Real analytic parameter dependence of solutions of differential equations (Q971965)

This highly interesting and deep paper is concerned with the classical problem when the tensor product of two surjective linear operators in locally convex spaces is again surjective. The question goes back to the classical theorem of Grothendieck stating that the tensor product of surjective linear operators in nuclear Fréchet spaces is always surjective. In the present paper, the problem is studied in the frame of (PLS)-spaces (and (PLN)-spaces), i.e., of countable projective limits of countable injective limits with compact (respectively, nuclear) linking maps. This covers most of the natural spaces of analysis including the spaces of (ultra) distributions or currents on \(C^\infty\)-manifolds, (ultra) differentiable or real analytic functions. The technical details of the solution are rather subtle and only some ideas can be given here: using the homological methods developed by Palamodov (see the book of [\textit{J. Wengenroth}, Derived functors in functional analysis. Berlin: Springer (2003; Zbl 1031.46001)] for an introduction to this topic), the surjectivity of the tensor mapping \(T\otimes Id_E\) can be translated into the vanishing of \(\text{Proj}^1\) for the vector valued kernels of the surjective linear mapping \(T\). This leads to a general characterization of the surjectivity of \(T\otimes \text{Id}_E\) by means of a subtle estimate connecting the dual norms of the kernel spectrum and \(E\). This solves the above problem in the most general case known so far and covers the case of surjective endomorphisms on (PLN)-spaces. The general condition is evaluated in the important case of operators \(T\) acting on distributions (or currents) with (weakly) real analytic parameters, i.e., where \(E\) is a space of real analytic functions. The condition then is equivalent to the fact that \(\text{ker}(T)\) satisfies the dual interpolation estimate for small \(\theta\) or, equivalently, \(\text{ker}(T)\) satisfies the condition \((P\overline{\overline{\Omega}})\) introduced by \textit{J. Bonet} and \textit{P. Domanski} [J. Funct. Anal. 230, 329--381 (2006; Zbl 1094.46006)]. For partial differential operators \(P(D)\) with constant coefficients on convex open sets in \(\mathbb R^d\) much more can be said: Using the Ehrenpreis-Palamodov Fundamental Principle it is shown that the dual interpolation estimate for the kernel is equivalent to a condition of Phragmen-Lindelöf type valid for plurisubharmonic functions on the complex characteristic variety of \(P\). Summarizing, we finally get the following concrete answers to the real analytic parameter dependence problem for \(P(D)\) on \(D'(\Omega)\) (notice that the answer is completely different from the solution for holomorphic or for \(C^\infty-\)parameter dependence [see \textit{J. Bonet} and \textit{P. Domanski}, Adv. Math. 217, 561--585 (2008; Zbl 1144.46057)]: the answer is negative for elliptic operators and general open \(\Omega\subset\mathbb R^d\), \(d>1\), for hypoelliptic operators on convex open \(\Omega\subset\mathbb R^d, d>1,\) and for the system of Cauchy-Riemann equations on a pseudoconvex domain \(\Omega\subset\mathbb C^d, d\geq 1\). The answer is positive for operators \(P(D)\) in two variables on convex open \(\Omega\subset\mathbb R^2\) iff \(P\) is hyperbolic, for operators of order two iff after some real linear change of variables it has the form \(\mu(\partial_1-a_1)^2+c\) or \(\mu[\sum_{j=1}^r(\partial_j-a_j)^2-\sum_{j=r+1}^{s-1}(\partial_j-a_j)^2]+\lambda i\partial_s+c\) for some \(\mu,c,a_j\in\mathbb C\), \(\lambda\in\mathbb R\), and \(1\leq r<s-1,\) and for homogeneous \(P\) on convex open sets \(\Omega\) iff \(P(D)\) has a continuous linear right inverse in \(C^\infty(\Omega)\) (or, equivalently, in \(D'(\Omega)\)). Thus, for these cases the answer is positive iff \(P(D)\) has a continuous linear right inverse in \(C^\infty(\Omega)\) (or, equivalently, in \(D'(\Omega)\)). The solution of the latter problem (of L. Schwartz) has been given by \textit{R. Meise, B. A. Taylor} and \textit{D. Vogt} in a series of papers starting with [Ann. Inst. Fourier (Grenoble) 40, 619--655 (1990; Zbl 0703.46025)].