Revista Matemática Iberoamericana

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Volume 27, Issue 3, 2011, pp. 953–976
DOI: 10.4171/RMI/659

Published online: 2011-12-06

Geometric-arithmetic averaging of dyadic weights

Jill Pipher[1], Lesley A. Ward[2] and Xiao Xiao[3]

(1) Brown University, Providence, USA
(2) University of South Australia, Mawson Lakes, Australia
(3) Brown University, Providence, USA

The theory of Muckenhoupt's weight functions arises in many areas of analysis, for example in connection with bounds for singular integrals and maximal functions on weighted spaces. We prove that a certain averaging process gives a method for constructing $A_p$ weights from a measurably varying family of dyadic $A_p$ weights. This averaging process is suggested by the relationship between the $A_p$ weight class and the space of functions of bounded mean oscillation. The same averaging process also constructs weights satisfying reverse Hölder ($RH_p$) conditions from families of dyadic $RH_p$ weights, and extends to the polydisc as well.

Keywords: Dyadic harmonic analysis, multiparameter harmonic analysis, dyadic weights, bounded mean oscillation, Muckenhoupt weights, $A_p$ weights, reverse Hölder weights, product weights, bidisc, polydisc

Pipher Jill, Ward Lesley, Xiao Xiao: Geometric-arithmetic averaging of dyadic weights. Rev. Mat. Iberoam. 27 (2011), 953-976. doi: 10.4171/RMI/659