Kniefacz Philipp, Schuster Franz E
Vienna University of Technology, Vienna, Austria.
J Geom Anal. 2021;31(7):7436-7454. doi: 10.1007/s12220-020-00544-6. Epub 2020 Oct 24.
A family of sharp Sobolev inequalities is established by averaging the length of -dimensional projections of the gradient of a function. Moreover, it is shown that each of these new inequalities directly implies the classical Sobolev inequality of Aubin and Talenti and that the strongest member of this family is the only affine invariant one among them-the affine Sobolev inequality of Lutwak, Yang, and Zhang. When , the entire family of new Sobolev inequalities is extended to functions of bounded variation to also allow for a complete classification of all extremal functions in this case.
通过对函数梯度的(n)维投影长度进行平均，建立了一族尖锐的索伯列夫不等式。此外，还表明这些新不等式中的每一个都直接蕴含了奥宾和塔伦蒂的经典索伯列夫不等式，并且该族中最强的成员是其中唯一的仿射不变不等式——卢特瓦克、杨和张的仿射索伯列夫不等式。当(n = 1)时，新的索伯列夫不等式族被扩展到有界变差函数，从而也能对这种情况下的所有极值函数进行完全分类。
