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

Victor Lie

Published 2019-02-11Version 1

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 methods are several key discretization techniques that intertwine with Littewood-Paley theory, elements of time-frequency analysis via Gabor-frame decompositions, shifted square-functions, and almost orthogonality arguments.