Logarithmic oscillatory multipliers and log-subdyadic square functions
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866913166090305536 |
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| author | Vergara, Vicente |
| author_facet | Vergara, Vicente |
| contents | We study Fourier multipliers with logarithmic oscillation at high frequency. The guiding example is the radial symbol \[
m_{γ,β}(ξ)
=
\bigl(\log(e+|ξ|)\bigr)^{-β}
e^{i(\log(e+|ξ|))^γ},
\qquad γ>1, \] whose natural frequency scale is smaller than dyadic but larger than every fixed power-subdyadic scale. We develop a square-function theory adapted to this logarithmic scale.
The main square-function result is a pointwise estimate for Fourier multiplier operators whose symbols satisfy a localized logarithmic Miyachi condition. We prove the corresponding log-subdyadic frequency decomposition, the associated decoupling and recoupling estimates, and the local multiplier estimate needed to control the operator. We also establish a high-frequency weighted $L^2$ multiplier estimate and derive unweighted $L^p$-boundedness for $1<p<\infty$ under the sufficient logarithmic decay condition \[
β>
d(γ-1)\left|\frac12-\frac1p\right|. \] The logarithmic model multiplier above satisfies the localized hypothesis in the high-frequency region. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_27746 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Logarithmic oscillatory multipliers and log-subdyadic square functions Vergara, Vicente Functional Analysis Primary 42B15, 42B20, Secondary 42B25, 42B35 We study Fourier multipliers with logarithmic oscillation at high frequency. The guiding example is the radial symbol \[ m_{γ,β}(ξ) = \bigl(\log(e+|ξ|)\bigr)^{-β} e^{i(\log(e+|ξ|))^γ}, \qquad γ>1, \] whose natural frequency scale is smaller than dyadic but larger than every fixed power-subdyadic scale. We develop a square-function theory adapted to this logarithmic scale. The main square-function result is a pointwise estimate for Fourier multiplier operators whose symbols satisfy a localized logarithmic Miyachi condition. We prove the corresponding log-subdyadic frequency decomposition, the associated decoupling and recoupling estimates, and the local multiplier estimate needed to control the operator. We also establish a high-frequency weighted $L^2$ multiplier estimate and derive unweighted $L^p$-boundedness for $1<p<\infty$ under the sufficient logarithmic decay condition \[ β> d(γ-1)\left|\frac12-\frac1p\right|. \] The logarithmic model multiplier above satisfies the localized hypothesis in the high-frequency region. |
| title | Logarithmic oscillatory multipliers and log-subdyadic square functions |
| topic | Functional Analysis Primary 42B15, 42B20, Secondary 42B25, 42B35 |
| url | https://arxiv.org/abs/2605.27746 |