# Error rate implying regularity

My question is a bit general/vague.

It is well known that the regularity of certain functions can be measured through the rate of decay of certain error quantity based on an approximation procedure (see, e.g., the famous result of DeVore, Jawerth and Popov for Besov regularity in terms of the rate of decay for the error of the best n-term approximation).

I would like to know if there is a standard well-developed theory which allows to obtain regularity results thanks to rates of decay of approximation.

Are there any general references in this direction ?

Many thanks in advance.

• I am not sure what you mean exactly, can you elaborate? For instance, many regularity spaces are defined in these terms, e.g., Holder (C^\alpha) functions are simply those that can be approximated in L^\infty by constant functions in every ball of radius r up to an error of O(r^\alpha). If you relax L^\infty to L^2 (in an appropriate sense) you get the space H^\alpha. The space C^{k,\alpha} is the space of functions which can, in every ball, be approximated in L^\infty by a polynomial of degree k up to an error of O(r^{k+\alpha}). – Scott Armstrong Jul 23 '20 at 19:58
• Since these are the very definitions of the regularity spaces, many arguments in elliptic regularity work like this. Google "Campanato iterations" or "epsilon regularity lemma" for example. Read any paper with "partial regularity" in the title. Read basically any paper of Caffarelli. Read the recent papers on the regularity of the Monge-Ampere equation/ optimal transportation by De Philippis & Figalli or Otto & collaborators. – Scott Armstrong Jul 23 '20 at 19:58
• If you want to see the Schauder estimates proved like this, see the book of Han-Lin. If you want to see the Calderon-Zygmund estimates proved like this, see Lemma 7.2 of my book with Kuusi-Mourrat. This is all very elliptic/parabolic, but perhaps only because it is what I know. – Scott Armstrong Jul 23 '20 at 19:59
• Thanks for the comments! Indeed, several functional spaces are termed directly in this way but what I am looking for is more in the spirit of approximation spaces and more specifically in the systematic identification/characterization of such spaces as known functional spaces. The way I am approximating my function is not necessarily classical (not a Taylor expansion) and therefore it might be not completely clear what is the underlying functional space associated with such an error rate. – user69642 Jul 24 '20 at 8:31