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Role of the fundamental solution in Hardy—Sobolev-type inequalities

Published online by Cambridge University Press:  12 July 2007

Adimurthi
Affiliation:
TIFR Centre, Indian Institute of Science Campus, PO Box 1234, Bangalore 560 012, India (aditi@math.tifrbng.res.in)
Anusha Sekar
Affiliation:
Department of Mathematics, University of Washington, Padelford C-138, Seattle, WA 98195-4350, USA (sekar@math.washington.edu)

Abstract

Let n ≥ 3, Ω ⊂ Rn be a domain with 0 ∈ Ω, then, for all the Hardy–Sobolev inequality says that and equality holds if and only if u = 0 and ((n − 2)/2)2 is the best constant which is never achieved. In view of this, there is scope for improving this inequality further. In this paper we have investigated this problem by using the fundamental solutions and have obtained the optimal estimates. Furthermore, we have shown that this technique is used to obtain the Hardy–Sobolev type inequalities on manifolds and also on the Heisenberg group.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 2006

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