Logarithmic oscillatory multipliers and log-subdyadic square functions

Abstract

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.