Global solutions for systems of strongly invariant operators on closed manifolds

We study the global hypoellipticity and solvability of strongly invariant operators and systems of strongly invariant operators on closed manifolds. Our approach is based on the Fourier analysis induced by an elliptic pseudo-differential operator, which provides a spectral decomposition of $L^2(M)$ into finite-dimensional eigenspaces. This framework allows us to characterize these global properties through asymptotic estimates on the matrix symbols of the operators. Additionally, for systems of normal strongly invariant operators, we derive an explicit solution formula and establish sufficient conditions for global hypoellipticity and solvability in terms of their eigenvalues.

💡 Research Summary

This paper investigates the global hypoellipticity and global solvability of strongly invariant operators and systems of such operators on closed (compact, boundary‑free) manifolds. The authors adopt a Fourier analysis framework generated by a fixed positive‑definite elliptic pseudo‑differential operator E of order ν > 0. The eigenvalues {λₖ} of E, together with an orthonormal basis of eigenfunctions {ϕₖℓ}, provide a spectral decomposition of L²(M) into finite‑dimensional eigenspaces E_{λₖ}. For any distribution u, its Fourier coefficients b_u(k)∈ℂ^{dₖ} encode the regularity of u: rapid decay of b_u(k) characterises smoothness, while polynomial growth corresponds to Sobolev regularity.

A linear operator P:C^∞(M)→L²(M) is called strongly invariant relative to E if it extends continuously to D′(M), commutes with E, and preserves each eigenspace E_{λₖ}. Equivalently, on each eigenspace there exists a matrix symbol σ_P(k)∈ℂ^{dₖ×dₖ} such that Pϕₖℓ = Σ_j σ_P(k)_{jℓ} ϕₖj. The symbol is assumed to have moderate growth, i.e. ‖σ_P(k)‖ ≤ C(1+λₖ)^{N/ν} for some N. The order of P is defined as the infimum of such N.

Global hypoellipticity (GH) means: if Pu∈C^∞(M) then u∈C^∞(M). Theorem 3.2 shows that a strongly invariant operator is GH iff there exist constants C>0, γ>0 such that for all sufficiently large k, ‖σ_P(k)ϕ‖ ≥ C(1+λₖ)^γ‖ϕ‖ for every unit vector ϕ∈E_{λₖ}. This lower bound guarantees that the symbol does not degenerate on high frequencies.

The paper then introduces almost global hypoellipticity (AGH): Pu∈C^∞ implies the existence of a smooth v with Pu = Pv. Proposition 3.3 proves that AGH holds precisely when there are constants t∈ℝ, C>0 such that ‖σ_P(k)ϕ‖ ≥ C(1+λₖ)^{t/ν}‖ϕ‖ for all ϕ orthogonal to ker σ_P(k). Under this condition, any distribution u with Pu∈H^s can be written as u = v + w where v∈ker P and w∈H^{s+t}, yielding the estimate ‖w‖{H^{s+t}} ≤ C^{-1}‖Pu‖{H^s}. Conversely, if the inequality fails, Proposition 3.4 constructs a counterexample where Pu is smooth but u cannot be smoothed modulo ker P. Consequently, Theorem 3.5 states that AGH ⇔ the existence of such a lower bound, and Theorem 3.6 establishes that global solvability (closed range of P:C^∞→C^∞) is equivalent to AGH.

When each σ_P(k) is a normal matrix, it can be diagonalised: σ_P(k)=diag(μ₁(k),…,μ_{dₖ}(k)). The minimal non‑zero singular value m(σ_P(k)) = min_{ℓ}|μ_ℓ(k)| replaces the norm estimate. Theorem 3.7 gives a clean criterion: P is globally solvable iff there exist C>0, γ>0 such that m(σ_P(k)) ≥ C(1+λₖ)^γ for all large k. This recovers known results for left‑invariant vector fields on compact Lie groups, where the eigenvalues are given by i⟨ξ,X⟩.

The authors extend the analysis to systems P = (P₁,…,P_n). Definition 4.1 mirrors the scalar case for GH, AGH, and solvability, now with the target space (C^∞)^n. The matrix symbol of the system is the block matrix σ_P(k) formed from the individual σ_{P_j}(k). The same lower‑bound philosophy applies: GH holds if each block satisfies a uniform growth condition; AGH and solvability are again equivalent to a uniform estimate on the minimal singular value of the whole block matrix.

A particularly important subclass is normal commuting systems: each P_j is normal and