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On the Best Constant in the Moser-Onofri-Aubin Inequality
Let be the 2-dimensional unit sphere and let denote the
nonlinear functional on the Sobolev space defined by where denotes Lebesgue
measure on , normalized so that . Onofri had
established that is non-negative on provided . In this note, we show that if is restricted to those that satisfy the Aubin condition: \int_{S^2}e^u x_j
dw=0\quad\text{for all}1\leq j\leq 3, then the same inequality continues to
hold (i.e., ) whenever for
some . The question of Chang-Yang on whether this remains true
for all remains open.Comment: 8 pages. Updated versions - if any - can be downloaded at
http://www.birs.ca/~nassif
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