Riemannian manifold, Sobolev space

Question

Let $(S, \sigma)$ be a Riemannian manifold, $dim \ S \leq 3$. I read in a paper:

Then $L^6(S, \sigma)$ is embedded in $H_1(S, \sigma)$.

My questions:

• What is the definition of $H_1$? I know that $H^1:= W^{1,2}$. Is it just a convention?
• Why do we have this embedding? I know about the Sobolev embedding but this would give requirements, when $H^m \subset L^p$.

Thanks in advance for your help!

Answer 1

Typically, $H^1(S,\sigma)$ is chosen to refer to the completion in the $W^{1,2}(S,\sigma)$-norm of the smooth functions on $(S,\sigma)$. The Sobolev space $W^{1,2}(S,\sigma)$ is very often defined by means of weak derivatives. Under assumptions on the regularity of the domain, one can show that $H^1(S,\sigma) = W^{1,2}(S,\sigma)$.

Concerning your second question, it is related to the Sobolev embedding which holds for manifolds like this wikipedia page says

The Sobolev embedding theorem holds for Sobolev spaces W k,p(M) on other suitable domains M. In particular (Aubin 1982, Chapter 2; Aubin 1976), [...].

An other reference on Sobolev Space on manifolds is Sobolev Spaces on Riemannian Manifolds by Emmanuel Hebey where you will find the embedding explicitly.