Sobolev embedding proof without Gagliardo–Nirenberg–Sobolev inequality or Morrey's inequality

Question

Hello Everybody,

Is there a proof of Sobolev embedding theorem without using the GNS or Morrey inequalities? If so, can you provide me with some references?

Background: I happened to attened a talk on Computational PDEs and Sobolev spaces. The speaker made a reference to a proof by S. L. Sobolev using polynomials. But, unfortunately he does not remember any references.

Thank you in advance.

Answer 1

There is an English translation of Sobolev's book containing his original proof:

Sobolev, S. L. Some applications of functional analysis in mathematical physics. Translated from the third Russian edition by Harold H. McFaden. With comments by V. P. Palamodov. Translations of Mathematical Monographs, 90. American Mathematical Society, Providence, RI, 1991.

Answer 2

I have no idea what Sobolev's proof using polynomials is, but I can suggest the following two proofs:

a) If all you want are the embedding theorms of a Sobolev space into an $L^p$ or $L^\infty$ space and do not need the sharp constant, then by far the easiest proof is using the original Gagliardo-Nirenberg inequality $$ \|f(x)\| _ \infty \le c\left(\Pi_{i=1}^n \|\partial_if\|_1\right)^{1/n} \le c\|\partial f\|_n $$ which can be proved using the fundamental theorem of calculus and the Fubini theorem.

Any other Sobolev inequality can be derived from this using the power rule for differentiation and the H\"older inequality.

This proof does use the simplest and original case of the Gagliardo-Nirenberg inequality, but the isoperimetric inequality plays no role at all.

b) A beautiful proof of the sharp first order Sobolev inequality using the Brenier map from mass transportation was given by Cordero-Erausquin, Nazaret, and Villani. The isoperimetric inequality is never used explicitly, but mass transportation is in a more general and powerful tool, one that implies the isoperimetric inequality.

Answer 3

Sobolev's original proof is different from the two approaches described by Deane. He uses certain integral formulae. You can read about this approach in the classic monograph

C. Morrey: Multiple Integrals in the Calculus of Variations, Springer