Weighted wave envelope estimates for the parabola
Abstract
In this paper, we extend the C\'ordoba-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 Lp-estimates for a class of Fourier multiplier operators and for solutions to free Schr\"odinger equation.
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