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

Preprint series: 98-02, Analysis

The paper is published: Dissertationes Math., Vol. 373, 1-59, 1998

MSC:
46E35 Sobolev spaces and other spaces of smooth'' functions, embedding theorems, trace theorems
46E30 Spaces of measurable functions ($L^p$-spaces, Orlicz spaces, Kothe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
41A46 Approximation by arbitrary nonlinear expressions; widths and entropy
35P15 Estimation of eigenvalues, upper and lower bounds
35P20 Asymptotic distribution of eigenvalues and eigenfunctions for PDO
35J70 Elliptic partial differential equations of degenerate type

Abstract: In previous papers we have studied compact embeddings of weighted function spaces on R^n, $id: H^s_q(w(x),R^n)\longrightarrow L_p(R^n)$, $s>0$, $1<q<= p<\infty$, $s- n/q+ n/p>0$, with, for example, $w(x)=\langle x\rangle^\alpha$, $\alpha>0$, or $w(x)=\log^\beta\langle x\rangle$, $\beta>0$, and $\langle x\rangle= (2+|x|^2)^{1/2}$. We have
determined the behaviour of their entropy numbers $e_k(id)$. Now we are interested in the limiting case $1/q= 1/p + s/n$. Let $w(x)=\log^\beta\langle x\rangle$, $\beta>0$.
Our results in some earlier paper imply that $id$ cannot be compact for any $\beta>0$, but replacing the target space $L_p(R^n)$ by some 'slightly' larger one, $L_p(log L)_{-a}(R^n)$, $a>0$, the respective embedding becomes compact and we can study its entropy numbers.
Finally we apply our result to estimate eigenvalues of the compact operator $\; B= b_2\circ b(\cdot, D)\circ b_1\;$ acting in some $L_p$ space, where $b(\cdot,D)$ belongs to some Hörmander class $\Psi^{-\varkappa}_{1,\gamma}$, $\varkappa>0$, $0\leq\gamma<1$, and $b_1, b_2$ are in (weighted) logarithmic Lebesgue spaces on R^n. Another application concerns the study of 'negative spectra' via the Birman-Schwinger principle.
The last part shows possible generalisations of the spaces $L_p(log L)_{-a}(X)$ on spaces of homogeneous type $(X, \delta, \mu)$.

Update: 1999-04-22

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