For dilation sets with bounded Assouad dimension α, the circular maximal function is proven of restricted weak type at the endpoint Q_{4,α}, and the fractal local smoothing estimate holds for an extended range of q.
Variation bounds for spherical averages over restricted dilates
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We study $L^p\rightarrow L^q(V^r_E)$ variation semi-norm estimates for the spherical averaging operator, where $E\subset [1,2]$.
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Endpoint estimates for the fractal circular maximal function and related local smoothing
For dilation sets with bounded Assouad dimension α, the circular maximal function is proven of restricted weak type at the endpoint Q_{4,α}, and the fractal local smoothing estimate holds for an extended range of q.