Logarithmic Sobolev spaces on R^n ; entropy numbers and some application.

Preprint series: 98-15, Analysis

The paper is published: Forum Math., 12, 257-313, 2000

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 earlier papers we have studied compact embeddings of weighted function spaces on R^n, $id_H: H^{s_1}_{p_1}(w(\cdot),R^n) \longrightarrow H^{s_2}_{p_2}(R^n)$, $s_1>s_2$, $1<p_1\leq p_2<\infty$, $s_1-n/p_1>s_2-n/p_2$, and $w(x)$ of the type $w(x)=(1+|x|)^\alpha (\log(2+|x|))^\beta$, $\alpha\geq 0$, $\beta\in\real$.
We have determined the asymptotic behaviour of the corresponding entropy numbers $e_k(id_H)$. Now we are interested in the limiting case $s_1-n/p_1=s_2-n/p_2$.
Let $w(x)=\log^\beta\langle x\rangle$, $\beta>0$. Then $id_H$ cannot be compact (for any $\beta>0$), but replacing the Sobolev spaces $H^{s_i}_{p_i}$, $i=1,2$, by their logarithmic counterparts, $H^{s_i}_{p_i}(\log H)_{a_i}$, $a_i\in\real$, $i=1,2$, one can prove compactness of the so modified embedding $id_{H,a}$ in some cases. In an earlier paper we have followed this idea, introducing logarithmic Lebesgue spaces $L_p(log L)_a(R^n)$ for this purpose.
We continue and extend these results now, and study the entropy numbers $e_k(id_{H,a})$.
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 $\Psi^{-\varkappa}_{1,\gamma}$, $\varkappa>0$, $0\leq\gamma\leq 1$, and $b_1, b_2$ are in (weighted) logarithmic Lebesgue spaces on R^n.

Keywords: logarithmic spaces, function spaces, entropy numbers