Limits of the \(A_p\) constants (Q2320069)

For the Muckenhoupt \(A_p\) and reverse Hölder \(RH_q\) classes of weights on an interval \(I_0\subset\mathbb R\), the author proves the four end-point limit relations \([w]_{A_r}=\lim_{p\to r}[w]_{A_p}\) for both \(r\in\{1,\infty\}\), as well as \([w]_{RH_\infty}=\lim_{q\to\infty}[w]_{RH_q}\) and \([w]_{RH_1}=\lim_{q\to 1}[w]_{RH_q}^{q/(q-1)}\). The definition of \([w]_{A_\infty}\) is the one by \textit{S. V. Khrushchev} [Proc. Am. Math. Soc. 90, 253--257 (1984; Zbl 0539.42009)] involving the exponential of the average of \(\log(1/w)\) and the definition of \([w]_{RH_1}\) is a condition related to \(L\log L\). The results for \(A_\infty\) and \(RH_1\) were previously known, also on \(\mathbb R^n\), but the author gives new proofs of all cases on intervals \(I_0\subset \mathbb R\). The proofs are based on his previous results characterising both the sharp exponents and the related constants in the self-improvement properties of both Muckenhoupt and reverse Hölder weights in terms of solutions of an explicit equation [\textit{A. Popoli}, Ann. Acad. Sci. Fenn., Math. 43, No. 2, 785--805 (2018; Zbl 1409.46026)].