Lebesgue classes and preparation of real constructible functions

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 $q>0$ and constructible functions $f$ and $\mu$ on $E\times\RR^n$, we prove a theorem describing the structure of the set of all $(x,p)$ in $E\times (0,\infty ]$ for which $y\mapsto f(x,y)$ is in $L^p(|\mu {|}_x^q)$, where $|\mu {|}_x^q$ is the positive measure on $\RR^n$ 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).