Timeline for Calderón's complex interpolation: what is the corresponding classical theorem?
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| Jun 4 at 15:02 | history | edited | YCor | CC BY-SA 4.0 |
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| S Jun 4 at 12:20 | history | suggested | The Amplitwist | CC BY-SA 4.0 |
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| Apr 13, 2017 at 12:57 | history | edited | CommunityBot |
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| Jan 2, 2012 at 9:22 | comment | added | TaQ | On pages 326−328, Williams considers C2 in connection with duality, specifically referring to pages 114, 116, 130, 148−156 in the article A. P. Calderón: "Intermediate spaces and interpolation, the complex method", Studia Math., 24, 1964, 113−190. Have you seen this? If the answer to your question can not be found there, maybe in this case Calderón had some original creativity of his own, and there is no corresponding "classical interpolation theorem". | |
| Jan 2, 2012 at 9:22 | comment | added | TaQ | @ Mark Kim: In other words, if I understand correctly what you wrote, you are asking for some historical motivation for creating Calderón's second interpolation method? For short, calling it C2, hence, the question is about the historical development of thoughts associated with C2. Shouldn't there then be a tag like "history", "reference request", or so? Accidentally, I just looked at the article Vernon Williams: "Generalized interpolation spaces", Trans. AMS, 156, 1971, 309−334. | |
| Jan 1, 2012 at 15:47 | comment | added | Mark Kim-Mulgrew | My question, then, concerns the motivation behind the second method. I do not know of any classical interpolation theorem that makes use of such restrictions on the derivative. Therefore, if the second method is a generalization of a "concrete theorem" (like Riesz-Thorin), then I don't know what it is. If it is not a generalization of a classical one, then I would like to know why anyone would consider the second method in the first place. In short, I am looking for an answer like Matthew Daws's comment, preferably with an added historical background. | |
| Jan 1, 2012 at 15:40 | comment | added | Mark Kim-Mulgrew | I understand that any old functor satisfying some conditions can be an interpolation method. Nevertheless, it appears to me that the "complex interpolation" functor and the "real interpolation" functor must have arisen as generalizations of Riesz-Thorin and Marcinkiewicz interpolation theorems, respectively. This, presumably, is why Hadamard three-lines lemma is built into the first complex interpolation method described above, and this would serve as a motivation behind the first method. That is, I "get" why one would consider the first method. | |
| Dec 31, 2011 at 23:28 | comment | added | TaQ | What do you mean by "an interpolation theorem that is a concrete realization of some interpolation method" ? In addition, what do you mean by "the abstraction in some interpolation method" ? In precise terms, one can define an interpolation method to be a functor from some category of pairs of Banach (or more general topological vector) spaces to the corresponding category of spaces. Shortly, I do not understand your question. | |
| Dec 28, 2011 at 8:59 | comment | added | Matthew Daws | Doesn't the 2nd form arise naturally when you try to compute the dual space of the 1st form? That's at least some motivation, but I don't really know the history... | |
| Dec 27, 2011 at 21:10 | history | asked | Mark Kim-Mulgrew | CC BY-SA 3.0 |