Sharp Sobolev Inequalities on the Sphere and the Moser--Trudinger Inequality

where de denotes normalized surface measure, V is the conformal gradient and q = (2n)/(n 2). A modern folklore theorem is that by taking the infinitedimensional limit of this inequality, one obtains the Gross logarithmic Sobolev inequality for Gaussian measure, which determines Nelson's hypercontractive estimates for the Hermite semigroup (see 8). One observes using conformal invariance that the above inequality is equivalent to the sharp Sobolev inequality on Rn for which boundedness and extremal functions can be easily calculated using dilation invariance and geometric symmetrization. The roots here go back to Hardy and Littlewood. The advantage of casting the problem on the sphere is that the role of the constants is evident, and one is led immediately to the conjecture that this inequality should hold whenever possible (for example, 2 < q < 0o if n = 2). This is in fact true and will be demonstrated in Section 2. A clear question at this point is What is the situation in dimension 2? Two important arguments (25, 26, 27) dealt with this issue, both motivated by geometric variational problems. Because q goes to infinity for dimension 2, the appropriate function space is the exponential class. Responding in part