A frequency localized maximum principle and its applications to time fractional equations with rough coefficients

We establish a frequency localized maximum principle for parabolic second-order partial differential equations with the Caputo time fractional derivative ${\mathrm{\partial }}_t^{\alpha }$ of order $\alpha \in (0,1)$. The leading coefficients are uniformly elliptic, measurable in time, and independent of space. We construct suitable equivalent norms on frequency localized subspaces of $L_{\mathrm{\infty }}({\mathbb{R}}^d)$, for which a frequency localized maximum principle holds. The corresponding result may fail in the usual $L_{\mathrm{\infty }}$ norm even for the classical heat equation. As an application, we prove existence, uniqueness, and regularity estimates in Sobolev-Zygmund spaces with Muckenhoupt weights in time for equations with spatially variable coefficients and zero initial data. The coefficients are strongly measurable in time with values in a Hölder-Zygmund space of positive order and uniformly bounded in this space.