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A characterization of the uniform strong type (1,1)(1,1) bounds for averaging operators

By J. Aldaz
We prove that in a metric measure space (X,d,μ)(X, d, \mu), the averaging operators Ar,μA_{r, \mu } satisfy a uniform strong type (1,1)(1,1) bound supr,μAr,μL1L1<\sup_{r, \mu} \|A_{r, \mu }\|_{L^1\to L^1} < \infty if and only if XX satisfies a certain geometric condition, the equal radius Besicovitch intersection property.
February 28, 2019
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A characterization of the uniform strong type $(1,1)$ bounds for averaging operators
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