Optimal Hardy Inequalities for Schrödinger Operators Based on Symmetric Stable Processes

Yusuke Miura

Journal of Theoretical Probability, 2023, vol. 36, issue 1, 134-166

Abstract: 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.

Keywords: Symmetric stable process; Dirichlet form; Hardy inequality; Girsanov transformation; 60J45; 60J75; 31C05 (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1007/s10959-022-01164-2

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