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Optimal Hardy Inequalities for Schrödinger Operators Based on Symmetric Stable Processes

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  • Yusuke Miura

Abstract

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

Suggested Citation

  • Yusuke Miura, 2023. "Optimal Hardy Inequalities for Schrödinger Operators Based on Symmetric Stable Processes," Journal of Theoretical Probability, Springer, vol. 36(1), pages 134-166, March.
    • Handle: RePEc:spr:jotpro:v:36:y:2023:i:1:d:10.1007_s10959-022-01164-2
    DOI: 10.1007/s10959-022-01164-2
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