Extremals for Sobolev and Moser inequalities in hyperbolic space (Q640723)

The authors first recall the Sobolev type inequalities in the hyperbolic plane \(\mathbb{H}^2\) for Grushin operators and the Hardy-Sobolev-Maz'ja inequality for the Hardy-Sobolev operator in \(\mathbb{R}^k_y\times \mathbb{R}^h_z \). When the functions are radial in \(y\), the Hardy-Sobolev-Maz'ja inequality transforms into the Sobolev type inequality in the hyperbolic space \(\mathbb{H}^n\). The extremals satisfy an elliptic equation in the space \(H^1(\mathbb{H}^n)\). They then recall some properties (hyperbolic symmetry, existence/nonexistence, uniqueness) of the solutions of the equation. In Section 3 they present a sharp Moser inequality in \(\mathbb{H}^2\). Finally, they present some results for the Moser functional on a conformal disc \(M_g\). The results include upper-boundedness of the functional, a Moser-Trudinger inequality in the critical case \(\alpha =4\pi\) and some conditions ensuring the achievement of maximum of the Moser functional via Lions's concentration-compactness lemma.