Singular multipliers on multiscale Zygmund sets

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.