Marstrand's projection theorem fails for the Assouad spectrum and quasi-Assouad dimension, with new almost-sure lower bounds from capacity profiles and upper bounds from tube-counting for planar sets.
Spherical maximal operators with fractal sets of dilations on radial functions
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abstract
For a given set of dilations $E\subset [1,2]$, Lebesgue space mapping properties of the spherical maximal operator with dilations restricted to $E$ are studied when acting on radial functions. In higher dimensions, the type set only depends on the upper Minkowski dimension of $E$, and in this case complete endpoint results are obtained. In two dimensions we determine the closure of the $L^p\to L^q$ type set for every given set $E$ in terms of a dimensional spectrum closely related to the upper Assouad spectrum of $E$.
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UNVERDICTED 2representative citing papers
A survey of conjectures and results on spherical maximal functions emphasizing problems with fractal dilation sets.
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On the Marstrand projection theorem for the Assouad spectrum
Marstrand's projection theorem fails for the Assouad spectrum and quasi-Assouad dimension, with new almost-sure lower bounds from capacity profiles and upper bounds from tube-counting for planar sets.
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Problems on spherical maximal functions
A survey of conjectures and results on spherical maximal functions emphasizing problems with fractal dilation sets.