Sharp Sobolev inequalities via projection averages

2019/11/29 by Kniefacz, Philipp, Schuster, Franz E.
#46E30 #46E35 #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1911.13075

Abstract

A family of sharp L^p Sobolev inequalities is established by averaging the length of i-dimensional projections of the gradient of a function. Moreover, it is shown that each of these new inequalities directly implies the classical L^p Sobolev inequality of Aubin and Talenti and that the strongest member of this family is the only affine invariant one among them -- the affine L^p Sobolev inequality of Lutwak, Yang, and Zhang. When p = 1, 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.

Related

• Sharp affine weighted L^p Sobolev type inequalities
• Sobolev improving for averages over curves in R⁴
• A sharp subelliptic Sobolev embedding theorem with weights
• Gradient stability for the Sobolev inequality: the case p≥ 2
• Sharp pointwise estimates for functions in the Sobolev spaces Hs(Rn)