Problems on averages and lacunary maximal functions

Details Problems on averages and lacunary maximal functions Problems on averages and lacunary maximal functions

2010-07-27

arxiv.org/abs/1007.4731v2

J\'ozef Marcinkiewicz Centenary Volume, Banach Center Publications, vol. 95 (2011), 235-250

Mathematics

Classical Analysis and ODEs

To appear in the Marcinkiewicz Centenary Volume (Banach Center Publications 95)

Scientific paper

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.

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