Problems on averages and lacunary maximal functions

Tom 95 / 2011

Andreas Seeger, James Wright Banach Center Publications 95 (2011), 235-250 MSC: Primary 42B25, Secondary 42B15, 42B30. DOI: 10.4064/bc95-0-11


We prove three results concerning convolution operators and lacunary maximal functions associated to dilates of measures. First we obtain an $H^1$ to $L^{1,\infty}$ bound for lacunary maximal operators under a dimensional assumption on the underlying measure and an assumption on an $L^p$ regularity bound for some $p>1$. Secondly, we obtain a necessary and sufficient condition for $L^2$ boundedness of lacunary maximal operator associated to averages over convex curves in the plane. Finally we prove an $L^p$ regularity result for such averages. We formulate various open problems.


  • Andreas SeegerDepartment of Mathematics
    University of Wisconsin-Madison
    480 Lincoln Drive
    Madison, WI 53706, USA
  • James WrightMaxwell Institute for Mathematical Sciences and the School of Mathematics
    University of Edinburgh
    JCMB, King's Buildings
    Mayfield Road,
    Edinburgh EH9 3JZ, Scotland

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