Square functions for bi-Lipschitz maps and directional operators

Abstract

First we prove a Littlewood-Paley diagonalization result for bi-Lipschitz perturbations of the identity map on the real line. This result entails a number of corollaries for the Hilbert transform along lines and monomial curves in the plane. Second, we prove a square function bound for a single scale directional operator. As a corollary we give a new proof of part of a theorem of Katz on direction fields with finitely many directions.2010 Mathematics Subject Classification. 42B25.

Maximal functions associated with families of homogeneous curves: Lp bounds for P ≤ 2

“…The second motivation comes from the above-mentioned work by Marletta and Ricci [8] on the maximal function for p > 2, and the third motivation comes from a curved version of the Stein-Zygmund vector-field problem concerning the L p boundedness of M (u(•)) and H (u(•)) where x → u(x) is a Lipschitz function. In this case the L p boundedness of M (u(•)) for the full range 1 < p < ∞ was proved by Guo et al [5], and the analogous result for H (u(•)) by Di Plinio et al [4]. We refer to the bibliography of Guo et al [6] for a list of related works.…”

A maximal function for families of Hilbert transforms along homogeneous curves

“…Moreover there is a sharp bound ≈ log(#U ) for lacunary sets of directions (see also Di Plinio and Parissis [9]) and there are other improvements for direction sets of Vargas type. 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.…”

“…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.…”

On logarithmic bounds of maximal sparse operators

“…Recent papers [17,18] have established variants of these results in higher dimensions. Other papers [8,9] consider certain Lipschitz versions. It would be interesting to study the analogous questions for both themes.…”