Circular average relative to fractal measures

Abstract

<p style='text-indent:20px;'>We prove new <inline-formula><tex-math id="M1">\begin{document}$ L^p $\end{document}</tex-math></inline-formula>–<inline-formula><tex-math id="M2">\begin{document}$ L^q $\end{document}</tex-math></inline-formula> estimates for averages over dilates of the circle with respect to fractal measures, which unify different types of maximal estimates for the circular average. Our results are consequences of <inline-formula><tex-math id="M3">\begin{document}$ L^p $\end{document}</tex-math></inline-formula>–<inline-formula><tex-math id="M4">\begin{document}$ L^q $\end{document}</tex-math></inline-formula> smoothing estimates for the wave operator relative to fractal measures. We also discuss similar results concerning the spherical averages.</p>