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Optimal Hardy Inequalities for Schrödinger Operators Based on Symmetric Stable Processes

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Abstract

Assume that \(\mathcal {L}^{\mu } :=\) \(-(-\Delta )^{\alpha /2}\) \(+ \mu \) is subcritical, where \((-\Delta )^{\alpha /2}\) is the fractional Laplacian and \(\mu \) is a positive smooth measure on \(\mathbb {R}^d\) in the Green-tight Kato class. In this paper, we probabilistically construct a Hardy-weight for a quadratic form \(\mathcal {E}^{\mu }\) associated with \(\mathcal {L}^{\mu }\) which is optimal in a certain sense. As a side product, we characterize the criticality and subcriticality of \(\mathcal {E}^{\mu }\) through Girsanov transformations.

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Data sharing is not applicable to this article as no new data were created or analyzed in this study.

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Acknowledgements

The author is grateful to the anonymous referees for their many helpful suggestions on this paper.

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  1. Hie-Miyawaki 26-13, Tsuruoka, 997-0824, Japan

    Yusuke Miura

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  1. Yusuke Miura
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Miura, Y. Optimal Hardy Inequalities for Schrödinger Operators Based on Symmetric Stable Processes. J Theor Probab 36, 134–166 (2023). https://doi.org/10.1007/s10959-022-01164-2

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  • DOI: https://doi.org/10.1007/s10959-022-01164-2

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