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).
