Mixed-norm estimates for a class of nonisotropic directional maximal operators and Hilbert transforms

For all d 2 and p ∈ (1,max(2, (d + 1)/2)], we prove sharp Lp to Lp(Lq ) estimates (modulo an endpoint) for a directional maximal operator associated to curves generated by the dilation matrices exp((log t)P), where P has real entries and eigenvalues with positive real part. For the corresponding Hilbert transform we prove an analogous result for all d 2 and p ∈ (1, 2]. As corollaries, we prove Lp bounds for variable kernel singular integral operators and Nikodym-type maximal operators taking averages over certain families of curved sets in Rd . © 2008 Elsevier Inc. All rights reserved