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

In [18] and [19] we have studied compact embeddings of weighted function spaces on Rn, id : Hs q (w(x),Rn) −→ Lp(R), s > 0, 1 < q ≤ p < ∞, s − n/q + n/p > 0, with, for example, w(x) = 〈x〉α, α > 0, or w(x) = logβ〈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) = logβ〈x〉, β > 0. Our results in [18] imply that id cannot be compact for any β > 0, but replacing the target space Lp(R) by some ’slightly’ larger one, Lp(log L)−a(R), a > 0, the respective embedding becomes compact and we can study its entropy numbers. Finally 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(X) on spaces of homogeneous type (X, δ, μ).