Timeline for Error rate implying regularity
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6 events
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| Jul 24, 2020 at 8:31 | comment | added | user69642 | 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. | |
| Jul 23, 2020 at 19:59 | comment | added | Scott Armstrong | 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. | |
| Jul 23, 2020 at 19:58 | comment | added | Scott Armstrong | 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. | |
| Jul 23, 2020 at 19:58 | comment | added | Scott Armstrong | 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}). | |
| Jul 22, 2020 at 11:11 | history | edited | YCor | CC BY-SA 4.0 |
removed capitals from title
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| Jul 22, 2020 at 11:09 | history | asked | user69642 | CC BY-SA 4.0 |