On extrapolation blowups in the $L^p$ scale

Yano's extrapolation theorem dated back to 1951 establishes boundedness properties of a subadditive operator $T$ acting continuously in $L_p$ for $p$ close to $1$ and/or taking $L_{\infty}$ into $L_p$ as $p\to1_{+}$ and/or $p\to\infty$ with norms blowing up at speed $(p-1)^{-\alpha}$ and/or $p^\beta$, $\alpha,\beta>0$. Here we give answers in terms of Zygmund, Lorentz-Zygmund and small Lebesgue spaces to what happens if $\|Tf\|_p\le c(p-r)^{-\alpha}\|f\|_p$ as $p\to r_+$ ($1<r<\infty$). The study has been motivated by current investigations of convolution maximal functions in stochastic analysis, where the problem occurs for $r=2$. We also touch the problem of comparison of results in various scales of spaces.