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Preprint Number 504

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504. Raf Cluckers, Daniel J. Miller
Lebesgue classes and preparation of real constructible functions

Submission date: 15 September 2012.


We call a function constructible if it has a globally subanalytic domain and can be expressed as a sum of products of globally subanalytic functions and logarithms of positively-valued globally subanalytic functions. For any q > 0 and constructible functions f and \mu on E x R^n, we prove a theorem describing the structure of the set of all (x,p) in E x (0,\infty] for which y \mapsto f(x,y) is in L^p(|\mu|_{x}^{q}), where |\mu|_{x}^{q} is the positive measure on R^n whose Radon-Nikodym derivative with respect to the Lebesgue measure is y\mapsto |\mu(x,y)|^q. We also prove a closely related preparation theorem for f and \mu. These results relate analysis (the study of L^p-spaces) with geometry (the study of zero loci).

Mathematics Subject Classification: 46E30, 32B20, 14P15 (Primary) 42B35, 03C64 (Secondary)

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Full text arXiv 1209.3439: pdf, ps.

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