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Resolvent near zero energy on Riemannian scattering (asymptotically conic) spaces, a Lagrangian approach

Andras Vasy

TL;DR

The paper develops a robust framework for resolvent estimates near zero energy on asymptotically conic spaces by marrying a Lagrangian regularity viewpoint with a resolved b-pseudodifferential calculus. A conjugation by $e^{-i\sigma/x}$ shifts the outgoing radial set to the zero section, and a blown-up, resolved calculus handles zero-energy degeneracies, yielding uniform Fredholm estimates for $0<|\sigma|\le\sigma_0$ on carefully chosen scattering-b Sobolev spaces. Central to the approach are the effective normal operator analysis and a detailed symbolic-positivity scheme, which together control both radial and indicial phenomena and extend invertibility from $\sigma\neq0$ to the $\sigma\to0$ limit. The treatment of potential zero-energy nullspaces via perturbations ensures the estimates remain valid in generality, highlighting the method’s applicability to Kerr-like and other geometries. Overall, the work provides a robust, operator-theoretic pathway to zero-energy resolvent bounds in singular geometric settings.

Abstract

We use a Lagrangian regularity perspective to discuss resolvent estimates near zero energy on Riemannian scattering, i.e. asymptotically conic, spaces, and their generalizations. In addition to the Lagrangian perspective we introduce and use a resolved pseudodifferential algebra to deal with zero energy degeneracies in a robust manner.

Resolvent near zero energy on Riemannian scattering (asymptotically conic) spaces, a Lagrangian approach

TL;DR

The paper develops a robust framework for resolvent estimates near zero energy on asymptotically conic spaces by marrying a Lagrangian regularity viewpoint with a resolved b-pseudodifferential calculus. A conjugation by shifts the outgoing radial set to the zero section, and a blown-up, resolved calculus handles zero-energy degeneracies, yielding uniform Fredholm estimates for on carefully chosen scattering-b Sobolev spaces. Central to the approach are the effective normal operator analysis and a detailed symbolic-positivity scheme, which together control both radial and indicial phenomena and extend invertibility from to the limit. The treatment of potential zero-energy nullspaces via perturbations ensures the estimates remain valid in generality, highlighting the method’s applicability to Kerr-like and other geometries. Overall, the work provides a robust, operator-theoretic pathway to zero-energy resolvent bounds in singular geometric settings.

Abstract

We use a Lagrangian regularity perspective to discuss resolvent estimates near zero energy on Riemannian scattering, i.e. asymptotically conic, spaces, and their generalizations. In addition to the Lagrangian perspective we introduce and use a resolved pseudodifferential algebra to deal with zero energy degeneracies in a robust manner.

Paper Structure

This paper contains 6 sections, 14 theorems, 218 equations, 4 figures.

Table of Contents

  1. Introduction and outline
  2. The operator
  3. Resolved b-algebra
  4. Symbolic estimates
  5. Normal operator
  6. Zero energy nullspace

Key Result

Theorem 1.1

Suppose that $|l'+1|<\frac{n-2}{2}$, and suppose that $P(0):H_{{\mathrm b}}^{\infty,l'}\to H_{{\mathrm b}}^{\infty,l'+2}$ has trivial nullspace, an assumption independent of $l'$ in this range. Suppose also that either $r>-1/2$, $l<-1/2$, or $r<-1/2$, $l>-1/2$. Let There exists $\sigma_0>0$ such that is invertible for $0<|\sigma|\leq\sigma_0$, $\operatorname{Im}\sigma\geq 0$, with this inverse b

Figures (4)

  • Figure 1: The resolved b-cotangent bundle, on the right, obtained by blowing up the corner $\overline{{}^{{\mathrm b}} T^*}_{\partial X}X\times\{0\}$ of $\overline{{}^{{\mathrm b}} T^*}X\times[0,1)_\sigma$, shown on the left.
  • Figure 2: The resolved b-double space, on the right, obtained by blowing up the corner given by the b-front face at $\sigma=0$ of the b-double space times $[0,1)_\sigma$, shown on the left.
  • Figure 3: Microsupport of the operator $B_1$ in \ref{['eq:br-fiber-infty']} on the resolved b-cotangent bundle on the left, resp. the operator $B_1$ in \ref{['eq:scr-fiber-infty']} on the scattering-b resolved cotangent bundle on the right. Both are shown as shaded regions.
  • Figure 4: The resolved b-cotangent bundle on the left, and its scattering-b resolution on the right obtained by blowing up the corner $x/\sigma=0$ at fiber infinity (nearest horizontal edges) of the resolved b-cotangent bundle. At the pseudodifferential operator level the symbolic calculus works at resolved b-fiber infinity which is the top (as well as bottom!) face on both pictures, as well as new face on the right picture, which corresponds to rescaled sc-decay.

Theorems & Definitions (35)

  • Theorem 1.1
  • Remark 1.2
  • Proposition 2.1
  • Remark 2.2
  • proof
  • Lemma 2.3: cf. Lemma 3.2 of Vasy:Limiting-absorption-lag
  • Definition 2.4
  • Theorem 2.5
  • Remark 2.6
  • Lemma 4.1
  • ...and 25 more