Spectral theory for fractal pseudodifferential operators
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866911886842265600 |
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| author | Triebel, Hans |
| author_facet | Triebel, Hans |
| contents | The paper deals with the distribution of eigenvalues of the compact fractal pseudodifferential operator $T^μ_τ$, \[ \big( T^μ_τf\big)(x) = \int_{\mathbb{R}^n} e^{-ixξ} \, τ(x,ξ) \, \big( fμ\big)^\vee (ξ) \, \mathrm{d} ξ, \qquad x\in \mathbb{R}^n, \] in suitable special Besov spaces $B^s_p (\mathbb{R}^n) = B^s_{p,p} (\mathbb{R}^n)$, $s>0$, $1<p<\infty$. Here $τ(x,ξ)$ are the symbols of (smooth) pseudodifferential operators belonging to appropriate Hörmander classes $Ψ^σ_{1, \varrho} (\mathbb{R}^n)$, $σ<0$, $0 \le \varrho \le 1$ (including the exotic case $\varrho =1$) whereas $μ$ is the Hausdorff measure of a compact $d$-set $Γ$ in $\mathbb{R}^n$, $0<d<n$. This extends previous assertions for the positive-definite selfadjoint fractal differential operator $(\mathrm{id} - Δ)^{σ/2} μ$ 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 \[ \mathrm{tr}_μ: \quad B^s_p (\mathbb{R}^n) \hookrightarrow L_p (Γ, μ), \quad s>0, \quad 1<p<\infty. \] 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_15814 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Spectral theory for fractal pseudodifferential operators Triebel, Hans Functional Analysis 46E35, 41A46, 28A80, 35P15 The paper deals with the distribution of eigenvalues of the compact fractal pseudodifferential operator $T^μ_τ$, \[ \big( T^μ_τf\big)(x) = \int_{\mathbb{R}^n} e^{-ixξ} \, τ(x,ξ) \, \big( fμ\big)^\vee (ξ) \, \mathrm{d} ξ, \qquad x\in \mathbb{R}^n, \] in suitable special Besov spaces $B^s_p (\mathbb{R}^n) = B^s_{p,p} (\mathbb{R}^n)$, $s>0$, $1<p<\infty$. Here $τ(x,ξ)$ are the symbols of (smooth) pseudodifferential operators belonging to appropriate Hörmander classes $Ψ^σ_{1, \varrho} (\mathbb{R}^n)$, $σ<0$, $0 \le \varrho \le 1$ (including the exotic case $\varrho =1$) whereas $μ$ is the Hausdorff measure of a compact $d$-set $Γ$ in $\mathbb{R}^n$, $0<d<n$. This extends previous assertions for the positive-definite selfadjoint fractal differential operator $(\mathrm{id} - Δ)^{σ/2} μ$ 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 \[ \mathrm{tr}_μ: \quad B^s_p (\mathbb{R}^n) \hookrightarrow L_p (Γ, μ), \quad s>0, \quad 1<p<\infty. \] 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. |
| title | Spectral theory for fractal pseudodifferential operators |
| topic | Functional Analysis 46E35, 41A46, 28A80, 35P15 |
| url | https://arxiv.org/abs/2405.15814 |