On the Best Constant in the Moser-Onofri-Aubin Inequality

Let S(2) be the 2-dimensional unit sphere and let J(alpha) denote the nonlinear functional on the Sobolev space H(1)(S(2)) defined by J(alpha)(u) = alpha/16 pi integral(S2) |del u|(2)d mu(0) + 1/4 pi integral(S2)ud mu(0)-ln integral(S2)e(u) d mu(0)/4 pi, where d mu(0) = sin theta d theta boolean AND d phi. Onofri had established that Ja is non-negative on H(1)(S(2)) provided alpha >= 1. In this note, we show that if J(alpha) is restricted to those u is an element of H(1)(S(2)) that satisfies the Aubin condition: integral(eu)(S2)x(j)d mu(0) = 0 for all 1 <= j <= 3, then the same inequality continues to hold (i.e., J(alpha)(u) = 0) whenever alpha >= 2/3 - epsilon(0) for some epsilon(0) > 0. The question of Chang-Yang on whether this remains true for all alpha >= 1/2 remains open.