On the Grushin operator and hyperbolic symmetry (Q2701606)
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scientific article
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | On the Grushin operator and hyperbolic symmetry |
scientific article |
Statements
On the Grushin operator and hyperbolic symmetry (English)
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19 February 2001
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Grushin operator
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differential operator with mixed homogeneity
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geometric symmetry
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Lie groups
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\(L^2\) Sobolev inequality
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hyperbolic symmetry
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conformal geometry
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The Grushin operator featured in the title of the paper is the differential operator with mixed homogeneity NEWLINE\[NEWLINE\Delta_G\equiv {\partial^2\over\partial t^2}+ 4t^2{\partial^2\over\partial x^2}.NEWLINE\]NEWLINE The purpose of the paper is to demonstrate the possible complexity of the geometric symmetry associated with operators defined on Lie groups. Use is made of the underlying \(\text{SL}(2,R)\) symmetry belonging to \(\Delta_G\) to compute the sharp constant for the associated \(L^2\) Sobolev inequality with the help of hyperbolic symmetry and conformal geometry.
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