Some logarithmic function spaces, entropy numbers, applications to spectral theory

Haroske Dorothee. Some logarithmic function spaces, entropy numbers, applications to spectral theory. Warszawa: Instytut Matematyczny Polskiej Akademi Nauk, 1998. <http://eudml.org/doc/271248>.

@book{HaroskeDorothee1998,
abstract = {AbstractIn [18] and [19] we have studied compact embeddings of weighted function spaces on ℝⁿ, $id: H^s_q(w(x),ℝⁿ) → Lₚ(ℝⁿ)$, s>0, 1 < q ≤ p< ∞, s-n/q+n/p > 0, with, for example, $w(x) = ⟨x⟩^α$, α > 0, or $w(x) = log^β⟨x⟩$, β > 0, and $⟨x⟩ = (2+|x|²)^\{1/2\}$. We have determined the behaviour of their entropy numbers eₖ(id). Now we are interested in the limiting case 1/q = 1/p + s/n. Let $w(x) = log^β⟨x⟩$, β > 0. Our results in [18] imply that id cannot be compact for any β > 0, but after replacing the target space Lₚ(ℝⁿ) by a “slightly” larger one, $Lₚ(log L)_\{-a\}(ℝⁿ)$, a > 0, the corresponding embedding becomes compact and we can study its entropy numbers. We apply our result to estimate eigenvalues of the compact operator B = b₂ ∘ b(·,D) ∘ b₁ acting in some Lₚ space, where b(·,D) belongs to some Hörmander class $Ψ^\{-ϰ\}_\{1,γ\}$, ϰ > 0, 0 ≤ γ < 1, and b₁, b₂ are in (weighted) logarithmic Lebesgue spaces on ℝⁿ. Another application concerns the study of “negative spectra” via the Birman-Schwinger principle. The last part shows possible generalisations of the spaces $Lₚ(log L)_\{-a\}(ℝⁿ)$ with ℝⁿ replaced by a space of homogeneous type (X,δ,μ).CONTENTSIntroduction.........................................................................................51. Non-limiting embeddings - a short review........................................82. The spaces Lₚ(log L)ₐ on ℝⁿ.........................................................12  2.1. The spaces Lₚ(log L)ₐ(Ω) and $L_\{p,q\}(log L)ₐ(Ω)$................12  2.2. The spaces $Lₚ(log L)_\{-a\}(ℝⁿ)$, a > 0...................................20  2.3. The spaces Lₚ(log L)ₐ(ℝⁿ), a > 0.............................................27  2.4. Hölder inequalities...................................................................32  2.5. Examples.................................................................................353. Entropy numbers, limiting embeddings.........................................374. Applications..................................................................................47  4.1. Eigenvalue distribution............................................................47  4.2. Negative spectrum...................................................................535. Homogeneous spaces..................................................................55References.......................................................................................581991 Mathematics Subject Classification: 46E35, 46E30, 41A46, 35P15, 35P20, 35J70.},
author = {Haroske Dorothee},
keywords = {negative spectra; compact embeddings; weighted function spaces; entropy numbers; estimate eigenvalues; Hörmander class; logarithmic Lebesgue spaces; Birman-Schwinger principle; space of homogeneous type},
language = {eng},
location = {Warszawa},
publisher = {Instytut Matematyczny Polskiej Akademi Nauk},
title = {Some logarithmic function spaces, entropy numbers, applications to spectral theory},
url = {http://eudml.org/doc/271248},
year = {1998},
}

TY - BOOK
AU - Haroske Dorothee
TI - Some logarithmic function spaces, entropy numbers, applications to spectral theory
PY - 1998
CY - Warszawa
PB - Instytut Matematyczny Polskiej Akademi Nauk
AB - AbstractIn [18] and [19] we have studied compact embeddings of weighted function spaces on ℝⁿ, $id: H^s_q(w(x),ℝⁿ) → Lₚ(ℝⁿ)$, s>0, 1 < q ≤ p< ∞, s-n/q+n/p > 0, with, for example, $w(x) = ⟨x⟩^α$, α > 0, or $w(x) = log^β⟨x⟩$, β > 0, and $⟨x⟩ = (2+|x|²)^{1/2}$. We have determined the behaviour of their entropy numbers eₖ(id). Now we are interested in the limiting case 1/q = 1/p + s/n. Let $w(x) = log^β⟨x⟩$, β > 0. Our results in [18] imply that id cannot be compact for any β > 0, but after replacing the target space Lₚ(ℝⁿ) by a “slightly” larger one, $Lₚ(log L)_{-a}(ℝⁿ)$, a > 0, the corresponding embedding becomes compact and we can study its entropy numbers. We apply our result to estimate eigenvalues of the compact operator B = b₂ ∘ b(·,D) ∘ b₁ acting in some Lₚ space, where b(·,D) belongs to some Hörmander class $Ψ^{-ϰ}_{1,γ}$, ϰ > 0, 0 ≤ γ < 1, and b₁, b₂ are in (weighted) logarithmic Lebesgue spaces on ℝⁿ. Another application concerns the study of “negative spectra” via the Birman-Schwinger principle. The last part shows possible generalisations of the spaces $Lₚ(log L)_{-a}(ℝⁿ)$ with ℝⁿ replaced by a space of homogeneous type (X,δ,μ).CONTENTSIntroduction.........................................................................................51. Non-limiting embeddings - a short review........................................82. The spaces Lₚ(log L)ₐ on ℝⁿ.........................................................12  2.1. The spaces Lₚ(log L)ₐ(Ω) and $L_{p,q}(log L)ₐ(Ω)$................12  2.2. The spaces $Lₚ(log L)_{-a}(ℝⁿ)$, a > 0...................................20  2.3. The spaces Lₚ(log L)ₐ(ℝⁿ), a > 0.............................................27  2.4. Hölder inequalities...................................................................32  2.5. Examples.................................................................................353. Entropy numbers, limiting embeddings.........................................374. Applications..................................................................................47  4.1. Eigenvalue distribution............................................................47  4.2. Negative spectrum...................................................................535. Homogeneous spaces..................................................................55References.......................................................................................581991 Mathematics Subject Classification: 46E35, 46E30, 41A46, 35P15, 35P20, 35J70.
LA - eng
KW - negative spectra; compact embeddings; weighted function spaces; entropy numbers; estimate eigenvalues; Hörmander class; logarithmic Lebesgue spaces; Birman-Schwinger principle; space of homogeneous type
UR - http://eudml.org/doc/271248
ER -