Logarithmic oscillatory multipliers and log-subdyadic square functions

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.