A note on dyadic coverings and nondoubling Calderón-Zygmund theory

2012/01/17 by Jose M. Conde, Conde, Jose M. · 1 citation
Mathematics · #42B20 #52C17 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #math.CA #msc:42B20 #msc:52C17

paper · pdf · doi:10.48550/arxiv.1201.3513

arxiv created 2012/03/15 · arxiv updated 2012/03/16

Abstract

We construct a family of n+1 dyadic filtrations in Rⁿ, so that every Euclidean ball B is contained in some cube Q of our family satisfying diam(Q) ≤ c_n diam(B) for some dimensional constant c_n. Our dyadic covering is optimal on the number of filtrations and improves previous results of Christ and Garnett/Jones by extending a construction of Mei for the n-torus. Based on this covering and motivated by applications to matrix-valued functions, we provide a dyadic nondoubling Calderón-Zygmund decomposition which avoids Besicovitch type coverings in Tolsa's decomposition. We also use a recent result of Hytönen and Kairema to extend our dyadic nondoubling decomposition to the more general setting of upper doubling metric spaces.

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