Weighted wave envelope estimates for the parabola

Abstract

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 applications, we obtain weighted $L^p$-estimates for a class of Fourier multiplier operators and for solutions to the free Schrödinger equation.

Strengthened results for general weights, added proof details, and corrected minor errors