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Certain averaging operators on Lebesgue spaces. (English) Zbl 1431.42024

Summary: In this paper, we study some multiplier operator \(\mu_{\gamma ,\alpha}\) raised from studying the \(L^p\)-approximation of the spherical mean \(S_t^{\gamma}.\) We obtain the optimal range of exponents \((\alpha ,\gamma ,p)\) such that \(\mu_{\gamma ,\alpha}\) is an \(L^p\) multiplier. As an application, we obtain the convergence rate for \(S_t^{\gamma}\left( f\right)\) in the \(L^p\) spaces.

MSC:

42B15 Multipliers for harmonic analysis in several variables
42B25 Maximal functions, Littlewood-Paley theory
41A35 Approximation by operators (in particular, by integral operators)
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
47B38 Linear operators on function spaces (general)
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References:

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