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$L^p$ estimates for joint quasimodes of two pseudodifferential operators whose characteristic sets have $k$-th order contact

Madelyne M. Brown, Melissa Tacy

TL;DR

This work proves sharp $L^p$ bounds for joint semiclassical quasimodes of two pseudodifferential operators whose characteristic sets meet with $k$-th order contact on a compact manifold. The authors flatten the first operator’s characteristic set with a Fourier integral operator, reduce to two simpler equations, and leverage a wavelet-frequency decomposition combined with a TT$^*$ argument and Van der Corput-type oscillatory estimates to derive bounds of the form $ rm{u_h}_{L^p} \\lesssim h^{-oldsymbol{ u(n,p,k)}} rm{u_h}_{L^2}$, where the exponent matches the Sogge bound for low to intermediate $p$ and gains a penalized term for large $p$ due to the $k$-th order contact. The approach generalizes Tacy’s $n=2$ results to higher dimensions and highlights how higher-order tangency yields improved concentration estimates in the high-$p$ regime, with sharpness demonstrated in model examples. The results have implications for spectral geometry and semiclassical analysis by refining eigenfunction concentration under joint constraint equations.

Abstract

On a smooth, compact, $n$-dimensional Riemannian manifold, we consider functions $u_h$ that are joint quasimodes of two semiclassical pseudodifferential operators $p_1(x,hD)$ and $p_2(x,hD)$. We develop $L^p$ estimates for $u_h$ when the characteristic sets of $p_1$ and $p_2$ meet with $k$-th order contact. This paper is the natural extension of the two-dimensional results from arXiv:1909.12559 to $n$ dimensions.

$L^p$ estimates for joint quasimodes of two pseudodifferential operators whose characteristic sets have $k$-th order contact

TL;DR

This work proves sharp bounds for joint semiclassical quasimodes of two pseudodifferential operators whose characteristic sets meet with -th order contact on a compact manifold. The authors flatten the first operator’s characteristic set with a Fourier integral operator, reduce to two simpler equations, and leverage a wavelet-frequency decomposition combined with a TT argument and Van der Corput-type oscillatory estimates to derive bounds of the form , where the exponent matches the Sogge bound for low to intermediate and gains a penalized term for large due to the -th order contact. The approach generalizes Tacy’s results to higher dimensions and highlights how higher-order tangency yields improved concentration estimates in the high- regime, with sharpness demonstrated in model examples. The results have implications for spectral geometry and semiclassical analysis by refining eigenfunction concentration under joint constraint equations.

Abstract

On a smooth, compact, -dimensional Riemannian manifold, we consider functions that are joint quasimodes of two semiclassical pseudodifferential operators and . We develop estimates for when the characteristic sets of and meet with -th order contact. This paper is the natural extension of the two-dimensional results from arXiv:1909.12559 to dimensions.

Paper Structure

This paper contains 13 sections, 6 theorems, 244 equations, 2 figures.

Table of Contents

  1. Introduction and Main Results
  2. A brief overview of $L^p$ estimates
  3. Statement of results
  4. Examples
  5. Simplifying Equations Using a Fourier Integral Operator
  6. Localisation and choosing coordinates
  7. A Fourier integral operator
  8. Incorporating the second equation
  9. Proof of Theorem \ref{['thrm:linfcontact']}
  10. A wavelet and frequency decomposition
  11. Obtaining the $L^p$ estimates
  12. $L^2$ estimates and interpolation
  13. Stationary Phase and Van der Corput

Key Result

Theorem 1.1

Suppose $u_h$ is a compactly microlocalised, strong joint quasimode of order $h$ for $P_1=p_1(x,hD)$ and $P_2=p_2(x,hD)$. Let $k\geq 1$ and odd. Furthermore assume $p_1(x,\xi)$ and $p_2(x,\xi)$ satisfy Then where

Figures (2)

  • Figure 1.1: Comparison of $\delta(n,p,k)$ from Theorem \ref{['thrm:linfcontact']} (in blue) for various values of $k$ with Sogge's estimate $\delta_{S}(n,p)$ from SoggeLp (in red).
  • Figure 2.1: A diagram of $k$th order contact between the two characteristic sets $\{p_1=0\}$ and $\{p_2=0\}$ (left) and the resulting hypersurfaces after flattening $\{p_1=0\}$ using the $W$ operator (right).

Theorems & Definitions (17)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Example 1.4: Sharpness of (\ref{['eqn:linf']}) for large $p$
  • Example 1.5: Sharpness of (\ref{['eqn:linf']}) for small $p$
  • Example 1.6
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • ...and 7 more