Sharp Sobolev Inequalities via Projection Averages
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Sharp Sobolev Inequalities via Projection Averages Philipp Kniefacz1 · Franz E. Schuster1 Received: 16 July 2020 / Accepted: 9 October 2020 © The Author(s) 2020
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. Keywords Sobolev inequalities · Isoperimetric inequalities · Affine invariant inequalities · Convex bodies Mathematics Subject Classification 46E35 · 26D15
1 Introduction The fruitful interplay between analysis and geometry is probably highlighted most prominently by the rich theory of Sobolev inequalities and, in particular, by its best known representative—the sharp L p Sobolev inequality in Rn . While the latter is often stated for functions from the Sobolev space W 1, p (Rn ) (consisting of L p functions with weak L p partial derivatives), a more natural setting for it is the larger homogeneous Sobolev space (see, e.g., [22, Chapter 11]) defined by ∗ W˙ 1, p (Rn ) := { f ∈ L p (Rn ) : ∇ f ∈ L p (Rn )},
B
Franz E. Schuster [email protected] Philipp Kniefacz [email protected]
1
Vienna University of Technology, Vienna, Austria
123
P. Kniefacz, F.E. Schuster
where 1 ≤ p < n and p ∗ = np/(n − p). The L p Sobolev inequality states that if f ∈ W˙ 1, p (Rn ), then Rn
1/ p ∇ f (x) p d x
≥ an, p f p∗ ,
(1.1)
where · denotes the standard Euclidean norm on Rn and we write f p for the usual L p norm of f in Rn . The exact value of the optimal constant an, p (see below) was first computed for p > 1 by Aubin [1] and, independently, by Talenti [35], who made critical use of the classical isoperimetric inequality to reduce (1.1) to a 1-dimensional problem. In the case p = 1, it was previously shown by Maz’ya [30] and Federer and Fleming [10] that the sharp L 1 Sobolev inequality is actually equivalent to the isoperimetric inequality. While the explicit knowledge of the optimal constant has proven beneficial in certain areas of mathematical physics, its importance is far outweighed by the classification of the extremal functions in (1.1). Apparently, it was known for some time that with the help of a rearrangement inequality of Brothers and Ziemer [3] all extremizers could be identified. However, the first explicit and selfcontained proof that equality holds in (1.1) for p > 1 if and only if there exist a, b > 0, and x0 ∈ Rn such that 1−n/ p f (x) = ± a + b(x − x0 ) p/( p−1)
(1.2)
was given by Cordero-Erausquin et al. [7] (and in a more general form). They also pointed out the disadvantage of consideri
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