Extreme cases of weak type interpolation

Abstract

We consider quasilinear operators $T$ of 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 $L_{\phi ,E}$ with the norm $\Vert \phi (t)f^{\ast }(t){\Vert }_{\tilde{E}}$, where $\tilde{E}$ is arbitrary rearrangement-invariant space with respect to the measure $dt/t$. A space $L_{\phi ,E}$ 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 $T:L_{\phi ,E}\to L_{\psi ,E}$.