The equivalence between pointwise Hardy inequalities and uniform fatness (Q652237)

Let \(1 \leq p < \infty\), \(\Omega\) a domain in \(\mathbb{R}^n\) and \(u \in C^{\infty}_0(\Omega)\). The inequality \[ |u(x)| \leq C d(x,\partial \Omega) M_{2d(x,\partial \Omega) }(|\nabla u|^p )(x)^{1/p}, \] \(x \in \Omega\), is called a pointwise \(p\)-Hardy-Littlewood inequality, see [\textit{P. Hajłasz}, Proc. Am. Math. Soc. 127, No. 2, 417--423 (1999; Zbl 0911.31005)] and [\textit{J. Kinnunen} and \textit{O. Martio}, Math. Res. Lett. 4, No. 4, 489--500 (1997; Zbl 0909.42014)]. Here \(M_t(f)\) stands for the restricted Hardy-Littlewood maximal function of \(f\). The classical \(p\)-Hardy-Littlewood inequality \[ \int_{\Omega}(|u(x)|/d(x,\partial \Omega))^p dx \leq C \int_{\Omega} |\nabla u|^p dx \] is the integrated counterpart of the pointwise inequality. The authors show that a domain \(\Omega\) admits the pointwise \(p\)-Hardy-Littlewood inequality iff \(\Omega' = \mathbb{R}^n \setminus \Omega\) is uniformly \(p\)-fat. The latter condition means that \(\mathrm{cap}_p(\Omega' \cap \overline{B}(x,r), B(x, 2r)) \geq c_o{\mathrm{cap}}_p( \overline{B}(x,r), B(x, 2r))\) for all \(x \in \partial \Omega\) and all \(r>0\) for some fixed \(c_o > 0\). By a result of \textit{J. L. Lewis} [Trans. Am. Math. Soc. 308, No. 1, 177--196 (1988; Zbl 0668.31002)] uniform \(p\)-fatness implies uniform \(q\)-fatness for some \(q\), \(1 < q <p\) and thus the domains having the pointwise \(p\)-Hardy-Littlewood inequality enjoy the same property. As a corollary it is shown that the pointwise \(p\)-Hardy-Littlewood inequality yields the classical \(p\)-Hardy-Littlewood inequality. Most of the work in the paper is done in metric measure spaces. Pointwise \(p\)-Hardy-Littlewood inequalities are also linked to certain Hausdorff content density conditions.