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

Haroske Dorothee

GDML_Books, (1998), p.

Résumé

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.......................................................................................58

1991 Mathematics Subject Classification: 46E35, 46E30, 41A46, 35P15, 35P20, 35J70.

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Zbl 0906.46027

EUDML-ID : urn:eudml:doc:271248