A unified approach to three themes in harmonic analysis ($1^{st}$ part)

TL;DR

This paper presents a unified approach to establishing $L^p$ bounds for several harmonic analysis operators along variable curves with non-zero curvature, covering linear, maximal, and bilinear transforms.

Contribution

It introduces a general framework that unifies the treatment of multiple harmonic analysis themes along variable curves with curvature, using a novel combination of discretization, Gabor frames, and time-frequency methods.

Findings

Proves $L^p$ boundedness for operators along variable curves with curvature.

Provides a unified method applicable to linear, maximal, and bilinear operators.

Establishes a new approach that simplifies and unifies previous disparate results.

Abstract

In the present paper and its sequel "A unified approach to three themes in harmonic analysis ($2^{nd}$ part)", we address three rich historical themes in harmonic analysis that rely fundamentally on the concept of non-zero curvature. Namely, we focus on the boundedness properties of (I) the linear Hilbert transform and maximal operator along variable curves, (II) Carleson-type operators in the presence of curvature, and (III) the bilinear Hilbert transform and maximal operator along variable curves. Our Main Theorem states that, given a general variable curve $\gamma(x,t)$ in the plane that is assumed only to be measurable in $x$ and to satisfy suitable non-zero curvature (in $t$) and non-degeneracy conditions, all of the above itemized operators defined along the curve $\gamma$ are $L^p$-bounded for $1<p<\infty$. Our result provides a new and unified treatment of these three themes. Moreover, it establishes a unitary approach for both the singular integral and the maximal operator versions within themes (I) and (III). At the heart of our approach stays a methodology encompassing three key ingredients: 1) discretization on the multiplier side that confines the phase of the multiplier to oscillate at the \emph{linear} level, 2) \emph{Gabor}-frame discretization of the input function(s) and 3) extraction of the \emph{cancelation} hidden in the non-zero curvature of $\gamma$ via $TT^{*}-$orthogonality methods and time-frequency \emph{correlation}.