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How does one figure out/prove the rate of convergence (in some norm) of mollifiers given a function bounded in some other norm (say Sobolev space, Besov space)? Also, is there a dimensional analysis heuristic which will predict what the rate will be?

For example, it is true that

$$||u - u^{(\epsilon)}||{L^3} u^{(\epsilon)}||_{L^3} \leq C \epsilon^\alpha ||u||{B_3^{\alpha,\infty}} |u||_{B_3^{\alpha,\infty}}$$

where the norm on the right hand side is a Besov space norm. (This fact is used in Constantin, E and Titi's paper on Onsager's Conjecture for Solutions to Euler's Equation).

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# Rate of convergence of smooth mollifiers

How does one figure out/prove the rate of convergence (in some norm) of mollifiers given a function bounded in some other norm (say Sobolev space, Besov space)? Also, is there a dimensional analysis heuristic which will predict what the rate will be?

For example, it is true that

$||u - u^{(\epsilon)}||{L^3} \leq C \epsilon^\alpha ||u||{B_3^{\alpha,\infty}}$

where the norm on the right hand side is a Besov space norm. (This fact is used in Constantin, E and Titi's paper on Onsager's Conjecture for Solutions to Euler's Equation).