Abstract:We prove a self-improvement property of a capacity density condition for a nonlocal Hajłasz gradient in complete geodesic spaces with a doubling measure. The proof relates the capacity density condition with boundary Poincaré 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 Hajłasz capacity density condition in terms of a strict up… Show more
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