$L^p$ Decay Estimates for Circular Means of Fractal Measures in ${\mathbb{R}}^2$

Abstract

In this paper, we investigate weighted $L^p\to L^q$ restriction estimates for the Fourier extension operator associated with planar curves of non-vanishing curvature. More precisely, we establish estimates of the form \begin{equation} \|Ef\|_{L^q(X)}\leq C_εR^ε\|f\|_p, \end{equation} where $X$ is an $\alpha$-dimensional set. As an application of this result, we derive a new $L^p$ decay estimate for circular means of Fourier transforms of fractal measures in ${\mathbb{R}}^2$. We also construct examples that give several upper bounds for the decay exponent when $p\in [1,2]$.