We investigate the validity, as well as the failure, of Sobolev-type inequalities on Cartan-Hadamard manifolds under suitable bounds on the sectional and the Ricci curvatures. We prove that if the sectional curvatures are bounded from above by a negative power of the distance from a fixed pole (times a negative constant), then all the L^p inequalities that interpolate between Poincaré and Sobolev hold for radial functions provided the power lies in the interval (− 2,0). The Poincaré inequality was established by H.P. McKean under a constant negative bound from above on the sectional curvatures. If the power is equal to the critical value − 2 we show that p must necessarily be bounded away from 2. Upon assuming that the Ricci curvature vanishes at infinity, the nonradial version of such inequalities turns out to fail, except in the Sobolev case. Finally, we discuss applications of the here established Sobolev-type inequalities to optimal smoothing effects for radial porous medium equations.

我们研究了在截面和Ricci曲率的适当范围内，在Cartan-Hadamard流形上Sobolev型不等式的有效性和失败性。我们证明，如果截面曲率由一个固定极点的距离的负幂（乘以一个负常数）从上方界定，则在Poincaré和Sobolev之间插值的所有L ^p不等式都适用于径向函数，只要该幂位于间隔（− 2,0）。庞加莱不等式是由HP McKean在截面曲率上从上方恒定的负边界处建立的。如果功率等于临界值− 2，则表明p必须将其限制在2之外。假设Ricci曲率在无穷大处消失，则这种不等式的非径向形式最终会失败，除非在Sobolev情况下。最后，我们讨论了这里建立的Sobolev型不等式在径向多孔介质方程的最佳平滑效果上的应用。