Holomorphic Toroidal Pseudodifferential Operators on the Polydisk
We give a necessary and sufficient triangular condition characterizing the toroidal pseudodifferential operators on $\mathbb T^d$ that preserve the positive-frequency cone $\mathbb N_0^d$ and hence act on holomorphic boundary values on the polydisk. For symbols of type $(1,0)$, we prove boundedness on holomorphic Besov and Triebel--Lizorkin spaces throughout the quasi-Banach range. For $S^m_{\rho,\delta}$, $0\leq\delta<\rho\leq1$, we obtain the critical loss $d(1-\rho)|1/p-1/2|$, together with endpoint estimates. A positive-cone oscillatory multiplier proves sharpness of the loss and necessity of the Besov endpoint condition $q\leq t$. As an application, we prove well-posedness for a first-order holomorphic differential operator on these scales.