Maximal operators and Hilbert transforms along variable non‐flat homogeneous curves

Abstract

We prove that the maximal operator associated with variable homogeneous planar curves ( t , u t α ) t ∈ R , α ≠ 1 positive, is bounded on L p ( R 2 ) for each p > 1 , under the assumption that u : R 2 → R is a Lipschitz function. Furthermore, we prove that the Hilbert transform associated with ( t , u t α ) t ∈ R , α ≠ 1 positive, is bounded on L p ( R 2 ) for each p > 1 , under the assumption that u : R 2 → R is a measurable function and is constant in the second variable. Our proofs rely on stationary phase methods, T T ∗ arguments, local smoothing estimates and a pointwise estimate for taking averages along curves.