Integral formulas for under/overdetermined differential operators via recovery on curves and the finite-dimensional cokernel condition I: General theory

Abstract: We introduce a new versatile method for constructing solution operators (i.e., right-inverses up to a finite rank operator) for a wide class of underdetermined PDEs $P u = f$, which are regularizing of optimal order and, more interestingly, whose integral kernels have certain prescribed support properties. By duality, we simultaneously obtain integral representation formulas (i.e., left-inverses up to a finite rank operator) for overdetermined PDEs $P^{\ast} v = g$ with analogous properties, which lead to Poincar\'e- or Korn-type inequalities. Our method applies to operators such as the divergence, linearized scalar curvature, and linearized Einstein constraint operators (which are underdetermined), as well as the gradient, Hessian, trace-free part of the Hessian, Killing, and conformal Killing operators (which are overdetermined). The starting point for our construction is a condition - dubbed the recovery on curves condition (RC) - that leads to Green's functions for $P$ supported on prescribed curves. Then the desired integral solution operators (and, by duality, integral representation formulas) are obtained by taking smooth averages over a suitable family of curves. This procedure generalizes the previous constructions of Bogovskii, Oh-Tataru, and Reshetnyak. We furthermore identify a simple algebraic sufficient condition for (RC), namely, that the principal symbol $p(x, \xi)$ of $P$ is full-rank for all non-zero complex vectors $\xi$ (as opposed to real, as in ellipticity). When the principal symbol has constant coefficients, this is equivalent to (RC) and also to the condition that the formal cokernel of $P$ (without any boundary conditions) is finite-dimensional; for this reason, we call it the finite-dimensional cokernel condition (FC). We give a short proof that all operators above satisfy (FC), and thus (RC). Various applications will be considered in subsequent papers.