riesz product measures, Cantor measures and certain measures on curves. We show that equicontractive, self-similar measures are L p -improving if and only if they satisfy a suitable linear independence property. Certain self-affine measures are also seen to be L p -improving.

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