Singular multipliers on multiscale Zygmund sets

TL;DR

This paper characterizes null sets for singular multipliers on multiscale Zygmund sets within Orlicz spaces, establishing endpoint bounds, weighted estimates, and sparse bounds, thus extending classical harmonic analysis results.

Contribution

It introduces a full characterization of null sets with multiscale Zygmund properties and develops new multi-frequency projection techniques for endpoint and weighted bounds.

Findings

Characterization of null sets with multiscale Zygmund property.

Sparse and quantitative weighted estimates for Fourier multipliers.

Answering Lerner's conjecture with a pointwise sparse bound.

Abstract

Given an Orlicz space $ L^2 \subseteq X \subseteq L^1$ on $[0,1]$, with submultiplicative Young function ${\mathrm{Y}_X}$, we fully characterize the closed null sets $\Xi$ of the real line with the property that H\"ormander-Mihlin or Marcinkiewicz multiplier operators $\mathrm{T}_m$ with singularities on $\Xi$ obey weak-type endpoint modular bounds on $X$ of the type \[ \left|\left\{x\in \mathbb R : |\mathrm{T}_m f(x)| >\lambda\right\}\right| \leq C \int_{\mathbb R} \mathrm{Y}_X \left(\frac{|f|}{\lambda}\right), \qquad \forall \lambda>0. \] These sets $\Xi$ are exactly those enjoying a scale invariant version of Zygmund's $(L\sqrt{\log L},{L^2})$ improving inequality with $X$ in place of the former space, which is termed multiscale Zygmund property. Our methods actually yield sparse and quantitative weighted estimates for the Fourier multipliers $\mathrm{T}_m$ and for the corresponding square functions. In particular, our framework covers the case of singular sets $\Xi$ of finite lacunary order and thus leads to modular and quantitative weighted versions of the classical endpoint theorems of Tao and Wright for Marcinkiewicz multipliers. Moreover, we obtain a pointwise sparse bound for the Marcinkiewicz square function answering a recent conjecture of Lerner. On the other hand, examples of non-lacunary sets enjoying the multiscale Zygmund property for each $X=L^p$, $1<p\leq 2$ are also covered. The main new ingredient in the proofs is a multi-frequency, multi-scale projection lemma based on Gabor expansion, and possessing independent interest.