4
$\begingroup$

I have not managed to find a reference for the following fact:

$H^s(\mathbb T)$ is a Banach algebra for $s>1/2$.

In particular, I need reference for the following inequality: $$ \|uv\|_{H^s} \,\le\, c\,\|u\|_{H^s}\|v\|_{H^s}, $$ for all $u,v\in H^s(\mathbb T)$, whenever $s>1/2$, and $c$ is a positive constant not depending on $u,v$.

It is very often assumed to hold, as an obvious fact, and no reference is provided.

Any known reference or proof?

Improve this question
$\endgroup$
1

1 Answer 1

Reset to default
4
$\begingroup$

This follows from two facts:

  • $(H^s \cap L^\infty) (\mathbb{T})$ is a Banach algebra (see for example in the framework of fractional spaces $W^{s, p}$ ($W^{s, 2} = H^s$) Bourgain, Brezis, Mironescu, Lifting in Sobolev spaces, 2000),

  • if $s > \frac{1}{2}$, then $H^{s} (\mathbb{T}) \subset L^\infty (\mathbb{T})$: if $u \in H^s (\mathbb{T})$ and $t \in \mathbb{T}$, then $$\vert u (t) \vert \le \sum_{k \in \mathbb{Z}} \vert c_k (u)\vert \le \Big(\sum_{k \in \mathbb{Z}} \frac{1}{(k^2 + 1)^s}\Big)^\frac{1}{2}\Big(\sum_{k \in \mathbb{Z}} \lvert c_k (u) \vert^2 (k^2 + 1)^s\Big)^\frac{1}{2} = C \Vert u \Vert_{H^s},$$ where $c_k (u)$ is the $k$-th Fourier coefficient of the function $u$. In general $W^{s, p} (M)$ is continuously embedded into $L^\infty (M)$ if $sp > \dim M$ (see for example Adams, Sobolev spaces, Academic Press, 1975, theorem 7.57).

More generally $W^{s, p} (M)$ is a Banach algebra if $sp > \dim M$.

Improve this answer
$\endgroup$
4
  • $\begingroup$ Can you provide a reference for your last assertion? Does one have that $W^{s,p}(M)\hookrightarrow L^\infty(M)$ if $sp>{\rm dim } M$? $\endgroup$
    – Delio Mugnolo
    Commented Jun 7, 2014 at 13:31
  • $\begingroup$ Adams, Sobolev spaces, Academic Press, 1975, theorem 7.57 $\endgroup$
    – Jean Van Schaftingen
    Commented Jun 10, 2014 at 3:49
  • $\begingroup$ But 7.57 is an imbedding result for Sobolev spaces on domains of $\mathbb R^n$! How does this implies the claimed property? I have checked a bit in the literature: It seems that Thm. 3.5 in Sobolev spaces on Riemannian manifolds by E. Hebey settles my question. $\endgroup$
    – Delio Mugnolo
    Commented Jun 10, 2014 at 9:35
  • $\begingroup$ This holds for a compact manifold: by Adams and local charts, the result holds on a neighbourhood of any point, and the manifold can be covered by finitely many such neighbourhoods. This should also work for manifolds with some “bounded geometry”. $\endgroup$
    – Jean Van Schaftingen
    Commented Jun 10, 2014 at 10:22

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .