Global analytic hypoellipticity and solvability of certain operators subject to group actions

On T × G, where T is a compact real-analytic manifold and G is a compact Lie group, we consider differential operators P which are invariant by left translations on G and are elliptic in T . Under a mild technical condition, we prove that global hypoellipticity of P implies its global analytichypoellipticity (actually Gevrey of any order s ≥ 1). We also study the connection between the latter property and the notion of global analytic (resp. Gevrey) solvability, but in a much more general setup. Introduction Two notoriously difficult problems in PDE theory are to determine whether a general linear differential operator P defined, say, on a compact manifold M , is globally hypoelliptic or globally solvable. In very general terms, given a reasonable space of functions F on M , the first problem means to determine if we can infer from the information Pu ∈ F that u itself belongs to F , where u is a priori taken in some larger space of (generalized) functions. The second one means to solve the equation Pu = f for “every” f ∈ F , with a solution u also in F – but a subtlety soon arises, for this is generally impossible as the global geometry of both M and P impose natural constraints on the right-hand side f ; we compromise by asking instead if we can always solve at least for those “admissible” f ∈ F . In order to make the situation more manageable, extra geometrical hypotheses may be imposed to the problem. A traditional one is to assume M endowed with a smooth action of a Lie group G which leaves P invariant; the action may be further assumed transitive (see e.g. [23, Chapter 5] and related works in the references therein) or free, in which case the operator induced by P on the orbit space M/G plays a key role. Here we are interested in the latter possibility, and deal with the simplest such situation: M is a product T × G, where T is a compact manifold and G is a compact Lie group, which acts freely on M by left-multiplication on the second factor alone (where G acts on itself). Invariance of P under this action allows us to put it in a global canonical form with separate variables (2.2), where the operator P0 induced by P on the orbit space T can be easily read off; this one will be further required to be elliptic in T . Operators P satisfying these properties essentially are said to belong to class T (Definition 2.2) and are our main focus. When M = Tt × T m x , where T N is the N -dimensional torus, the invariance of P under the action of T means that P = P (t,Dt, Dx) and there is a vast literature about global hypoellipticity and solvability of such operators in this ambient. Requiring the ellipticity condition on P0 imposes certain restrictions, but still there are important classes of such operators, for instance sums of squares of certain real vector fields (see e.g. [11, 13, 1]). As for the remaining ingredient – the space of functions F on which our questions will be posed – we are mainly interested in the space of real-analytic functions (for which we of course need to require further regularity of all the objects involved): global analytic hypoellipticity and solvability of P are then addressed. Actually, our approach applies to the more general Gevrey classes of functions G s of order s ≥ 1 (see e.g. [21]) of which real-analytic functions are a special case (s = 1), and one can even guess that the program could be carried forward for other ultradifferentiable classes of Roumieu type (even quasianalytic ones). As far as the regularity problem is concerned, we consider the following question: when does global hypoellipticity (i.e. in the smooth setup) implies global analytic hypoellipticity? Positive answers for 2020 Mathematics Subject Classification. 35A01, 35H10 (primary), 35R01, 35R03 (secondary).