Global Hypoellipticity and Solvability with Loss of Derivatives on the Torus
André Pedroso Kowacs, Alexandre Kirilov
TL;DR
This work addresses global hypoellipticity and solvability with loss of derivatives for Fourier multiplier operators on the torus. It develops precise GH-$r$ and GS-$r$ criteria via symbol estimates, finite-zero conditions, and closed-range characterizations, yielding a cohesive index theory that ties regularity to the operator's intrinsic order. The paper then uncovers deep number-theoretic connections, showing how irrationality measures govern derivative loss in vector fields on $\mathbb{T}^2$ and in the wave operator on tori, with concrete computations for $e$, Champernowne constants, and $\Gamma(1/4)$. These results provide sharp, computable bounds linking Diophantine properties to regularity, and they extend to higher-dimensional wave-type operators, highlighting when the loss of derivatives is inevitable and how arithmetic affects solvability.
Abstract
This paper provides a complete characterization of global hypoellipticity and solvability with loss of derivatives for Fourier multiplier operators on the $n$-dimensional torus. We establish necessary and sufficient conditions for these properties and examine their connections with classical notions of global hypoellipticity and solvability, particularly in relation to the closedness of the operator's range. As an application, we explore the interplay between these properties and number theory in the context of differential operators on the two-torus. Specifically, we prove that the loss of derivatives in the solvability of the vector field $\partial_{x_1} - α\partial_{x_2}$ is precisely determined by the well-known irrationality measure $μ(α)$ of its coefficient $α$. Furthermore, we analyze the wave operator $\partial_{x_1}^2 - η^2 Δ_{\mathbb{T}^n}$ and show how the loss of derivatives depends explicitly on the parameter $η> 0$.