Logarithmic oscillatory multipliers and log-subdyadic square functions

Abstract

We develop square-function estimates for Fourier multipliers whose local oscillation scale is \[ \rho(R)=\frac{R}{(\log R)^{\gamma-1}}, \qquad \gamma>1. \] This scale lies strictly between the dyadic scale and every fixed power-subdyadic scale at high frequency. For high-frequency symbols satisfying a localized Sobolev condition on balls of radius comparable to $\rho(R)$, we prove a pointwise square-function estimate and a weighted $L^2$ multiplier inequality. After adjoining a smooth compactly supported low-frequency part, we derive unweighted $L^p$ bounds. The weighted estimate is governed by a logarithmic geometric maximal operator whose $L^r$ threshold is necessary apart from the equality case. As a model application, consider \[ L(\xi)=\frac12\log(e^2+|\xi|^2), \qquad m_{\gamma,\beta}(\xi)=L(\xi)^{-\beta}e^{iL(\xi)^\gamma}. \] For $p=2$, the associated multiplier is bounded on $L^2$ for every $\beta\geq0$. For $1 d(\gamma-1)\left|\frac12-\frac1p\right|. \]