Journal of the Mathematical Society of Japan

Superharmonic functions of Schrödinger operators and Hardy inequalities

Yusuke MIURA

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Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1554364814

Digital Object Identifier
doi:10.2969/jmsj/79597959

Mathematical Reviews number (MathSciNet)
MR3985615

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

Keywords
symmetric Markov process Dirichlet form superharmonic function excessive function Hardy inequality

Citation

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. https://projecteuclid.org/euclid.jmsj/1554364814


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