Convergence of two-dimensional weighted integrals (Q2781377)

This paper considers the question of convergence of integrals of the form \(\int\int_{B} \frac{|g(x,y)|^\varepsilon}{|f(x,y)|^\delta} dy dx\) in which \(B\) is a small ball centered at the origin, \(f,g\) are real-analytic functions in the plane and \(\varepsilon,\delta\) are positive. The problem has its origins in questions of growth rates of real-analytic functions and convergence of oscillatory integrals. The unweighted case (\(g=1\)) was resolved by \textit{D. H. Phong, E. M. Stein}, and \textit{J. A. Sturm} [Am. J. Math. 121, 519-554 (1999; Zbl 1015.26031)]. This present paper represents, in part, an attack on the unweighted case in higher dimensions by means of a reduction through the use of symmetries to the weighted case in two dimensions. Techniques of \textit{D. H. Phong} and \textit{E. M. Stein} [Acta Math. 179, 105-152 (1997; Zbl 0896.35147)]are extended here to deal with the latter case. The main result uses a notion of Newton distance \(\delta_0(g,f,\varepsilon;\varphi)\) of \(f\) and \(g\) associated to a transformation \(\varphi\) to which a generalized notion of a Newton diagram -- the boundary of the convex hull determined by the indices of the nonzero terms of a Taylor expansion -- can be applied. Then \(\delta_0(f,g,\varepsilon)=\min\{\delta_0(g,f,\varepsilon;\varphi)\}\) where the minimum is taken over a suitable class of transformations \(\varphi\). The main result is that, if \(f,g\) are real-analytic, complex-valued functions satisfying \(f(0)=g(0)=0\) then there is a small neighborhood \(B\) of the origin such that the weighted integral above converges whenever \(\delta<\delta_0(f,g,\varepsilon)\) and diverges otherwise. The point is that, in principle, the Newton distance is a computable quantity once \(f,g\) are fixed.