Abstract
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:
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
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].
Mathematics Subject Classification (2010). Primary 42B20, 42B25. Secondary 46B70, 47B38.
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Ortiz-Caraballo, C., Pérez, C., Rela, E. (2013). Improving Bounds for Singular Operators via Sharp Reverse Höolder Inequality for \( A_\infty \) . In: Almeida, A., Castro, L., Speck, FO. (eds) Advances in Harmonic Analysis and Operator Theory. Operator Theory: Advances and Applications, vol 229. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0516-2_17
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