(Université Claude Bernard Lyon 1)
On a second order Caffarelli-Kohn-Nirenberg inequality
Abstract: Using an appropriate change of variable, if the inequality
`(\int_{\mathbb R^d} |x|^{-bp} | u|^p dx)^{2/p} \le C_{a,b}\int_{\mathbb R^d}| x|^{-2a}|\Delta u|^2 dx`
is scale invariant, then it can be reduced to the standard Sobolev embedding theorem `H^{2} \to L^p` in the cylinder `\mathbb R\times \mathbb S^{d-1}`. I will discuss symmetry and positivity issues for optimizers of the inequality.