Global hypoellipticity for a class of periodic Cauchy operators

Fernando de Ávila Silva

doi:10.1016/j.jmaa.2019.123650

paper

Global hypoellipticity for a class of periodic Cauchy operators

arXiv:1907.11670 · doi:10.1016/j.jmaa.2019.123650

Abstract

This note presents an investigation on the global hypoellipticity problem for Cauchy operators on $T^{n + 1}$ belonging to the class \linebreak $L = \prod_{j = 1}^{m} (D_{t} + c_{j} (t) P_{j} (D_{x}))$ , where $P_{j} (D_{x})$ is a pseudo-differential operator on $T^{n}$ and $c_{j} = c_{j} (t)$ , a smooth, complex valued function on $T$ . The main goal of this investigation consists in establishing connections between the global hypoellipticity of the operators $L$ and its normal form $L_{0} = \prod_{j = 1}^{m} (D_{t} + c_{0, j} P_{j} (D_{x}))$ . In order to do so, the problem is approached by combining Hörmander's and Siegel's conditions on the symbols of the operators $L_{j} = D_{t} + c_{j} (t) P_{j} (D_{x})$ .

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