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Weak convergence of measures and weak type $ (1,q)$ of maximal convolution operators

Authors: Filippo Chiarenza and Alfonso Villani
Journal: Proc. Amer. Math. Soc. 97 (1986), 609-615
MSC: Primary 42B20; Secondary 28A33
MathSciNet review: 845974
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Abstract: Let $ {G^ * }$ be the maximal convolution operator associated with a sequence of $ {L^1}$ kernels. We show that if $ {G^ * }$ is of weak type $ (1,q)$, $ 1 \leq q < \infty $, over a subset $ {\mathcal N}$ of $ {\mathcal M}$ (the space of all finite positive Borel measures on $ {{\bf {R}}^h}$ endowed with the weak topology), then $ {G^ * }$ is of weak type $ (1,q)$ over the closed cone in $ {\mathcal M}$ generated by $ {\mathcal N}$. As a particular case we obtain a well-known result by de Guzman.

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