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Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality. (English) Zbl 0826.58042
The principal results contained here include:
(1) extension to higher dimensions of the sharp Moser-Trudinger inequality due to Onofri for the 2-sphere,
(2) conformal invariance of the higher-dimensional inequality and characterization of extremal functions in terms of conformal factors,
(3) equivalence with sharp entropy estimates for a logarithmic fractional integral corresponding to the Green’s function for higher-order elliptic operators,
(4) application to the geometric variational problem of determining extremal values for the functional determinant of the conformal Laplacian on the 4-sphere under a conformal deformation with fixed volume, and
(5) extension of the classical Sobolev inequalities on the sphere to a one-parameter family of higher-order Sobolev inequalities.

MSC:
58J60 Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.)
58J05 Elliptic equations on manifolds, general theory
58J52 Determinants and determinant bundles, analytic torsion
35B65 Smoothness and regularity of solutions to PDEs
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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