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An extension problem, trace Hardy and Hardy's inequalities for the Ornstein–Uhlenbeck operator

Pritam Ganguly, Ramesh Manna and Sundaram Thangavelu

Vol. 16 (2023), No. 5, 1205–1244
Abstract

We study an extension problem for the Ornstein–Uhlenbeck operator L = Δ + 2x + n, and we obtain various characterisations of the solution of the same. We use a particular solution of that extension problem to prove a trace Hardy inequality for L from which Hardy’s inequality for fractional powers of L is obtained. We also prove an isometry property of the solution operator associated to the extension problem. Moreover, new Lp Lq estimates are obtained for the fractional powers of the Hermite operator.

Keywords
extension problem, trace Hardy inequality, Hardy's inequality, Ornstein–Uhlenbeck operator
Mathematical Subject Classification
Primary: 26D10, 35J15
Secondary: 26A33, 33C45, 35A23, 43A80
Milestones
Received: 17 December 2020
Revised: 19 September 2021
Accepted: 28 October 2021
Published: 12 August 2023
Authors
Pritam Ganguly
Institut für Mathematik
Universität Paderborn
Paderborn
Germany
Department of Mathematics
Indian Institute of Science
Bangalore
India
Ramesh Manna
School of Mathematical Sciences
National Institute of Science Education and Research
Bhubaneswar
an OCC of Homi Bhabha National Institute
Jatni, 752050
India
Sundaram Thangavelu
Department of Mathematics
Indian Institute of Science
Bangalore
India

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