No abstract

We prove that the maximal operator associated with variable homogeneous planar curves (t, ut 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, ut 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.

We prove variation-norm estimates for certain oscillatory integrals related to Carleson's theorem. Bounds for the corresponding maximal operators were first proven by Stein and Wainger. Our estimates are sharp in the range of exponents, up to endpoints. Such variation-norm estimates have applications to discrete analogues and ergodic theory. The proof relies on square function estimates for Schrödinger-like equations due to Lee, Rogers and Seeger. In dimension one, our proof additionally relies on a local smoothing estimate. Though the known endpoint local smoothing estimate by Rogers and Seeger is more than sufficient for our purpose, we also give a proof of certain local smoothing estimates using Bourgain-Guth iteration and the Bourgain-Demeter ℓ 2 decoupling theorem. This may be of independent interest, because it improves the previously known range of exponents for spatial dimensions n ≥ 4.

Let H (u) be the Hilbert transform along the parabola (t, ut 2 ) where u ∈ R. For a set U of positive numbers consider the maximal func-We obtain an (essentially) optimal result for the L p operator norm of H U when 2 < p < ∞. The results are proved for families of Hilbert transforms along more general nonflat homogeneous curves.

For a polynomial P of degree greater than one, we show the existence of patterns of the form (x, x + t, x + P (t)) with a gap estimate on t in positive density subsets of the reals. This is an extension of an earlier result of Bourgain. Our proof is a combination of Bourgain's approach and more recent methods that were originally developed for the study of the bilinear Hilbert transform along curves.