Singular Radon transforms and maximal functions under convexity assumptions

Abstract

We prove variable coefficient analogues of results in [5] on Hilbert transforms and maximal functions along convex curves in the plane.

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

“…This corresponds to the case of Γ α u when α = 1. Apart from the work [24] that has been mentioned already, one more relevant result from this body of literature is due to Seeger and Wainger [32]. In this paper the variable curve (t, u(x, y) · [t] α ) t∈R appears as a special case of the more general curve Γ(x, y, t) which satisfies some convexity and doubling hypothesis uniformly in (x, y).…”

The non-resonant bilinear Hilbert--Carleson operator

“…• the presence of the term φk1,k2 (j 1 ) φl1,l2 (j 2 ) provides a polynomial decay in terms of the size of the parameters j 1 , j 2 thus allowing us to sum in j 1 and j 2 in (113); 91 The above explains why, for x ∈ I pr k , the right-hand side in (113) can be reduced to the (dominant) term Sn,v k,pr [0, 0](f ℓ+1 , g ℓ−1 ) and moreover, why the latter can be further reduced to the expression Sn,v k (f ℓ+1 , g ℓ−1 ) which, given the imposed range for x, is well approximated 92 by Sn,v k (f ℓ+1,r , g ℓ−1,r )(x), thus certifying (114).…”

“…Thus in [16] and [5] the authors prove the expected L p range for the situation γ(x, y, t) = P (x)γ(t), where P is a polynomial and γ is smooth and obeys some suitable nonvanishing curvature condition. In a different direction, this time analyzing the behavior of H Γ under the assumption that γ(x, y, t) obeys (x, y, t)-smoothness and non-zero curvature in t hypotheses, we have: in the nilpotent setting the work in [18], and in the context of singular Radon transforms (and their maximal analogues) i) along differentiable submanifolds the work in [22], or ii) along variable curves in a diffeomorphism invariant setting, the work in [92].…”

“…In this section we focus on obtaining an explicit form for the kernel K u,v,n,p m,k,ℓ by exploiting the time-frequency localization of the wavepackets in relation to the oscillatory factor e ica(ξ−η) a ′ λ(x) − 1 a−1 in (92). Notice that the change of variables ξ → k η in(92) together with the definition of the wave-packets (89) allows us to rewrite the above as…”

$L^p(\mathbb{R}^2)$-boundedness of Hilbert Transforms and Maximal Functions along Plane Curves with Two-variable Coefficients

“…For the general two-variable case, we would like to mention more useful results besides the relevant results introduced at the beginning of this paper. Seeger and Wainger [34] obtained the L p (R 2 )-boundedness of H ∞ U,γ and M ∞ U,γ for p ∈ (1, ∞), where U(x 1 , x 2 )γ(t) was written as Γ(x 1 , x 2 , t), under some convexity and doubling hypothesis about Γ. Recently, for γ(t) := [t] α (where 0 < α < 1 or α > 1), Di Plinio, Guo, Thiele and Zorin-Kranich [16] obtained the L p (R 2 )-boundedness, p ∈ (1, ∞), of H ε 0 U,γ from some positive constant ε 0 and a Lipschitz function U : R 2 → R satisfying U Lip 1, where Jones's beta numbers from [25] play an important role in their proof.…”