Sharpness of the asymptotic interpolation exponents

Determine whether the exponents in the asymptotic interpolation estimates of Theorem \ref{thm:liminf}, namely the exponents defined in \eqref{eq:thetas}, are sharp for parameter values other than those treated by the Hardy-operator example.

Background

The paper proves asymptotic Marcinkiewicz interpolation estimates for quasi-sublinear operators acting between weak Lebesgue spaces. The estimates control lower and upper asymptotic distribution quantities of the output using corresponding quantities for the input, with interpolation exponents \theta and \Theta determined by the parameters 0<q<p<r\leq\infty.

The Hardy operator example establishes optimality of the lower-limit exponent in the special case q=1 and r=\infty, showing that \theta=1-1/p cannot generally be improved for those parameters. The authors explicitly state that they have not investigated whether the formulas for the exponents are sharp for other parameter choices. Establishing this would characterize the optimal dependence of the asymptotic interpolation bounds across the remaining parameter range.