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In [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).
In [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).
CONTENTS
Introduction.........................................................................................5
1. Non-limiting embeddings - a short review........................................8
2. 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.................................................................................35
3. Entropy numbers, limiting embeddings.........................................37
4. Applications..................................................................................47
4.1. Eigenvalue distribution............................................................47
4.2. Negative spectrum...................................................................53
5. Homogeneous spaces..................................................................55
References.......................................................................................58
Introduction.........................................................................................5
1. Non-limiting embeddings - a short review........................................8
2. 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.................................................................................35
3. Entropy numbers, limiting embeddings.........................................37
4. Applications..................................................................................47
4.1. Eigenvalue distribution............................................................47
4.2. Negative spectrum...................................................................53
5. Homogeneous spaces..................................................................55
References.......................................................................................58
Słowa kluczowe
Tematy
Kategoryzacja MSC:
- 35P15: Estimation of eigenvalues, upper and lower bounds
- 46E30: Spaces of measurable functions ( L p -spaces, Orlicz spaces, K\"othe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
- 35J70: Degenerate elliptic equations
- 46E35: Sobolev spaces and other spaces of ``smooth'' functions, embedding theorems, trace theorems
- 35P20: Asymptotic distribution of eigenvalues and eigenfunctions
- 41A46: Approximation by arbitrary nonlinear expressions; widths and entropy
Miejsce publikacji
Warszawa
Copyright
Seria
Rozprawy Matematyczne
tom/nr w serii:
373
Liczba stron
59
Liczba rozdzia³ów
Opis fizyczny
Dissertationes Mathematicae, Tom CCCLXXIII
Daty
wydano
1998
otrzymano
1996-10-26
poprawiono
1997-07-08
Twórcy
autor
- Mathematisches Institut, Friedrich-Schiller-Universität Jena, D-07740 Jena, Germany, haroske@minet.uni-jena.de
Bibliografia
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1991 Mathematics Subject Classification: 46E35, 46E30, 41A46, 35P15, 35P20, 35J70.
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