We consider quasilinear operators $T$ of {\it joint weak type}
$(a,b;p,q)$ (in the sense of [Bennett, Sharpley: Interpolation
of operators, Academic Press, 1988]) and study their properties
on spaces $L_{\varphi,E}$ with the norm $\|\varphi(t)f^*(t)
\|_{\tilde E}$, where $\tilde E$ is arbitrary rearrangement-invariant
space with respect to the measure $dt/t$. A space $L_{\varphi,E}$
is said to be ``close" to one of the endpoints of interpolation if
the corresponding Boyd index of this space is equal to $1/a$ or to
$1/p$. For all possible kinds of such ``closeness", we give sharp
estimates for the function $\psi(t)$ so as to obtain that every
$T:L_{\varphi,E}\to L_{\psi,E}$.