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The carr\'e du champ method is a powerful technique for proving interpolationinequalities with explicit constants in presence of a non-trivial metric on amanifold. The method applies to some classical Gagliardo-Nirenberg-Sobolevinequalities on the sphere, with optimal constants. Very nonlinear regimesclose to the critical Sobolev exponent can be covered using nonlinear parabolicflows of porous medium or fast diffusion type. Considering power law weights isa natural question in relation with symmetry breaking issues forCaffarelli-Kohn-Nirenberg inequalities, but regularity estimates for a completejustification of the computation are missing. We provide the first example of acomplete parabolic proof based on a nonlinear flow by regularizing thesingularity induced by the weight. Our result is established in the simplifiedframework of a diffusion built on the ultraspherical operator, which amounts toreduce the problem to functions on the sphere with simple symmetry properties.

Parabolic methods for ultraspherical interpolation inequalities