Best Constants for Other Geometric Inequalities on the Heisenberg Group

Abstract

As the point of departure for this final chapter, we return to the equivalence of the isoperimetric inequality with the geometric (L1-) Sobolev inequality. As shown in Section 7.1, the best constant for the isoperimetric inequality agrees with the best constant for the geometric (L1-) Sobolev inequality. Recall that in the context of the Heisenberg group, the Lp-Sobolev inequalities take the form
$$ \left\| u \right\|_{4p/(4 - p)} \leqslant Cp(\mathbb{H})\left\| {\nabla _0 u} \right\|_p , u \in C_0^\infty (\mathbb{H}). $$
(9.1)
In this chapter we discuss sharp constants for other analytic/geometric inequalities in the Heisenberg group and the Grushin plane. These include the Lp-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}.

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© Birkhäuser Verlag AG 2007

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