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Local solvability of second order differential operators with double characteristics I: Necessary conditions

Detlef Mueller

TL;DR

The paper studies local solvability of doubly characteristic second-order differential operators and shows that, for a broad class, local solvability at a point implies essential dissipativity at that point. The authors reduce the solvability question to a real-quadratic problem on a finite-dimensional symplectic space: with real quadratics $Q_A$, $Q_B$ and $Q_C = \{Q_A,Q_B\}$, determine when a common zero of $Q_A$ and $Q_B$ occurs with $Q_C \neq 0$, in line with Hörmander's necessary condition. They develop a geometric and semi-algebraic analysis of intersections of ellipsoidal sublevel sets of $Q_A$ and $Q_B$, proving containment or transversal-intersection results under various rank hypotheses, and show that under large maxrank and independence, $C$ lies in $\mathrm{span}\{A,B\}$, giving a solvability obstruction. Using Hörmander's criterion, they deduce a non-solvability (nowhere locally solvable) result for left-invariant, doubly characteristic operators on two-step nilpotent groups (notably the Heisenberg group), with the analysis extendable to higher-step nilpotent groups via a 2-step quotient; Part II will complete the solvability picture.

Abstract

This is a the first in a series of two articles devoted to the question of local solvability of doubly characteristic second order differential operators. For a large class of such operators, we show that local solvability at a given point implies "essential dissipativity" of the operator at this point. By means of Hoermander's classical necessary condition for local solvability, the proof is reduced to the following question, whose answer forms the core of the paper: Suppose that $Q_A$ and $Q_B$ are two real quadratic forms on a finite dimensional symplectic vector space, and let $Q_C:=\{Q_A,Q_B\}$ be given by the Poisson bracket of $Q_A$ and $Q_B.$ Then $Q_C$ is again a quadratic form, and we may ask: When can we find a common zero of $Q_A$ and $Q_B$ at which $Q_C$ does not vanish? The second paper, in combination with the first one, will give a fairly comprehensive picture of what rules local solvability of invariant second order operators on the Heisenberg group.

Local solvability of second order differential operators with double characteristics I: Necessary conditions

TL;DR

The paper studies local solvability of doubly characteristic second-order differential operators and shows that, for a broad class, local solvability at a point implies essential dissipativity at that point. The authors reduce the solvability question to a real-quadratic problem on a finite-dimensional symplectic space: with real quadratics , and , determine when a common zero of and occurs with , in line with Hörmander's necessary condition. They develop a geometric and semi-algebraic analysis of intersections of ellipsoidal sublevel sets of and , proving containment or transversal-intersection results under various rank hypotheses, and show that under large maxrank and independence, lies in , giving a solvability obstruction. Using Hörmander's criterion, they deduce a non-solvability (nowhere locally solvable) result for left-invariant, doubly characteristic operators on two-step nilpotent groups (notably the Heisenberg group), with the analysis extendable to higher-step nilpotent groups via a 2-step quotient; Part II will complete the solvability picture.

Abstract

This is a the first in a series of two articles devoted to the question of local solvability of doubly characteristic second order differential operators. For a large class of such operators, we show that local solvability at a given point implies "essential dissipativity" of the operator at this point. By means of Hoermander's classical necessary condition for local solvability, the proof is reduced to the following question, whose answer forms the core of the paper: Suppose that and are two real quadratic forms on a finite dimensional symplectic vector space, and let be given by the Poisson bracket of and Then is again a quadratic form, and we may ask: When can we find a common zero of and at which does not vanish? The second paper, in combination with the first one, will give a fairly comprehensive picture of what rules local solvability of invariant second order operators on the Heisenberg group.

Paper Structure

This paper contains 7 sections, 24 theorems, 300 equations.

Table of Contents

  1. Introduction
  2. On the intersection of two real quadrics
  3. The form problem
  4. Auxiliary results
  5. The case where no stratum of ${\cal N}$ spans ${\mathbb R}^n.$
  6. The case where at least one stratum of ${\cal N}$ spans ${\mathbb R}^n.$
  7. Applications to non-solvability of doubly characteristic differential operators

Key Result

Theorem 1.1

Suppose there is some \xi_0\in {\mathbb R}^n\setminus\{0\} such that where a:={\rm Re}\, p_k and b:={\rm Im}\, p_k. Then L is not locally solvable at x_0.

Theorems & Definitions (28)

  • Theorem 1.1: Hörmander
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • Theorem 2.2
  • Lemma 2.3
  • Proposition 2.4
  • Proposition 2.5
  • Theorem 2.6
  • Remark 2.7
  • ...and 18 more