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Erschienen in:
2007 | OriginalPaper | Buchkapitel
Best Constants for Other Geometric Inequalities on the Heisenberg Group
Erschienen in: An Introduction to the Heisenberg Group and the Sub-Riemannian Isoperimetric Problem
Verlag: Birkhäuser Basel
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As the point of departure for this final chapter, we return to the equivalence of the isoperimetric inequality with the geometric (
L
1
-) Sobolev inequality. As shown in Section 7.1, the best constant for the isoperimetric inequality agrees with the best constant for the geometric (
L
1
-) Sobolev inequality. Recall that in the context of the Heisenberg group, the
L
p
-Sobolev inequalities take the form
9.1
$$ \left\| u \right\|_{4p/(4 - p)} \leqslant Cp(\mathbb{H})\left\| {\nabla _0 u} \right\|_p , u \in C_0^\infty (\mathbb{H}). $$
In this chapter we discuss sharp constants for other analytic/geometric inequalities in the Heisenberg group and the Grushin plane. These include the
L
p
-Sobolev inequality (9.1) in the case
p
= 2, the Trudinger inequality (9.14), which serves as a natural substitute for (9.1) in the limiting case
p
= 4, and the Hardy inequality (9.24), a weighted inequality of Sobolev type on the domain ℍ \ {
o
}.