Entropy of Sobolev embeddings of radial functions and radial eigenvalues of Schrödinger operators on isotropic manifolds

Abstract

Key words Sobolev embeddings, entropy numbers, hyperbolic spaces, Schrödinger operator MSC (2000) 46E35, 41A46, 47B06We consider Sobolev embeddings between Sobolev and Besov spaces of radial functions on noncompact symmetric spaces of rank one. An asymptotic behaviour of entropy numbers of the compact embeddings is described. The estimates are used for investigation of the negative spectrum of Schrödinger type operators.Let X be a connected Riemannian manifold and let o be a fixed point of X. We call a function f (x) radial if its value at x depends only on a distance of x to the point o. It was noticed in late seventies of the last century that radiality implies compactness of Sobolev embeddings. First, it was proved for the first order Sobolev spaces of radial functions defined on R d by Strauss [27] and Coleman, Glazer, and Martin [3], cf. also Berestycki and Lions [1]. Later the corresponding theory was developed for other type of function spaces, cf. [22], more general symmetry conditions, cf. [19] and [25], as well as on function spaces defined on Riemannian manifolds, cf. [11], [13], and [24]. On manifolds the symmetry conditions are expressed in terms of invariance with respect to the action of a compact group of isometries.Isotropic Riemannian manifolds seem to be of special interest here since on such manifolds radial means invariant with respect to the action of isotropy group of the point o. A Riemannian manifold X is called isotropic if for each y ∈ X the linear isotropy group acts transitively on the unit sphere in the tangent space T y X. It is well-known that a noncompact Riemannian manifold is isotropic if and only if it is either an Euclidean space or a noncompact Riemannian globally symmetric space of rank one.The quantitative approach to the compactness of Sobolev embeddings of radial Sobolev and Besov spaces in terms of entropy numbers was developed in [17]. The n-th (dyadic) entropy number of a bounded linear operator T : X → Y between two Banach spaces is defined aswhere B X and B Y stand for the closed unit balls in X and Y , respectively. For basic properties of entropy numbers and more background we refer to the literature, see e.g. [2], [5], [16] or [20]. The operator T is compact if and only if lim n→∞ e n (T ) = 0. The quantitative approach to compactness means that we describe an asymptotic behaviour of e n (T ) as n tends to infinity. Let RH s p R d , s ∈ R, 1 < p < ∞, denote a subspace of the fractional Sobolev space H s p R d consisted of radial distributions. In a similar way we define a radial Besov space RB s p,q R d , 1 ≤ p, q ≤ ∞. The Sobolev embeddings RH s0 p0 R d → RH s1 p1 R d and RB s0 p0,q0 R d → RB s1 p1,q1 R

Approximation numbers of Sobolev embeddings of radial functions on isotropic manifolds

“…The proof of this theorem can be found in [26], but it is similar to the proof of theorem about the traces on a straight line in the Euclidean case, cf. [17].…”

“…It can be proved that related trace and extension operators give us an isomorphisms of some radial and weighted Besov spaces, cf. [26].…”

“…Asymptotic behaviour of entropy e n and approximation numbers a n of the embeddings of radial functions on R n was studied in [17] and [25] respectively. The behaviour of the entropy numbers in the case of symmetric spaces of rank one was described in [26].…”

Approximation and entropy numbers of compact Sobolev embeddings

“…This reminds more the behavior of the Sobolev embeddings on bounded domains on R d than the embeddings of radial spaces on R d , cf. [45]. Now we consider the embeddings of weighted function spaces.…”

Eigenvalue distributions of some non-isotropic degenerate elliptic operators

“…This attracted afterwards some attention in diverse settings. We mention in particular [12][13][14] and the references given there. In all cases C in (1.7) is smooth and the singularity behavior originates exclusively from b 1 , b 2 , typically belonging to some L r -spaces.…”