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Maximal averages over flat radial hypersurfaces. (English) Zbl 0961.42015

Consider the operator \[ Af := \sup_{t > 0} f * d\sigma_t \] where \(d\sigma_t\) is a normalized surface measure on some smooth compact hypersurface \(tS\). If this surface has finite type then this operator is bounded on a non-trivial range of \(L^p\). The author studies the infinite type case, in which one can only expect Orlicz space estimates. In the case when the surface is a surface of revolution the author obtains essentially sharp criteria on the Orlicz spaces on which A is bounded.

MSC:

42B25 Maximal functions, Littlewood-Paley theory
53A05 Surfaces in Euclidean and related spaces
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