On Sharp Higher Order Sobolev Embeddings

Abstract

We present an elementary unified and self-contained proof of sharp Sobolev embedding theorems. We introduce a new function space and use it to improve the limiting Sobolev embedding theorem due to Brézis and Wainger.

Some notes on embedding for anisotropic Sobolev spaces

“…Indeed, there are many mathematicians, e.g., Bastero, Milman, Ruiz, Martín, Pustylnik, who have researched this kind of question and found many meaningful and important results. The first result we get, Theorem 3.5, can be regarded as an extension of Theorem 1.2 and Corollary 1.3 in [11]. The second result is Theorem 3.7, which can be regarded as a generalization of Theorem 3 in [6] in the background of the rearrangement invariant spaces.…”

“…Hence, by Theorem 3.4, we get that if α X > r/n, α X < 1 (equivalent to the Pcondition in [11]), then…”

“…By Lemma 2.6 and Lemma 2.7 in [11], we know that if α X > r/n (equivalent to the Q(r)-condition there), then for all measurable functions f with f * * (∞) = 0,…”

Symmetrization inequalities and Sobolev embeddings

“…Following [23] and [26] we shall now construct the range spaces for our generalized Sobolev embedding theorem. Suppose that X and Y are r.i. spaces, and let s ∈ R. We define…”

“…In [2], [23] it is shown that (1.4) implies the following improvement to the Hansson-Brézis-Wainger-Maz ya 2 limiting version of the Sobolev embedding theorem:…”