MHB How Does Sobolev Space Boundedness Relate to Different Norms in $R^n$?

[Danny2]

If $ r<s<t $ then for any $ ϵ>0 $there exists $ C>0 $ such that $ ∥f∥_{(s)}≤ϵ∥f∥_{(t)}+C∥f∥_{(r)} $for all $f∈H_t $

Can you please tell me how to start thinking of this problem? I really feel stuck and don't know where to start!

 

[Euge]

Welcome, Danny! (Wave)

Some information is missing in the problem statement. What is the domain of $f$?

 

[Danny2]

$f$ is a tempered distribution

 

[Euge]

What I mean is this: $f\in H^t(X)$ for some space $X$ -- what is $X$? It makes a difference, since, e.g., integration on the torus is different from integration on $\Bbb R^n$.

 

[Danny2]

$R^n$