Global hypoellipticity and solvability for a class of evolution operators in time-periodic weighted Sobolev spaces

Abstract We study the hypoellipticity and solvability properties of a class of time-periodic evolution operators, with coefficients globally defined on $$\mathbb {R}^d$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> </mml:math> and growing polynomially with respect to the space variable. To this aim, we introduce a class of time-periodic weighted Sobolev spaces, whose elements are characterised in terms of suitable Fourier expansions associated with elliptic operators.