Endpoint estimates for the fractal circular maximal function and related local smoothing

Sharp $L^p$–$L^q$ estimates for the spherical maximal function over dilation sets of fractal dimensions, including the endpoint estimates, were recently proved by Anderson–Hughes–Roos–Seeger. More intricate $L^p$–$L^q$ estimates for the fractal circular maximal function were later established in the sharp range by Roos–Seeger, but the endpoint estimates have been left open, particularly when the fractal dimension of the dilation set lies in $[1\mathrm{/}2,1)$. In this work, we prove these missing endpoint estimates for the circular maximal function. We also study the closely related $L^p$–$L^q$ local smoothing estimates for the wave operator over fractal dilation sets, which were recently investigated by Beltran–Roos–Rutar–Seeger and Wheeler. Making use of a bilinear approach, we also extend the range of $p,q$, for which the optimal estimate holds.