Sharp inequalities and geometry manifolds

Sharp constants for function-space inequalities over a manifold encode information about the geometric structure of the manifold. An important example is the Moser-Trudinger inequality where limiting Sobolev behavior for critical exponents provides significant understanding of geometric analysis for conformal deformation on a Riemannian manifold[5,6]. From the overall perspective of the conformal group acting on the classical spaces, it is natural to consider the extension of these methods and questions in the context of SL(2, R), the Heisenberg group, and other Lie groups. Among the principal tools used in this analysis are the linear and multilinear operators mapping L-p(M) to L-q(M) defined by the Stein-Weiss integral kernels which extend the Hardy-Littlewood-Sobolev fractional integrals K(x, y) = \x\(-alpha) \x - y\(-lambda) \y\(-beta), (1) conformal geometry, and the notion of equimeasurable geodesic radial decreasing rearrangement. To illustrate these ideas, four model problems will be examined here: (1) logarithmic Sobolev inequality and the uncertainty principle, (2) SL(2, R) and atrial symmetry in fluid dynamics, (3) Stein-Weiss integrals on the Heisenberg group, and (4) Morpurgo's work on zeta functions and trace inequalities of conformally invariant operators.