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Perturbations of globally hypoelliptic pseudo-differential operators on $\mathbb{R}^n$

Pedro Meyer Tokoro

TL;DR

Addresses the stability of S-globally hypoellipticity for globally hypoelliptic pseudo-differential operators on R^n under lower-order perturbations. Develops a Weyl-Hörmander type calculus with temperate weights, defines Hypo(M,M0;Φ,Ψ) and Sobolev spaces H(M), and uses the Planck function h = Φ^{-1}Ψ^{-1} together with the strong uncertainty principle. The main result shows that if P is globally hypoelliptic and A has strictly smaller growth in the Planck sense (M0 Ṁ^{-1} ≳ h^{-ε}), then P + A remains globally hypoelliptic, with a precise condition relating M0 and Mtilde. Applications to Shubin and SG classes illustrate the stability, while counterexamples in Hörmander classes demonstrate the limitations of extending the result to classical symbol classes.

Abstract

This paper demonstrates the stability of the global regularity for a class of pseudo-differential operators under lower-order perturbations. We establish that if an operator has a globally hypoelliptic symbol, its global regularity (in the sense of Schwartz functions and tempered distributions) is preserved when perturbed by operators of sufficiently lower order. This result applies in particular to operators within the Shubin and SG classes. Furthermore, we discuss why this stability result does not hold in the standard Hörmander classes.

Perturbations of globally hypoelliptic pseudo-differential operators on $\mathbb{R}^n$

TL;DR

Addresses the stability of S-globally hypoellipticity for globally hypoelliptic pseudo-differential operators on R^n under lower-order perturbations. Develops a Weyl-Hörmander type calculus with temperate weights, defines Hypo(M,M0;Φ,Ψ) and Sobolev spaces H(M), and uses the Planck function h = Φ^{-1}Ψ^{-1} together with the strong uncertainty principle. The main result shows that if P is globally hypoelliptic and A has strictly smaller growth in the Planck sense (M0 Ṁ^{-1} ≳ h^{-ε}), then P + A remains globally hypoelliptic, with a precise condition relating M0 and Mtilde. Applications to Shubin and SG classes illustrate the stability, while counterexamples in Hörmander classes demonstrate the limitations of extending the result to classical symbol classes.

Abstract

This paper demonstrates the stability of the global regularity for a class of pseudo-differential operators under lower-order perturbations. We establish that if an operator has a globally hypoelliptic symbol, its global regularity (in the sense of Schwartz functions and tempered distributions) is preserved when perturbed by operators of sufficiently lower order. This result applies in particular to operators within the Shubin and SG classes. Furthermore, we discuss why this stability result does not hold in the standard Hörmander classes.

Paper Structure

This paper contains 7 sections, 7 theorems, 64 equations.

Table of Contents

  1. Introduction
  2. Global pseudo-differential operators
  3. The stability theorem
  4. Applications
  5. Shubin classes
  6. SG classes
  7. Hörmander classes

Key Result

Theorem 2.1

Assume the strong uncertainty principle and suppose that $P$ has symbol in $\mathrm{Hypo}(M,M_0;\Phi,\Psi)$. Then, there exists an operator $Q\in OPS(M_0^{-1};\Phi,\Psi)$ and regularizing operators $S_1,S_2$ such that where $I$ is the identity operator. In this case, we say that $Q$ is a parametrix of $P$.

Theorems & Definitions (16)

  • Theorem 2.1
  • Remark 1
  • Definition 2.2
  • Corollary 2.3
  • Definition 2.4
  • Proposition 2.5
  • Proposition 2.6
  • Theorem 3.1
  • proof
  • Remark 2
  • ...and 6 more