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Transactions of the American Mathematical Society

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Weak type estimates for a singular convolution operator on the Heisenberg group


Author: Loukas Grafakos
Journal: Trans. Amer. Math. Soc. 325 (1991), 435-452
MSC: Primary 43A80; Secondary 22E30, 42B20
DOI: https://doi.org/10.1090/S0002-9947-1991-1024772-X
MathSciNet review: 1024772
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Abstract: On the Heisenberg group $ {\mathbb{H}^n}$ with coordinates $ (z,t) \in {\mathbb{C}^n} \times \mathbb{R}$, define the distribution $ K(z,t) = L(z)\delta (t)$, where $ L(z)$ is a homogeneous distribution on $ {\mathbb{C}^n}$ of degree $ - 2n$ , smooth away from the origin and $ \delta (t)$ is the Dirac mass in the $ t$ variable. We prove that the operator given by convolution with $ K$ maps $ {H^1}({\mathbb{H}^n})$ to weak $ {L^1}({\mathbb{H}^n})$.


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DOI: https://doi.org/10.1090/S0002-9947-1991-1024772-X
Article copyright: © Copyright 1991 American Mathematical Society

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