New results on Fourier multipliers on $L^p$: a perspective through unimodular symbols
María Jesús Carro, Alberto Salguero-Alarcón
TL;DR
This work establishes a unifying framework linking unimodular Fourier multipliers to general $L^p$-multipliers via an exponential-growth bound for $||e^{itm}||_{\mathcal{M}_p}$. By extending Stein's analytic families of operators to accommodate exponential border growth, the authors derive boundedness of the multiplier derivative at the interpolation endpoint and transfer these estimates to weighted and endpoint spaces. The main contributions include a precise characterization of multiplier boundedness in terms of unimodular growth, and new results for homogeneous rough operators, singular integrals along curves, and oscillatory integrals, with weighted extensions and explicit endpoint-space identifications. This framework provides a cohesive, technically robust method to obtain $L^p$ and weighted $L^p$ bounds for a broad class of operators in harmonic analysis and dispersive PDE contexts, and clarifies the endpoint behavior via Calderón-type interpolation spaces.
Abstract
The paper focuses on the behaviour of unimodular Fourier multipliers with exponential growth in the context of weighted $L^p$-spaces. Our main result shows that much of the general theory of multipliers is approachable through the theory of unimodular multipliers. Indeed, we show that a bounded measurable function $m$ is a multiplier on $L^p$ for $1\leq p<\infty$ if and only if $e^{itm}$ is a multiplier on $L^p$ and its multiplier norm admits an exponential bound of the form $e^{c|t|^s}$ for suitable $c>0$ and $0<s<1$. We then apply this principle to obtain new results related to the boundedness of homogeneous rough operators, singular operators along curves and oscillatory integrals. A key ingredient in our study is an extension of the classical Stein's theorem on analytic families of operators that studies the behaviour of the derivative operator when $θ\to 0$.