Spectral theory for fractal pseudodifferential operators

Abstract

The paper deals with the distribution of eigenvalues of the compact fractal pseudodifferential operator \(T^\mu _\tau \),

$$\begin{aligned} \big ( T^\mu _\tau f\big )(x) = \int _{{{\mathbb {R}}}^n} e^{-ix\xi } \, \tau (x,\xi ) \, \big ( f\mu \big )^\vee (\xi ) \, {\mathrm d}\xi , \qquad x\in {{\mathbb {R}}}^n, \end{aligned}$$

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 Hörmander classes \(\Psi ^\sigma _{1, \delta } ({{\mathbb {R}}}^n)\), \(\sigma <0\), \(0 \le \delta \le 1\) (including the exotic case \(\delta =1\)) 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 \((\textrm{id}- \Delta )^{\sigma /2} \mu \) 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

$$\begin{aligned} \textrm{tr}\,_\mu : \quad B^s_p ({{\mathbb {R}}}^n) \hookrightarrow L_p (\Gamma , \mu ), \quad s>0, \quad 1<p<\infty . \end{aligned}$$

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.