Real analytic parameter dependence of solutions of differential equations

Abstract

We consider the problem of real analytic parameter dependence of solutions of the linear partial differential equation $P(D)u=f$, i.e., the question if for every family $(f_{\lambda })\subseteq {\mathcal{D}}^{\prime }(\Omega )$ of distributions depending in a real analytic way on $\lambda \in U$, $U$ a real analytic manifold, there is a family of solutions $(u_{\lambda })\subseteq {\mathcal{D}}^{\prime }(\Omega )$ also depending analytically on $\lambda$ such that

where $\Omega \subseteq {\mathbb{R}}^d$ an open set. For general surjective variable coefficients operators or operators acting on currents over a smooth manifold we give a solution in terms of an abstract “Hadamard three circle property” for the kernel of the operator. The obtained condition is evaluated giving the full solution (usually in terms of the symbol) for operators with constant coefficients and open (convex) $\Omega \subseteq {\mathbb{R}}^d$ if $P(D)$ is of one of the following types: 1) elliptic, 2) hypoelliptic, 3) homogeneous, 4) of two variables, 5) of order two or 6) if $P(D)$ is the system of Cauchy–Riemann equations. An analogous problem is solved for convolution operators of one variable. In all enumerated cases, it follows that the solution is in the affirmative if and only if $P(D)$ has a linear continuous right inverse which shows a striking difference comparing with analogous smooth or holomorphic parameter dependence problems.

The paper contains the whole theory working also for operators on Beurling ultradistributions ${\mathcal{D}}_{(\omega )}^{\prime }$. We prove a characterization of surjectivity of tensor products of general surjective linear operators on a wide class of spaces containing most of the natural spaces of classical analysis.