We prove three results concerning convolution operators and lacunary maximal functions associated to dilates of measures. First we obtain an H¹ to bound for lacunary maximal operators under a dimensional assumption on the underlying measure and an assumption on an regularity bound for some p > 1. Secondly, we obtain a necessary and sufficient condition for L² boundedness of lacunary maximal operator associated to averages over convex curves in the plane. Finally we prove an regularity result for such averages. We formulate various open problems.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-bc95-0-11,
author = {Andreas Seeger and James Wright},
title = {Problems on averages and lacunary maximal functions},
journal = {Banach Center Publications},
volume = {95},
year = {2011},
pages = {235-250},
zbl = {1241.42015},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc95-0-11}
}
Andreas Seeger; James Wright. Problems on averages and lacunary maximal functions. Banach Center Publications, Tome 95 (2011) pp. 235-250. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc95-0-11/