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

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The discussion centers on the relationship between Sobolev space boundedness and different norms in R^n, specifically addressing the inequality relating norms for functions in Sobolev spaces. It highlights that for given parameters r, s, and t, there exists a constant C that connects the norms of functions across these spaces. The importance of the domain of the function f is emphasized, noting that it affects the analysis, particularly distinguishing between integration on different domains like the torus versus R^n. The conversation indicates that understanding the context of the problem is crucial for tackling the inequality effectively. Overall, the discussion underscores the need for clarity on the function's domain to proceed with the problem.
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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!
 
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Welcome, Danny! (Wave)

Some information is missing in the problem statement. What is the domain of $f$?
 
$f$ is a tempered distribution
 
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 $\Bbb R^n$.
 
$R^n$
 

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