An Extrapolation Theorem in the setting of Hausdorff capacities
Aniruddha Deshmukh
TL;DR
This work extends Rubio de Francia extrapolation to the setting of Hausdorff capacities by developing capacitary Muckenhoupt weights $A_{p,\beta}$, establishing Hölder-type inequalities for Choquet integrals, and constructing $A_{1,\beta}$ weights via the Hausdorff maximal operator $M_{\beta}$. It then proves a Rubio de Francia-type extrapolation theorem for capacitary $L^p$ spaces, showing boundedness transfer from $L^{p_0}$ to $L^p$ for $p<p_0$ (with $w\in A_{1,\beta}$) and for all $p>1$ (with $w\in A_{p,\beta}$), using Jones factorization and a duality/interpolation strategy anchored in capacity-specific tools. The results rely on the dyadic Hausdorff content, capacity-based maximal bounds, and the self-improving properties of capacitary weights, thereby enriching the analytical toolkit available for nonlinear potential theory. Overall, the paper broadens the scope of extrapolation techniques to nonlinear capacity settings, enabling robust norm inequalities in capacitary harmonic analysis and potential theory.
Abstract
In this article, we prove an analogue of the Rubio de Francia's extrapolation theorem in the setting of Hausdorff capacities. We prove the result using techniques analogous to those in the classical setting and using the recently developed theory of capacitary Muckenhoupt weights.