Abstrakt
The Calderón theorem states that every quasilinear operator, which is bounded both from $L^{p_1,1}$ to $L^{q_1,\infty}$, and from $L^{p_2,1}$ to $L^{q_2,\infty}$ 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 $\Lambda$ 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 $\Lambda(X_1)$ to $\Lambda(Y_1)$, and from $\Lambda(X_2)$ to $M(Y_2)$ using duality arguments, thus, we need to study the operators bounded both from $\Lambda(X_1)$ 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.