Singular multipliers on multiscale Zygmund sets
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| Format: | Preprint |
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2024
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| author | Bakas, Odysseas Ciccone, Valentina Di Plinio, Francesco Fraccaroli, Marco Parissis, Ioannis Vitturi, Marco |
| author_facet | Bakas, Odysseas Ciccone, Valentina Di Plinio, Francesco Fraccaroli, Marco Parissis, Ioannis Vitturi, Marco |
| contents | 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 $Ξ$ of the real line with the property that Hörmander-Mihlin or Marcinkiewicz multiplier operators $\mathrm{T}_m$ with singularities on $Ξ$ obey weak-type endpoint modular bounds on $X$ of the type
\[
\left|\left\{x\in \mathbb R : |\mathrm{T}_m f(x)| >λ\right\}\right| \leq C \int_{\mathbb R} \mathrm{Y}_X \left(\frac{|f|}λ\right), \qquad \forall λ>0.
\] These sets $Ξ$ 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 $Ξ$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_17521 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Singular multipliers on multiscale Zygmund sets Bakas, Odysseas Ciccone, Valentina Di Plinio, Francesco Fraccaroli, Marco Parissis, Ioannis Vitturi, Marco Classical Analysis and ODEs Functional Analysis 42B20 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 $Ξ$ of the real line with the property that Hörmander-Mihlin or Marcinkiewicz multiplier operators $\mathrm{T}_m$ with singularities on $Ξ$ obey weak-type endpoint modular bounds on $X$ of the type \[ \left|\left\{x\in \mathbb R : |\mathrm{T}_m f(x)| >λ\right\}\right| \leq C \int_{\mathbb R} \mathrm{Y}_X \left(\frac{|f|}λ\right), \qquad \forall λ>0. \] These sets $Ξ$ 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 $Ξ$ 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. |
| title | Singular multipliers on multiscale Zygmund sets |
| topic | Classical Analysis and ODEs Functional Analysis 42B20 |
| url | https://arxiv.org/abs/2406.17521 |