Weighted wave envelope estimates for the parabola

In this paper, we extend the Córdoba-Fefferman square function estimate for the parabola to a weighted setting. Our weighted square function estimate is derived from a weighted wave envelope estimate for the parabola. The bounds are formulated in terms of families of multiscale tubes together with weight parameters that quantify the distribution of the weight. As an application, we obtain some weighted L^p-estimates for a class of Fourier multiplier operators and for solutions to free Schrödinger equation.

💡 Research Summary

This paper develops a weighted version of the Córdoba‑Fefferman square‑function estimate for the parabola and shows how it can be used to obtain weighted L^p bounds for Fourier multipliers and solutions of the free Schrödinger equation.
The authors begin by fixing the truncated parabola P = {(t, t^2) : |t| ≤ 1} and its R^{-1}‑neighbourhood N_{R^{-1}}P. For a function f whose Fourier support lies in this neighbourhood they decompose f = Σ_θ f_θ, where the pieces f_θ are associated with finitely overlapping parallelograms θ of dimensions R^{-1/2} × R^{-1}. The orthogonality of the pieces in L^2 is classical; the curvature of the parabola yields additional L^p orthogonality for p > 2, leading to the classical square‑function estimate
  ‖f‖{L^4(ℝ^2)} ≤ C ‖( Σ_θ |f_θ|^2 )^{1/2}‖{L^4(ℝ^2)}.
The goal of the paper is to replace the unweighted L^p norm on the left by a weighted norm ‖f‖_{L^p(Hdx)} with a non‑negative weight H. To capture how H is distributed relative to the geometry of the parabola, the authors introduce a multiscale decomposition. For each dyadic scale s ∈