2013 | OriginalPaper | Chapter
Improving Bounds for Singular Operators via Sharp Reverse Höolder Inequality for
Authors : Carmen Ortiz-Caraballo, Carlos Pérez, Ezequiel Rela
Published in: Advances in Harmonic Analysis and Operator Theory
Publisher: Springer Basel
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In this expository article we collect and discuss some recent results on different consequences of a Sharp Reverse Hölder Inequality for
$$ A\infty $$
weights. For two given operators
T
and
S
, we study
$$ L^{p}(w) $$
bounds of Coifman– Fefferman type:
$$ \parallel T\;f \parallel_{L^p}(w)\;\leq \; c_{n,w,p}\parallel S\;f \parallel _{L^p}(w),$$
that can be understood as a way to control T by S.
We will focus on a
quantitative
analysis of the constants involved and show that we can improve classical results regarding the dependence on the weight
w
in terms of Wilson’s
$$ A\infty $$
constant
$$ [w]A_{\infty}\; := \; {\rm sup_Q}\frac{1}{w(Q)}\int_{Q}{M({w_\mathcal{X}}_Q)} .$$
We will also exhibit recent improvements on the problem of finding sharp constants for weighted norm inequalities involving several singular operators. In the same spirit as in [10], we obtain mixed
$$ A_{1}-A_{\infty}$$
estimates for the commutator [b,T] and for its higher–order analogue
$$ T^{k}_{b}$$
. A common ingredient in the proofs presented here is a recent improvement of the Reverse Hölder Inequality for
$$ A_{\infty}$$
weights involving Wilson’s constant from [10].