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, 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.

Square functions for bi-Lipschitz maps and directional operators

“…Adding curvature to the picture by defining where r α may be interpreted either as |r| α or sgn(r)|r| α , we may argue similarly as above but remove the conditionality thanks to the results in [Guo+16]. We obtain Corollary 1.9.…”

A maximal function for families of Hilbert transforms along homogeneous curves

“…Another motivation for our work comes from the recent papers [16], [8] which take up the curved cases and analyze the linear operator f → H (u(·)) f for special classes of measurable functions x → u(x). [16] covers the case when u(x) depends only on x 1 and [8] covers the case where u is Lipschitz. The analogous questions for variable lines are still not completely resolved (cf.…”

“…The growth condition in terms of log N(U ) is only relevant for the maximal function sup u∈U |S u f | for which we prove L p bounds for all 1 < p < ∞. Here we use the Chang-Wilson-Wolff inequality, together with a variant of an approximation argument in [16]. It turns out that the full maximal operators associated to the T u ± are bounded in L p (R 2 ) for 2 < p < ∞.…”

“…As now u ν = 2 n(ν)b for a strictly increasing sequence {n(ν)} we see by another application of Littlewood-Paley theory that (7.9) is a consequence of the inequality This is proved as in [16] by using a superposition of shifted maximal operators, in a vector-valued setting. To analyze the situation we recall how R u j,ℓ was formed (namely by rescaling T u j,ℓ , then see §2).…”

“…Here we have used Lemma 3.3 with g = T n f and the fact that the operators T n and P (2) k,b commute; q will be chosen so that 1 < q < p. We shall now use an idea in [16] and approximate the operators T n by a convolution operator acting in the first variable. Define T (1) by…”