We prove a sharp quantitative version of the p-Sobolev inequality for any 1<p<n, with a control on the strongest possible distance from the class of optimal functions. Surprisingly, the sharp exponent is constant for p≤2, while it depends on p for p>2. 

　　Publication: 

　　Duke Mathematical Journal, 2022, Volume 171, Issue 12, pp: 2407-2459. 

　　Author: 

　　Alessio Figalli 

　　ETH Zu ？rich, Department of Mathematics, Ra ？mistrasse 101, 8092, Zu ？rich, Switzerland 

　　Yi Ru-Ya Zhang 

　　ETH Zu ？rich, Department of Mathematics, Ra ？mistrasse 101, 8092, Zu ？rich, Switzerland 

　　Academy of Mathematics and Systems Science, The Chinese Academy of Sciences, Beijing 100190, China 

　　Email: yzhang@amss.ac.cn