Holomorphic Toroidal Pseudodifferential Operators on the Polydisk

Abstract

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 $S_{\rho ,\delta }^m$, $0\le \delta <\rho \le 1$, 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 $q\le t$. As an application, we prove well-posedness for a first-order holomorphic differential operator on these scales.