Limited-Range Multilinear Off-Diagonal Extrapolation and Weighted Transference Principle
Jonas Sauer
TL;DR
The paper tackles weighted extrapolation in $L^{p}$ spaces on σ-finite measure spaces and locally compact abelian groups, addressing multilinear off-diagonal and mixed-norm settings. It advances the theory by decoupling weight exponents from spatial exponents (subject to a basic consistency condition) to achieve full-range extrapolation, including $(0,1]$, with explicit bounds and off-diagonal behavior. It provides a limited-range multilinear extrapolation framework, extends it to mixed-norm and vector-valued contexts with ${\mathcal{R}}$-bounds, and establishes a weighted de Leeuw-type transference principle for transferring Fourier multiplier bounds between groups under $A_p$-type weights, including endpoint cases. Together, these results unify and sharpen weighted extrapolation and transference in broad non-Euclidean settings, offering explicit, sharp-type bounds and applicability to sigma-finite spaces.
Abstract
Multilinear $L^p$ extrapolation results are established in a limited-range, multilinear, and off-diagonal setting for mixed-norm Lebesgue spaces over $σ$-finite measure spaces. Integrability exponents are allowed in the full range $(0,\infty]$. We detach the exponents for the weight classes completely from the exponents for the initial and target spaces for the extrapolation except for the basic consistency condition. This enables to cover the full range $(0,\infty]$ for all integrability exponents and provides new insights into the dependency of the extrapolated bounds on the weight characteristic. Certain endpoint results are new even for $\mathbb{R}^d$. Additionally, in the setting of compact abelian groups, a weighted transference principle is established.