Weighted inequalities for real-analytic functions in \(\mathbb R^ 2\).
This paper is a continuation in a series of papers by the author concerning the question of convergence of integrals of the form \(\int\int_{B} \frac{| g(x,y)| ^\epsilon| }{| f(x,y)| ^\delta}\, dy\,dx\) in which \(B\) is a small ball centered at the origin, \(f,g\) are real-analytic functions in the plane vanishing at the origin and \(\epsilon,\delta\) are positive [see \textit{M. Pramanik}, Trans. Am. Math. Soc. 354, 1651--1665 (2002; Zbl 1025.42007)]. The problem has its origins in questions of growth rates of real-analytic functions and convergence of oscillatory integrals and is considered as hopeful means of addressing problems of convergence of certain unweighted integrals, i.e., ones in which \(g=1\), in higher dimensions by means of symmetry reductions. In the present paper the author considers the question of determining conditions under which, for fixed \(p\), the weight \(| g| ^{\epsilon}/| f| ^\delta \) defines a weight in the Muckenhoupt class \(A_p\) in a small neighborhood of the origin in \(\mathbb R^2\). The univariate version is simpler. In that case, \(f(x)\sim x^K\) and \(g(x)\sim x^L\) for suitable \(K,L\) near \(x=0\). Then \(| g| ^{\epsilon}/| f| ^\delta \in A_p\) near \(x=0\) if and only if \(-1<\epsilon L-\delta K <p-1\). In the plane one applies the Weierstrass preparation theorem to represent \[ f(x,y)= x^\alpha y^\beta \prod (y-r_\nu (x)) \] where the product is taken over expansions in the \(x\)-coordinate near the zeros of \(f\). A similar expansion applies to \(g\). The author derives a necessary and sufficient condition on \(f,g,\epsilon,\delta\) in order that \(| g| ^{\epsilon}/| f| ^\delta \in A_p\) near \(x=0\) in terms of the orders of the zeroes of the \(r_\nu\) at \(x=0\). This is quantified in terms of a {weighted Newton distance} whose utility is derived from its definition in terms of the sort of change of coordinates that arises in the reduction from unweighted to weighted integrals.
scientific article; zbMATH DE number 1777204