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

TL;DR
This paper develops a detailed structure theorem for constructible functions and measures, linking their $L^p$-integrability properties with geometric and analytic aspects of globally subanalytic functions.
Contribution
It introduces a new structure theorem describing the $L^p$-space membership of constructible functions with respect to parametrized measures, connecting analysis and geometry.
Findings
Characterization of the set where functions are in $L^p$ spaces
A preparation theorem for constructible functions and measures
Bridging analysis of $L^p$ spaces with geometric zero loci
Abstract
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 and constructible functions and on , we prove a theorem describing the structure of the set of all in for which is in , where is the positive measure on whose Radon-Nikodym derivative with respect to the Lebesgue measure is . We also prove a closely related preparation theorem for and . These results relate analysis (the study of -spaces) with geometry (the study of zero loci).
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