Endpoint extrapolation from weak to strong type

Determine whether the endpoint extrapolation conclusion can be extended from boundedness of the sub-linear operator T from L^1 to L^{1,\infty} to boundedness from L^{p_0,\infty}(\omega) to L^{p_0}(\omega) for a class of weights such as A_1\cap RH_{q_0'} .

Background

The endpoint extrapolation theorem in the paper proves that, under suitable weighted weak-type hypotheses, a sub-linear operator is bounded from L^1 to L^{1,\infty}. The authors raise the possibility of a stronger endpoint result involving an arbitrary lower endpoint p_0 and weighted Lorentz-type input control.

The proposed weight class is A_1\cap RH_{q_0'}. The paper explicitly leaves the validity of this strengthening unresolved and does not investigate it further.