Using the optimal mass transport method and a suitable quasi-conformal mapping, we study the sharp weighted isoperimetric, Sobolev, Gagliardo–Nirenberg and Caffarelli–Kohn–Nirenberg inequalities. The class of weight functions under consideration includes all nonnegative homogeneous weights satisfying a concavity condition that is equivalent to a usual curvature-dimension bound and the nonnegativity of a Bakry–Émery Ricci tensor. Though our densities are not radial in general, the optimizers are radially symmetric.

中文翻译：

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夏普加权等距和Caffarelli–Kohn–Nirenberg不等式

使用最佳质量传输方法和合适的准保形映射，我们研究了加权加权等参线，Sobolev，Gagliardo-Nirenberg和Caffarelli-Kohn-Nirenberg不等式。所考虑的权重函数类别包括满足凹面条件的所有非负均质权重，该凹度条件等效于通常的曲率维边界和Bakry-ÉmeryRicci张量的非负性。尽管我们的密度通常不是径向的，但优化器是径向对称的。