Spectral theory for fractal pseudodifferential operators

TL;DR

This paper investigates the eigenvalue distribution of fractal pseudodifferential operators on Besov spaces, extending previous results from differential operators to fractal measures and more general symbol classes.

Contribution

It extends spectral analysis of fractal differential operators to a broader class of pseudodifferential operators with symbols in Hörmander classes on Besov spaces.

Findings

Eigenvalue distribution characterized for fractal pseudodifferential operators

Extension of spectral results from differential to pseudodifferential operators

Estimates for entropy numbers of trace operators on Besov spaces

Abstract

The paper deals with the distribution of eigenvalues of the compact fractal pseudodifferential operator $T^\mu_\tau$, \[ \big( T^\mu_\tau f\big)(x) = \int_{\mathbb{R}^n} e^{-ix\xi} \, \tau(x,\xi) \, \big( f\mu \big)^\vee (\xi) \, \mathrm{d} \xi, \qquad x\in \mathbb{R}^n, \] in suitable special Besov spaces $B^s_p (\mathbb{R}^n) = B^s_{p,p} (\mathbb{R}^n)$, $s>0$, $1<p<\infty$. Here $\tau(x,\xi)$ are the symbols of (smooth) pseudodifferential operators belonging to appropriate H\"{o}rmander classes $\Psi^\sigma_{1, \varrho} (\mathbb{R}^n)$, $\sigma <0$, $0 \le \varrho \le 1$ (including the exotic case $\varrho =1$) whereas $\mu$ is the Hausdorff measure of a compact $d$-set $\Gamma$ in $\mathbb{R}^n$, $0<d<n$. This extends previous assertions for the positive-definite selfadjoint fractal differential operator $(\mathrm{id} - \Delta)^{\sigma/2} \mu$ based on Hilbert space arguments in the context of suitable Sobolev spaces $H^s (\mathbb{R}^n) = B^s_2 (\mathbb{R}^n)$. We collect the outcome in the {Main Theorem} below. Proofs are based on estimates for the entropy numbers of the compact trace operator \[ \mathrm{tr}_\mu: \quad B^s_p (\mathbb{R}^n) \hookrightarrow L_p (\Gamma, \mu), \quad s>0, \quad 1<p<\infty. \] We add at the end of the paper a few personal reminiscences illuminating the role of Pietsch in connection with the creation of approximation numbers and entropy numbers.