The properties of rare maximal functions (i.e. Hardy-Littlewood maximal functions associated with sparse families of intervals) are investigated. A simple criterion allows one to decide if a given rare maximal function satisfies a converse weak type inequality. The summability properties of rare maximal functions are also considered.
It is known that all continuous orbital measures, on a compact, connected, classical simple Lie group or its Lie algebra satisfy a dichotomy: either or is purely singular to Haar measure. In this note we prove that the same dichotomy holds for the dual situation, continuous orbital measures on the complex group . We also determine the sharp exponent such that any -fold convolution product of continuous -bi-invariant measures on is absolute continuous with respect to Haar measure.
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