Limits of the Ap constants

The constant [w]Ajavax.xml.bind.JAXBElement@42fee7bb of a weight w in the Muckenhoupt class A∞=⋃p>1Ap can be seen as limit of the [w]Ajavax.xml.bind.JAXBElement@5d3111dc constant as p tends to infinity in virtue of a theorem proved by Sbordone and Wik in [21]. A parallel theorem holds true if we look at weights v in the Gehring class RH1=⋃q>1RHq, being A∞=RH1. In this note we prove that corresponding theorems hold true, in dimension one, for the smallest limit classes A1 and RH∞. We show that the constants [w]Ajavax.xml.bind.JAXBElement@59bb311d and [v]RHjavax.xml.bind.JAXBElement@5b81dd19 can be obtained from the constants [w]Ajavax.xml.bind.JAXBElement@6ef73aed and [v]RHjavax.xml.bind.JAXBElement@3b1d791e, passing to the limits respectively for p→1 and q→∞. Our method relies on some recent sharp integrability results which highlight some new symmetries between the classes. Furthermore we provide a simplified proof in R for the original theorems in A∞ and RH1.