The Hajłasz Capacity Density Condition is Self-improving
Canto, J., & Vähäkangas, A. V. (2022). The Hajłasz Capacity Density Condition is Self-improving. Journal of Geometric Analysis, 32(11), Article 276. https://doi.org/10.1007/s12220-022-00979-z
Published inJournal of Geometric Analysis
DisciplineMatematiikkaAnalyysin ja dynamiikan tutkimuksen huippuyksikköMathematicsAnalysis and Dynamics Research (Centre of Excellence)
© The Author(s) 2022
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 upper bound on the upper Assouad codimension of the underlying set, which shows the self-improvement property of the Hajłasz capacity density condition.
PublisherSpringer Science and Business Media LLC
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Additional information about fundingOpen Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature. J.C. is supported by supported by the Ministerio de Economía y Competitividad (Spain) through Grant Nos. PID2020-113156GB-I00 and SEV-2017-0718, and by Basque Government through Grant Nos. IT-641-13 and BERC 2018-2021 and “Ayuda para la formación de personal investigador no doctor".
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