article

Title: Calderon-type theorems for operators of non-standard endpoint behaviour

Authors: Amiran Gogatishvili and Lubos Pick

Issue: Volume 58 (2009), Issue 4, 1831-1852

Abstract:

$The fundamental interpolation theorem of Calder\'on states that a quasilinear operator satisfying, for $1\leq p_0,q_0,p_1,q_1\leq\infty$\[T:L^{p_0,1}\to L^{q_0,\infty}\quad\mbox{and}\quad T:L^{p_1,1}\to L^{q_1,\infty},\] is bounded from a rearrangement-invariant space into another one if and only if an appropriate one-dimensional integral operator is bounded between their respective representation spaces. We establish a Calder\'on-type theorem for operators satisfying \[T:L^{p_0,1}\to L^{q_0,\infty}\quad\textup{and}\quad T:L^{p_1,\infty}\to L^{\infty}, \] and apply this result to the fractional maximal operator. We next prove a Calder\'on-type theorem for operators satisfying \[T:L^1\to L^{q_0,1}\quad\mbox{and}\quad T:L^{p_1,1}\to L^{q_1,\infty}, \] which has applications to sharp Sobolev embeddings and to boundary trace embeddings.$