Journal of the Mathematical Society of Japan

Superharmonic functions of Schrödinger operators and Hardy inequalities

Yusuke MIURA

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Given a Dirichlet form with generator $\mathcal{L}$ and a measure $\mu$, we consider superharmonic functions of the Schrödinger operator $\mathcal{L} + \mu$. We probabilistically prove that the existence of superharmonic functions gives rise to the Hardy inequality. More precisely, the $L^2$-Hardy inequality is derived from Itô's formula applied to the superharmonic function.

Article information

J. Math. Soc. Japan, Volume 71, Number 3 (2019), 689-708.

Received: 14 January 2018
First available in Project Euclid: 4 April 2019

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Mathematical Reviews number (MathSciNet)

Primary: 31C25: Dirichlet spaces
Secondary: 31C05: Harmonic, subharmonic, superharmonic functions 60J25: Continuous-time Markov processes on general state spaces

symmetric Markov process Dirichlet form superharmonic function excessive function Hardy inequality


MIURA, Yusuke. Superharmonic functions of Schrödinger operators and Hardy inequalities. J. Math. Soc. Japan 71 (2019), no. 3, 689--708. doi:10.2969/jmsj/79597959.

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