论文标题
超球插值不平等的抛物线方法
Parabolic methods for ultraspherical interpolation inequalities
论文作者
论文摘要
CarréduChamp方法是一种强大的技术，用于证明在有歧管上有非平凡度量的情况下，具有明确常数的插值不平等。该方法适用于某些经典的Gagliardo-Nirenberg-Sobolev在球体上具有最佳常数。可以使用多孔培养基或快速扩散类型的非线性抛物线流覆盖靠近关键Sobolev指数的非常非线性的机制。考虑力量权重是Caffarelli-Kohn-Nirenberg不平等的对称性破坏问题的自然问题，但是丢失了对计算的完全合理性的规律性估计。我们通过将重量引起的奇异性正规化为基于非线性流的完整抛物线证明的第一个例子。我们的结果是在基于超球操作员建立的扩散的简化框架中建立的，该框架以简单的对称属性将问题减少到在球体上的功能。
The carré du champ method is a powerful technique for proving interpolation inequalities with explicit constants in presence of a non-trivial metric on a manifold. The method applies to some classical Gagliardo-Nirenberg-Sobolev inequalities on the sphere, with optimal constants. Very nonlinear regimes close to the critical Sobolev exponent can be covered using nonlinear parabolic flows of porous medium or fast diffusion type. Considering power law weights is a natural question in relation with symmetry breaking issues for Caffarelli-Kohn-Nirenberg inequalities, but regularity estimates for a complete justification of the computation are missing. We provide the first example of a complete parabolic proof based on a nonlinear flow by regularizing the singularity induced by the weight. Our result is established in the simplified framework of a diffusion built on the ultraspherical operator, which amounts to reduce the problem to functions on the sphere with simple symmetry properties.
