An Introduction to the Heisenberg Group and the Sub-Riemannian Isoperimetric Problem pp 191-202 | Cite as

# Best Constants for Other Geometric Inequalities on the Heisenberg Group

Chapter

## Abstract

As the point of departure for this final chapter, we return to the equivalence of the isoperimetric inequality with the geometric ( In this chapter we discuss sharp constants for other analytic/geometric inequalities in the Heisenberg group and the Grushin plane. These include the

*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$$
\left\| u \right\|_{4p/(4 - p)} \leqslant Cp(\mathbb{H})\left\| {\nabla _0 u} \right\|_p , u \in C_0^\infty (\mathbb{H}).
$$

(9.1)

*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*}.## Keywords

Heisenberg Group Sobolev Inequality Isoperimetric Inequality Hardy Inequality Carnot Group
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© Birkhäuser Verlag AG 2007