Fractional Hardy inequalities and capacity density

We prove that a pointwise fractional Hardy inequality implies a fractional Hardy inequality, defined via a Gagliardo-type seminorm. The proof consists of two main parts. The first one is to characterize the pointwise fractional Hardy inequality in terms of a fractional capacity density condition. The second part is to show the deep open-endedness or self-improvement property of the fractional capacity density, which we accomplish in the setting of a complete geodesic space equipped with a doubling measure. These results are new already in the standard Euclidean setting.