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

Abstract

For all d ⩾ 2 and p ∈ ( 1 , max ( 2 , ( d + 1 ) / 2 ) ] , we prove sharp L p to L p ( L q ) 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 L p bounds for variable kernel singular integral operators and Nikodym-type maximal operators taking averages over certain families of curved sets in R d .