Nonstandard Calder\'on-type theorems

We establish Calder\'on-type theorems for operators bounded on nonstandard end-point Lorentz spaces \begin{equation*} T\colon L^{p_0, q_0}\to L^{p_1, q_1}\quad\text{and}\quad T\colon L^{q, 1}\to L^\infty \end{equation*} and the improvement of target spaces which is intimately connected with this. The emphasis will be placed on the cases $q_0=q_1$ and $q_1=\infty$.