Weighted inequalities for real-analytic functions in ℝ2

Let ƒ and g be real-analytic functions near the origin in ℝ2. Given 1 < p < ∞, we obtain a characterization of the set of positive numbers ∈ and δ that ensures\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\frac{{|g|^\varepsilon }}{{|f|^\delta }} \in A_p (K)$$ \end{document} for some small neighborhood K of the origin. A notion of stability is introduced in relation to Ap weights and a counterexample is presented to show that the two-dimensional weighted problem, unlike its analog in dimension one, is not stable.