Dual of the Choquet spaces with general Hausdorff content
Volume 266 / 2022
Abstract
Let $\lambda :{\mathcal D}\to (0,\infty ]$ be a set function defined on the extended dyadic cubes ${\mathcal D}\subset \mathbb {R}^{n}$ satisfying a certain continuity property. We denote by $H^{\lambda }$ the general Hausdorff content. We define the fractional maximal function of a (signed) Radon measure $\mu $ by $$M_{\lambda }\mu (x)=\sup _{Q\in {\mathcal D}}{\bf 1}_{Q}(x)\frac {|\mu |(Q)}{\lambda (Q)}, \quad x\in \mathbb {R}^{n}.$$ We verify that the dual of the Choquet space $L^{1}(H^{\lambda })$ is the set of all Radon measures $\mu $ satisfying $$\|M_{\lambda }\mu \|_{L^{\infty }(H^{\lambda })} \lt \infty ,$$ and the dual of $L^{p}(H^{\lambda })$, $1 \lt p \lt \infty $, is the set of all Radon measures $\mu $ satisfying $$\|M_{\lambda }\mu \|_{L^{p’}(H^{\lambda })} \lt \infty ,\quad p’=\frac {p}{p-1}.$$
