Nonstandard Calderón-type theorems
David Kubíček
TL;DR
The paper develops Calderón-type interpolation for operators bounded on nonstandard end-point Lorentz spaces, addressing mappings of the form $T: L^{p_0,q_0} \to L^{p_1,q_1}$ and $T: L^{q,1} \to L^ fty$, with emphasis on the diagonal case $q_0=q_1$ and the endpoint case $q_1=\infty$. The core approach expresses the Calderón operator via one-dimensional Hardy-type components $R_{q_0}$ and $S$, and introduces new rearrangement-invariant spaces $Y^{\langle p,q_1\rangle}$ (and the nonlinear $Y_{q_0,q_1}$) to capture improved target information. The main results establish a diagonal Calderón-type theorem: boundedness from a rearrangement-invariant space $X$ to $Y^{\langle p_1,q_0\rangle}$ is equivalent to the boundedness of $R_{q_0}$ and $S$ on representation spaces, with extensions to $q_0<q_1$ (including $q_1=\infty$ and $1\le q_0<q_1<\infty$) via $Y_{q_0,q_1}$ and extrapolation arguments. The findings clarify when Calderón-type bounds hold in nonstandard Lorentz settings and demonstrate that both Calderón suboperators are generally required, enabling potential applications to fractional maximal and Riesz potential operators on nonstandard measures.
Abstract
We establish Calderón-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 $q_1=\infty$.