Spectral theory for fractal pseudodifferential operators

Abstract

The paper deals with the distribution of eigenvalues of the compact fractal pseudodifferential operator $T_{\tau }^{\mu }$, \[ \big( T^μ_τf\big)(x) = \int_{\mathbb{R}^n} e^{-ixξ} \, τ(x,ξ) \, \big( fμ\big)^\vee (ξ) \, \mathrm{d} ξ, \qquad x\in \mathbb{R}^n, \] in suitable special Besov spaces $B_p^s({\mathbb{R}}^n)=B_{p,p}^s({\mathbb{R}}^n)$, $s>0$, $1<p<\infty$. Here $\tau (x,\xi )$ are the symbols of (smooth) pseudodifferential operators belonging to appropriate Hörmander classes ${\Psi }_{1,\varrho }^{\sigma }({\mathbb{R}}^n)$, $\sigma <0$, $0\le \varrho \le 1$ (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 $H^s({\mathbb{R}}^n)=B_2^s({\mathbb{R}}^n)$. We collect the outcome in the {Main Theorem} below. Proofs are based on estimates for the entropy numbers of the compact trace operator \[ \mathrm{tr}_μ: \quad B^s_p (\mathbb{R}^n) \hookrightarrow L_p (Γ, μ), \quad s>0, \quad 1<p<\infty. \] 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.