Abstract:We study Fourier multipliers with logarithmic oscillation at high frequency. The guiding example is the radial symbol \[
m_{\gamma,\beta}(\xi)
=
\bigl(\log(e+|\xi|)\bigr)^{-\beta}
e^{i(\log(e+|\xi|))^\gamma},
\qquad \gamma>1, \] 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 \[
\beta>
d(\gamma-1)\left|\frac12-\frac1p\right|. \] The logarithmic model multiplier above satisfies the localized hypothesis in the high-frequency region.
From: Vicente Vergara [view email]
[v1]
Tue, 26 May 2026 22:43:58 UTC (18 KB)
此内容由惯性聚合(RSS阅读器)自动聚合整理，仅供阅读参考。 原文来自 — 版权归原作者所有。