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A Hardy–Sobolev inequality for the twisted Laplacian

Published online by Cambridge University Press:  11 January 2017

Tata Institute of Fundamental Research, Centre for Applicable Mathematics, Sharada Nagar, Chikkabommasandra, Bangalore 560065, India (;
P. K. Ratnakumar
Harish-Chandra Research Institute, Chhatnag Road, Jhunsi, Allahabad 211019, India ( and Homi Bhabha National Institute, Training School Complex, Anushakti Nagar, Mumbai 400085, India
Vijay Kumar Sohani
Department of Mathematics, Indian Institute of Science, Bangalore 560012, India (


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.

Research Article
Copyright © Royal Society of Edinburgh 2017 

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