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Sufficient Conditions for Solvability of Operators of Subprincipal Type

Nils Dencker

TL;DR

This work addresses local solvability of linear pseudodifferential operators with real principal symbols vanishing to second order on nonradial involutive manifolds, under a subprincipal-type sign condition $\operatorname{Sub_r}(\Psi)$. It develops a microlocal normal form and a second microlocalization, then employs Wick quantization and a carefully designed multiplier to derive a priori estimates that yield microlocal solvability with a $5/2$ derivative loss. The analysis hinges on invariant properties under symplectic changes of variables and elliptic conjugations, and on a detailed treatment of symbol classes and weights near sign-changing sets. An appendix extends the solvability framework to quasilinear PDEs, illustrating broader applicability of the method.

Abstract

In this paper we show that condition $\operatorname{Sub_r}(Ψ)$ on the subprincipal symbol is sufficient for local solvability of linear pseudodifferential operators of real subprincipal type. These are the operators having real principal symbol, which is of principal type and vanishes of second order on an involutive manifold where the subprincipal symbol is of principal type. Condition $\operatorname{Sub_r}(Ψ)$ is a condition on the sign changes of the imaginary part of the subprincipal symbol, which has previously been shown by the author to be necessary for local solvability of linear pseudodifferential operators of real subprincipal type. In the appendix, we study the local solvability of quasilinear second order partial differential operators of real principal type.

Sufficient Conditions for Solvability of Operators of Subprincipal Type

TL;DR

This work addresses local solvability of linear pseudodifferential operators with real principal symbols vanishing to second order on nonradial involutive manifolds, under a subprincipal-type sign condition . It develops a microlocal normal form and a second microlocalization, then employs Wick quantization and a carefully designed multiplier to derive a priori estimates that yield microlocal solvability with a derivative loss. The analysis hinges on invariant properties under symplectic changes of variables and elliptic conjugations, and on a detailed treatment of symbol classes and weights near sign-changing sets. An appendix extends the solvability framework to quasilinear PDEs, illustrating broader applicability of the method.

Abstract

In this paper we show that condition on the subprincipal symbol is sufficient for local solvability of linear pseudodifferential operators of real subprincipal type. These are the operators having real principal symbol, which is of principal type and vanishes of second order on an involutive manifold where the subprincipal symbol is of principal type. Condition is a condition on the sign changes of the imaginary part of the subprincipal symbol, which has previously been shown by the author to be necessary for local solvability of linear pseudodifferential operators of real subprincipal type. In the appendix, we study the local solvability of quasilinear second order partial differential operators of real principal type.
Paper Structure (9 sections, 40 theorems, 328 equations)

This paper contains 9 sections, 40 theorems, 328 equations.

Table of Contents

  1. Introduction
  2. The Preparation
  3. The microlocal estimate
  4. The symbol classes
  5. Properties of the symbol
  6. The Weight function
  7. The Wick quantization
  8. The multiplier estimate
  9. Solvability of quasilinear PDE

Key Result

Proposition 1.13

If $P \in \Psi^m$ has real principal symbol that vanishes of order $2$ at an involutive manifold $\Sigma _2$ with nondegenerate Hessian and is of real subprincipal type, then $P$ satisfies condition $\operatorname{Sub_r} (\Psi)$ if and only if $\operatorname{Im} p_s$ does not change sign on the leav

Theorems & Definitions (108)

  • Example 1.1
  • Example 1.2
  • Example 1.3
  • Example 1.4
  • Example 1.5
  • Example 1.6
  • Remark 1.7
  • Remark 1.8
  • Definition 1.9
  • Definition 1.10
  • ...and 98 more