spaces of smooth functions for linear hyperbolic PDE

Question

Which classes of (scalar or systems of) linear first or second order hyperbolic PDEs $Lf=g$ in $n$ variables and spaces $S$ of functions have the property that there is a retarded Green's function $G:S\to S$ with $LG=1$. Retarded means that for every $g\in S$ whose support is disjoint from the past causal cone of $x$, the support of the retarded solution $g=Gf$ of $Lf=g$ is disjoint from the past causal cone of $x$.

When can one choose $S$ as a space of smooth (i.e., $C^\infty$) functions?

Where can I read about the mathematical tools for studying this and similar questions?

A particular case I am interested in is hyperbolic $L$ of the form $$L=\pmatrix{\alpha & (a-d)^* \cr b-d & B},$$ where $d$ is the exterior derivative, ${}^*$ is the Minkowski adjoint, $\alpha$ is a scalar field on Minkowski space, $a,b$ are covector fields, and $B$ is a matrix field mapping covectors to covectors. Hyperbolicity is defined as in Wikipedia.

Answer 1

For normally hyperbolic operators (those whose principal symbol is the same as for the wave operator, but possibly acting on a vector bundle, the theory of fundamental solutions/Green functions (as distributions that would be acting on smooth functions) is very well developed in

Bär, Christian; Ginoux, Nicolas; Pfäffle, Frank, Wave equations on Lorentzian manifolds and quantization., ESI Lectures in Mathematics and Physics. Zürich: European Mathematical Society Publishing House (ISBN 978-3-03719-037-1/pbk). viii, 194 p. (2007). ZBL1118.58016. arXiv:0806.1036

Of course, the condition of global hyperbolicity on the (Lorentzian) metric associated to the wave-like principal symbol is crucial there. The simplest space of smooth functions that is closed under the action of the retarded Green function is $C_+^\infty$, the space of smooth functions with retarded support, consisting of functions whose support is contained in $J^+(K)$, the future causal influence set of some compact subset $K$. But there is a also a larger space $C_{pc}^\infty$ with the same property, the space of smooth functions with past compact support, consisting of functions whose support has compact intersection with $J^-(K)$ for any compact $K$. These and other natural support restrictions are conveniently described in

Sanders, Ko, A note on spacelike and timelike compactness, Classical Quantum Gravity 30, No. 11, Article ID 115014, 10 p. (2013). ZBL1272.83015. arXiv:1211.2469

Going beyond normally hyperbolic equations, the following reference shows that the basic properties of Green functions that are needed for the discussion above are also shared by symmetric hyperbolic systems (or those that can be reduced to them)

Bär, Christian, Green-hyperbolic operators on globally hyperbolic spacetimes, Commun. Math. Phys. 333, No. 3, 1585-1615 (2015). ZBL1316.58027. arXiv:1310.0738

I don't know as much about the case of sub-$C^\infty$ regularity. If that's what you are interested in, perhaps others can give more information.

UPDATE: The example system given in the updated question is just the first order form of a normally hyperbolic equation whose principal symbol is the same as for $d^* B^{-1} d$ (so normal hyperbolicity here is with respect to a metric that is composed of the background metric and the operator $B$). I'm assuming here that the inverse $B^{-1}$ exists and that the equation does not lead to any integrability conditions (e.g., no new equations of order 1 or lower appear after applying the exterior derivative $d$ to the second row of $L$). Otherwise, it's not even clear that the system is indeed hyperbolic.

Speaking at a higher level of generality, you might say that $L$ belongs to the class of generalized normally hyperbolic operators. At least that's the terminology that I used in a recent paper, for lack of a better one in the literature (AFAIK). See Lemma 3 and the definition preceding it in

García-Parrado, Alfonso; Khavkine, Igor, Conformal Killing initial data, J. Math. Phys. 60, No. 12, 122502, 13 p. (2019). ZBL1435.83020. arXiv:1905.01231

More specifically, if we let $\square_B = d^* B^{-1} d$, there should exist $$ M = \pmatrix{ 1 & -d^* \\ B^{-1} d & B^{-1} (\square_B - d d^*)} + \text{l.o.t}, $$ such that $$ L M = \pmatrix{ \square_B & 0 \\ 0 & \square_B } + \text{l.o.t} . $$ You see that $L M$ is normally hyperbolic in the usual way and hence has a retarded Green function $G_{LM}$. The retarded Green function for $L$ is then just $G_L = M G_{LM}$. Hence $L$ is Green hyperbolic in the sense of Bär.

N.B.: Naively, the upper right corner of $L M$ might actually be $O(\partial^2)$, instead of $O(\partial)$, and hence contribute to the principal symbol. The point of the lower order terms in $M$ is to try to cancel that contribution to the principal symbol. If such a cancellation is impossible (basically when the adjoint operator $L^*$ has non-trivial integrability conditions), then the principal symbol of $LM$ will only be upper triangular, with $\square_B$ on the diagonal. It should still be possible to construct $G_{LM}$ then, by exploiting the upper triangular structure.

Answer 2

I think that you should read L. Gaarding's seminal paper Linear hyperbolic partial differential equations with constant coefficients, Acta Math 85:1-62 (1951). It explains why hyperbolicity is the appropriate condition for the $C^\infty$-well-posedness of the Cauchy problem.