Calderón‐type theorems for operators with non‐standard endpoint behavior on Lorentz spaces

The Calderón theorem states that every quasilinear operator, which is bounded both from \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$L^{p_1,1}$\end{document} to \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$L^{q_1,\infty }$\end{document} , and from \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$L^{p_2,1}$\end{document} to \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$L^{q_2,\infty }$\end{document} for properly ordered values of p 1 , p 2 , q 1 , q 2 , is bounded on some rearrangement‐invariant space if and only if the so‐called Calderón operator is bounded on the corresponding representation space. We will establish Calderón‐type theorems for non‐standard endpoint behavior, where Lorentz Λ and M spaces will be the endpoints of the interpolation segment. Two distinctive types of non‐standard behavior are to be discussed; we’ll explore the operators bounded both from Λ( X 1 ) to Λ( Y 1 ), and from Λ( X 2 ) to M ( Y 2 ) using duality arguments, thus, we need to study the operators bounded both from Λ( X 1 ) to M ( Y 1 ), and from M ( X 2 ) to M ( Y 2 ) first. For that purpose, we evaluate Peetre's K ‐functional for varied pairs of Lorentz spaces.