Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Local Solvability of Quasilinear Pseudodifferential Operators of Real Principal Type

Nils Dencker

TL;DR

The paper proves local solvability for quasilinear pseudodifferential operators whose principal symbol $p_m$ is real and of principal type. The core method is a microlocal reduction to a first-order normal form, followed by solving microlocalized linearized problems with elliptic Fourier integral operators and precise Sobolev estimates. It constructs a local smooth solution $u$ to $P(u,\partial^\alpha u, x, D)u=f$ matching prescribed jets at $x_0$, with the real-valued data case yielding a real solution; the solution is generally non-unique. This work extends Hörmander’s solvability theory to a broad class of nonlinear pseudodifferential operators by combining microlocal normal form reduction, smoothing remainder control, and a fixed-point/degree argument in a localized, parametric setting.

Abstract

In this paper we prove local solvability of quasilinear pseudodifferential operators which has homogeneous principal symbol of real principal type. This generalizes Theorem A.1 in arXiv:2403.19054, which treats the case of quasilinear partial differential operators of order 2. The proof is by microlocalization to first order model operators.

Local Solvability of Quasilinear Pseudodifferential Operators of Real Principal Type

TL;DR

The paper proves local solvability for quasilinear pseudodifferential operators whose principal symbol is real and of principal type. The core method is a microlocal reduction to a first-order normal form, followed by solving microlocalized linearized problems with elliptic Fourier integral operators and precise Sobolev estimates. It constructs a local smooth solution to matching prescribed jets at , with the real-valued data case yielding a real solution; the solution is generally non-unique. This work extends Hörmander’s solvability theory to a broad class of nonlinear pseudodifferential operators by combining microlocal normal form reduction, smoothing remainder control, and a fixed-point/degree argument in a localized, parametric setting.

Abstract

In this paper we prove local solvability of quasilinear pseudodifferential operators which has homogeneous principal symbol of real principal type. This generalizes Theorem A.1 in arXiv:2403.19054, which treats the case of quasilinear partial differential operators of order 2. The proof is by microlocalization to first order model operators.
Paper Structure (7 sections, 9 theorems, 79 equations)

This paper contains 7 sections, 9 theorems, 79 equations.

Table of Contents

  1. Introduction
  2. The Normal Form
  3. Solutions of the Microlocalized Equation
  4. A Calculus Lemma
  5. The Microlocal Estimate
  6. Solutions of the Linearized Equation
  7. Proof of Theorem \ref{['appthm']}

Key Result

Theorem 1.2

Let $X$ be a manifold and $P \in \Psi^m(X)$ be a properly supported nonlinear operator given by psymbol so that $p_j (v,x, \xi) \in S^j(T^*X)$ depends $C^\infty$ on $\partial^\alpha v(x) \in C^\infty$, $| \alpha | < m$, and the principal symbol $p_m$ is homogeneous of degree $m \in \mathbf Z_+$.

Theorems & Definitions (22)

  • Definition 1.1
  • Theorem 1.2
  • Remark 1.3
  • Definition 2.1
  • Remark 2.2
  • Proposition 2.3
  • proof
  • Remark 2.4
  • Proposition 3.1
  • Corollary 3.2
  • ...and 12 more