The Hajłasz capacity density condition is self-improving
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We prove a self-improvement property of a capacity density condition for a nonlocal Haj lasz gradient in complete geodesic spaces. The proof relates the capacity density condition with boundary Poincar´e inequalities, adapts Keith–Zhong techniques for establishing local Hardy inequalities and applies Koskela–Zhong arguments for proving self-improvement properties of local Hardy inequalities. This leads to a characterization of the Haj lasz capacity density condition in terms of a strict upper bound on the upper Assouad codimension of the underlying set, which shows the self-improvement property of the Haj lasz capacity density condition.