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A Hardy–Sobolev inequality for the twisted Laplacian

Part of: Partial differential equations

Published online by Cambridge University Press:  11 January 2017

Adimurthi ,
P. K. Ratnakumar  and
Vijay Kumar Sohani
Adimurthi
Affiliation:
Tata Institute of Fundamental Research, Centre for Applicable Mathematics, Sharada Nagar, Chikkabommasandra, Bangalore 560065, India (adiadimurthi@gmail.com; aditi@math.tifrbng.res.in)
P. K. Ratnakumar
Affiliation:
Harish-Chandra Research Institute, Chhatnag Road, Jhunsi, Allahabad 211019, India (ratnapk@hri.res.in) and Homi Bhabha National Institute, Training School Complex, Anushakti Nagar, Mumbai 400085, India
Vijay Kumar Sohani
Affiliation:
Department of Mathematics, Indian Institute of Science, Bangalore 560012, India (vijaykumarsohani@gmail.com)
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We prove a strong optimal Hardy–Sobolev inequality for the twisted Laplacian on ℂn . The twisted Laplacian is the magnetic Laplacian for a system of n particles in the plane, corresponding to the constant magnetic field. The inequality we obtain is strong optimal in the sense that the weight cannot be improved. We also show that our result extends to a one-parameter family of weighted Sobolev spaces.

Keywords

Hardy–Sobolev inequalityfundamental solutiontwisted Laplacianspecial Hermite expansion

MSC classification

Secondary: 35A23: Inequalities involving derivatives and differential and integral operators, inequalities for integrals 35A08: Fundamental solutions 33C50: Orthogonal polynomials and functions in several variables expressible in terms of special functions in one variable
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 147 , Issue 1 , February 2017 , pp. 1 - 23
Copyright
Copyright © Royal Society of Edinburgh 2017