Problems on averages and lacunary maximal functions
Tom 95 / 2011
Banach Center Publications 95 (2011), 235-250
MSC: Primary 42B25, Secondary 42B15, 42B30.
DOI: 10.4064/bc95-0-11
Streszczenie
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.
