Let $T$ be a sublinear operator such that $(T\chi_E)^*(t)\,{\le}\, h(t, |E|)$ for some positive function $h(t,s)$ and every measurable set $E$. Then it is shown that under some conditions on the operator $T$, this restricted weak type estimate can be extended to the set of all functions $f\,{\in}\, L^1$ such that $\n f.\infty\,{\le}\, 1$, in the sense that $(Tf)^*(t)\,{\le}\, h(t, \n f.1)$. This inequality allows strong type estimates for $T$ to be obtained on several classes of spaces, such as logarithmic spaces and Lorentz spaces. A similar problem for operators $T$ acting on radial functions is also studied.