Global Analytic Hypoellipticity in the Presence of Symmetry

A linear partial differential operator, L, is said to be globally analytic hypoelliptic on some real analytic manifold M without boundary if, for any u ∈ D′(M) such that Lu ∈ C(M), one has u ∈ C(M). It is of some interest to determine under what circumstances this property holds, especially for sums of squares of vector fields satisfying the bracket hypothesis of Hörmander, and for related operators such as b arising in complex analysis in several variables. Examples are known (and discussed at the end of this note) in which global analytic hypoellipticity holds [C1], [C2], [CoH], [DT1], [DT2], yet analytic hypoellipticity in the more usually studied local sense fails [Ch1], [Ch2], [HH]. Those examples all possess certain global symmetries. We have recently shown [Ch3] that global analytic hypoellipticity does not always hold for certain of these operators, so that symmetry hypotheses are no mere crutch eventually to be discarded, as might have been hoped. Our aim here is to show how global analytic hypoellipticity follows quite directly from the presence of certain symmetry, emphasizing the general nature of the result and structure of the argument. Throughout this note, “analytic” means “real analytic”, and is denoted by C. The symbols L, C∞, C refer to sections of whichever vector bundle is relevant. Our results are formulated for sections of vector bundles, rather than for functions, because that situation arises for b, but the reader will lose no ideas by assuming all functions to be scalar-valued. Let there be given an analytic manifold Ω without boundary, not necessarily compact, a compact subset K ⊂ Ω, and a linear partial differential operator L mapping sections of one analytic vector bundle over Ω to sections of a second such bundle. We consider the following notion of regularity.