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 $$m_{\gamma ,\beta }(\xi )=(\log (e+\mathrm{\mid }\xi \mathrm{\mid }){)}^{-\beta }{e}^{i(\log (e+\mathrm{\mid }\xi \mathrm{\mid }){)}^{\gamma }},\gamma >1,$$ 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 $1<p<\mathrm{\infty }$ under the sufficient logarithmic decay condition $$\beta >d(\gamma -1)\mid \frac{1}{2}-\frac{1}{p}\mid \mathrm{.}$$ The logarithmic model multiplier above satisfies the localized hypothesis in the high-frequency region.