Extrapolation theorem on $L^p$ spaces over infinite measure space: An approach $p\searrow p_0$

After S.Yano's classical work [10], several extrapolation theorems on $L^p$ spaces have been proved to approach to the "singular" spaces, $L^1$ and $L^\infty$. We sometimes find such "singularity" on $L^{p_0} , 1 < p_0 <\infty$. In this paper, we shall prove some extrapolation theorem in the case $p\searrow p_0$.