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arXiv · 1007.4731

Problems on averages and lacunary maximal functions

Andreas Seeger·James Wright
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Abstract

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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Andreas Seeger, James Wright. 2010-07-27. Problems on averages and lacunary maximal functions. https://arxiv.org/abs/1007.4731

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