Approximation numbers of Sobolev embeddings of radial functions on isotropic manifolds

We regard the compact Sobolev embeddings between Besov and Sobolev spaces of radial functions on noncompact symmetric spaces of rank one. The asymptotic formula for the behaviour of approximation numbers of these embeddings is described.


Introduction
Approximation numbers measure the closeness by which a bounded operator may be approximated by linear maps of finite range, whereas entropy numbers measure compactness of the operator by means of finite coverings of an image of the unit ball.Both, approximation and entropy numbers, of compact Sobolev embeddings between function spaces of Sobolev and Besov type on the Euclidean case have been investigated by several authors: D. E. Edmunds and H. Triebel cf.[4], [5], D. Haroske and H. Triebel [10], [11] , D. Haroske [9], and A.M. Caetano [1].Some of these results are described in [6], where one can find also the applications to the spectral theory of the pseudo-differential operators.
It was noticed in late seventies of the last century by W. A. Strauss [27] and S. Coleman, V. Glazer, and A. Martin [3] that radiality implies compactness of Sobolev embeddings, cf. also [19] and [13], [12] for corresponding results on Riemannian manifolds.W. Sickel and the first named author found the necessary and sufficient conditions for compactness of the embeddings of radial Besov and Triebel-Lizorkin spaces, cf.[21].Later the corresponding theory was developed for more general symmetry conditions, cf.[24] as well as for function spaces defined on Riemannian manifolds, [22].On manifolds the symmetry conditions are expressed in terms of invariance with respect to the action of a compact group of isometries.
If X is a connected d-dimensional Riemannian manifold and o is a fixed point of X, then we call a function f radial if its value at x depends only on a distance of x to the point o.Isotropic Riemannian manifolds seems 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 necessary and sufficient conditions for compactness of Sobolev embeddings for Besov and Sobolev function spaces defined on the Euclidean space R d with d ≥ 2 , are the same as for the functions on the noncompact Riemannian symmetric space of rank one cf.[21], [22].Namely the embeddings RH s0 p0 (X) → RH s1 p1 (X) and RB s0 p0,q0 (X) → RB s1 p1,q1 (X) (1) are compact if and only if p 0 < p 1 and s 0 − d p0 > s 1 − d p1 .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].
Even though the necessary and sufficient conditions for compactness of Sobolev embedding are the same for R d and X the asymptotic behaviour of corresponding approximation (as well as entropy) numbers is quite different.
We put 1 p = 1 p0 − 1 p1 and t = min{p 0 , p 1 } .The main result of the paper describes the behaviour of the approximation sequence for embeddings (1) as follows Whereas, for the compact embeddings (2) we have In both cases (p 0 , p

Approximation numbers.
We recall the definition of approximation numbers and corresponding operator ideal quasi-norms, that will be used widely in the paper.We refer to books of B. Carl and I. Stephani [2] and A. Pietsch [20] for details, proofs and more information.Let B 0 and B 1 be two complex Banach spaces and let T : B 0 → B 1 be a bounded linear operator.The k th approximation number a k (T ) of the operator T : B 0 → B 1 is the infimum of all numbers T − A where A runs over the collection of all continuous linear operators A : B 0 → B 1 of rank smaller than k .So, where rank(A) denote the dimension of the range A(B 0 ).
Approximation numbers a k (T ) form a decreasing sequence with a 1 (T ) = T .If this sequence converges to zero then the operator T is compact.The opposite implication is generally not true.It may happen that lim k→∞ a k (T ) > 0 for some compact T if B 1 fails to have the approximation property.The approximation numbers have in particular the following properties: Later on we use the notation of operator ideals, cf.[2] and [20] for details.Here we recall only just what we need for the proofs.Given a bounded linear operator T and a positive real number s we put s,∞ (T ) is an example of an operator ideal quasi-norm This means in particular that there exists a number 0 < ≤ 1 such that ( 4) s,∞ j The following lemma concerning approximation numbers of embeddings of finite dimensional complex sequence spaces is essentially due to Gluskin [8] cf. also [6,Corollary 3.2.3]and [1] Lemma 1.Let N, k ∈ N.
, then there is a positive constant C independent of N and k such that where Moreover if k ≤ N 4 then in both cases we have an equivalence.

Function spaces, embeddings and traces.
We recall the definition of Sobolev spaces and Besov spaces on Riemannian symmetric space.There are several possible approaches.Here we present the definition in terms of a heat semi-group.We assume that the reader is familiar with the definition and elementary properties of function from fractional Sobolev spaces H s p and Besov spaces B s p,q on R n .All we need can be found in [29].Our notation related to symmetric spaces is standard and can be found, for example, in [14] or [15].Let G be noncompact connected semisimple Lie group with finite center of real rank one.Let K be its maximal compact subgroup and let X = G/K be an associated symmetric spaces of dimension d.The Killing form of G induces a G-invariant Riemannian metric on X.So X is simply connected homogeneous Riemannian manifolds and G acts transitively on X as a group of isometries.If o = eK then K is an isotropy group of o.The assumption that G is a group of rank one implies that the group K acts transitively on any sphere S(o, r) = {x ∈ X : |x| = r} , |x| denotes the Riemannian distance from x to o.
Let Δ denote the Laplace operator on X.The heat semi-group Let S 1 (X) denote the L 1 -Schwartz space on X and S 1 (X) be the corresponding space of distributions.For convenience we put The definition of the Besov spaces is independent of k up to norm equivalence, cf.[22].One can give an equivalent norm for the Sobolev spaces in term of heat semi-group, cf.ibidem.
(2) The spaces H s p (X) and B s p,q (X) are Banach spaces.If s is a positive integer then the space H s p (X) coincides with the classical Sobolev space defined in term of gradient of order s.
(3) If s > 0 then H 0,k f p can be replaced by f p in the definition of Besov spaces.
(4) The above definition coincides with the definition of H s p -B s p,q spaces on a Riemannian manifold with bounded geometry by the uniform localization principle, cf.[22].The last approach is due to H.Triebel, [30].
(5) The Besov spaces are real interpolation spaces of the Sobolev spaces.More precisely X) and the norms are equivalent.
(6) Most of the results, we quote for Sobolev spaces, is also true, mutatis mutandis, for more general Triebel-Lizorkin function spaces F s p,q .
Since the group K acts transitively on spheres centered at o, a function is radial if and only if is invariant with respect to the action of K.So the notation of radiality can be extended to distribution.The distribution f is called radial if So for any possible s, p, q we can put The spaces RH s p (X) are defined in the similar way.The both spaces are close subspaces of B s p,q (X) and H s p (X) respectively, so they are Banach spaces.
The following theorem describing the compactness of the Sobolev embeddings was proved in [22] are compact if and only if p 0 < p 1 and s 0 − d p0 > s 1 − d p1 (with the restriction 1 < p 0 < p 1 < ∞ for Sobolev spaces.) We will calculate the approximation numbers of the above embeddings.For future use we need also some information on traces of radial Besov spaces on geodesic rays starting at origin o.The traces can be described in terms of weighted Besov spaces on the ray.
It can be proved that related trace and extension operators give us an isomorphisms of some radial and weighted Besov spaces, cf.[26].
To present this result in details we will need a positive weight function (the exact behaviour near zero is not important).Here ϑ is a positive constant depending on X.It is sufficient for us to regard the weighted Besov spaces B s p,q (R, v p ) with positive smoothness s > 0. They may be defined as follows.The function Let γ : (−∞, ∞) → X be a geodesic parametrized by the arc length such that γ(0) = o.It should be clear that we can define a weighted Besov space on the geodesic by pulling back to the weighted Besov space defined on R, i.e.
We are interested in a trace on γ((0, ∞)).For f continuous on X we put To describe the traces of radial function on geodesic ray out of the origin we need the following spaces.Definition 2. Let 0 < τ < ∞, 1 ≤ p, q ≤ ∞, and s > 0 .Then we put Theorem 2. Let 1 ≤ p, q ≤ ∞, s > 0 and τ > 0 .There exist continuous operators such that tr • ext = id and ext • tr = id.
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].

Approximation numbers of the Sobolev embeddings
Let us remind that 1 p = 1 p0 − 1 p1 and t = min{p 0 , p 1 } , as before.The main result of the paper reads as follows The formula (7) follows from ( 6) by elementary embeddings and properties of approximation numbers.The proof of ( 6) is presented in Section 3.1.
(2) The estimates of the approximation numbers described in the above theorem are independent of the dimension d of the underlying space X.This is in some contrast with the behaviour of the approximation numbers for radial function spaces defined on R d , where we have dependence on d in all cases as it was proved in [25], cf.(3).Moreover the asymptotic behaviour of the approximation numbers in ( 6) and ( 7) is the same as in the case of bounded domains in one dimensional Euclidean space R, cf.[6].
The same phenomenon occurs for entropy numbers e k of the embeddings (6) and (7).In this case we have as it was proved in [26].

3.1
The proof of the main theorem.The proof of the main theorem is based on the same idea as that for entropy numbers (see [26]).We divide the origin space into two parts: the local part near origin o and the global part out of the origin.Since X is the symmetric space of real rank one the analysis of the local part can be reduced to corresponding embeddings of function spaces defined on the Euclidean ball.On the other hand the analysis of the global part can be reduced to embeddings of weighted function spaces on the real line due to the trace theorem cf.Theorem 2.
So to prove Theorem 3 we divide the identity operator id : RB s0 p0,q0 (X) → RB s1 p1,q1 (X) into two parts in the following way.Let φ ∈ C ∞ 0 (X), φ(x) = 1 for x ∈ B(o, 1  2 ), supp φ ⊂ B(o, 1) , we put id and in consequence by additivity of approximation numbers Moreover, it should be clear that where Here the spaces RB s p,q B(o, 1) are defined by ( 10) whereas the spaces RB s p,q X \ B(o, 1) were introduced in Definition 2. Thus to estimate a k (id) it is sufficient to estimate a k (Id 1 ) and a k (Id 2 ).
The last approximation numbers are estimated from above in Subsection 3.1.2 On the other hand, the space RB si pi,qi X \ B(o, 1) is isomorphic to the space B si pi,qi γ[1, ∞), v pi , cf. ( 5) and Theorem 2. By standard arguments where 1 p = 1 p0 − 1 p1 .Both the lower and upper estimates of the last approximation numbers are known at least for specialist.Indeed, it follows easily from the estimates for polynomial weights, cf.Lemma 3 below and Proposition 2. This estimates will finish the estimates of a k (id) from above as well as give us the estimates from below since we have the following commutative diagram: Here T φ denotes an operator The approximation numbers a n B s0 p0,q0 R, v p → B s1 p1,q1 R are described in Subsection 3.1.3.But first we need information about approximation numbers of some weighted sequence spaces.

3.1.1.
Approximation numbers of some sequence spaces.We will need weighted spaces of double indexed sequences.Let 1 ≤ p, q ≤ ∞.Let δ ≥ 0 and let w : N 0 × N 0 → R + denote a weight function.In particular for the positive α > 0 we put (13) w α (j, k) = (1 + k) α and w α (j, k) = (1 + 2 −j k) α and v α (j, k) = exp(α2 −j k) We will work with the following weighted sequence spaces (14) In future considerations it will be useful to regard the following subspaces of the above sequence spaces: γ ∈ N. Now we formulate the main result of this subsection.
holds, where and Let P j : Λ → Λ be the canonical projection onto j -level, i.e. for λ = λ k, we put Monotonicity arguments and elementary properties of the approximation numbers yield ( 17) and where D σ is a diagonal operator defined by the sequence σ = (1+ ) −α/p .
Step 2. The estimate from above.
Substep 2.1.First we regard the case 1 ≤ p 0 < p 1 ≤ 2 or 2 ≤ p 0 < p 1 ≤ ∞.Using ( 17) and ( 18) we find ( 19) We have ( 20) cf. [20, p. 108].Now using (20) and the elementary properties of approximation numbers we get In consequence ( 21) Under the assumption 1/r > α/p we conclude from ( 21) and ( 22) that ( 23) where the constant C depends on γ but is independent of j .Now, for given M ∈ N 0 let ( 24) Hence ( 4), ( 19), ( 23) and ( 24) yield: ( 25) with the constant c 2 independent of M , if 1 r > α p + δ .Hence for every δ > 0 we have In a similar way to (25) we obtain This implies the estimate from above for 1 We put N = γ2 j .Let us choose K ∈ N such that 2 K−1 ≤ N < 2 K and let Π i : N p0 → N p0 be a projection defined in the following way Multiplicativity of the approximation numbers yields Let r be real positive number.Lemma 1 implies We choose r such that 1 r > 1 2 and that 1 r − 1 2 + 1 t − α p > 0. The formulas ( 4) and (28) yield The constant c 2 is independent of K .This implies (26).In the similar way if tα 2p < 1 r < 1 2 , then the formulas ( 4) and ( 29) yield and the constant c 2 is independent of K .This implies (27).(23) we get and in consequence In the similar way Now, (30) and the first case of (31) imply the estimates from above for δ ]. Then α p < 1 t so by the second case of (31) we have By standard argument the above estimates hold for any positive integer k .Moreover, (30) implies since δ + α p ≤ 1 t .Now (32) and (33) give us the estimates from above in the remaining case.
Step 3. We estimate the approximation numbers from below.The space q1 γ2 j p1 is isomorphic to q1 γ2 j p1 (w α ) and the isomorphism is given by the diagonal operator a j,i → (1+i) −α/p1 a j,i .Moreover the inverse operator a j,i → (1 + i) α/p1 a j,i is an isomorphism of the space q0 2 jδ γ2 j p0 (w α ) onto q0 2 jδ γ2 j p0 (w β ) with β = αp0 p .This implies the equivalence of approximation numbers .
Thus we can regard the following commutative diagram where an operator R : γ2 j p0 → q0 2 jδ γ2 j p0 (w β ) is given by and P is a projection.It should be clear that R ≤ c 2 j(δ+ α p ) and P = 1.So, by Lemma 1, This finishes the proof.
The estimates from below can be proved in the way similar to Step 3 in the proof of the last lemma.We regard the following commutative diagram t we take k = 2 j−2 and N = 2 j .So by Lemma 1 and (34) we get ] and N as above.Using once again (34) and Lemma 1 we get

Approximation numbers for function spaces: local analysis.
Now we apply the estimates for approximation numbers of embeddings of sequence spaces to function spaces, first we regard the local part near origin.We use the Epperson-Frazier approach to radial Besov spaces and their construction of the radial ϕ-transform, cf.[7].We apply the construction with normalization described [17] that is different to the original one.For any possible s and p Epperson and Frazier constructed two families of radial functions ϕ (s,p) (convergence in S (R n )), cf.[7].Moreover the following theorem holds for radial Besov spaces.
The last theorem and the next proposition reduce the estimates from above of approximation numbers of Sobolev embeddings of the function spaces to the estimates of approximation numbers of embeddings of the sequence spaces defined in ( 14)-( 16).The following lemma was proved in [26] Lemma 4. Let 1 ≤ p 0 < p 1 ≤ ∞, and and the following diagram is commutative RB s0 p0,q0 B e (0, 1) The last lemma has an important consequence.Since α is at our disposal and can be as large as we want, approximation numbers of the embeddings of the above function spaces can be estimated from above by approximation numbers of sequence spaces of the type q (2 jδ 2 j+2 p (w d−1 )).Proposition 1. Suppose 1 ≤ p 0 < p 1 ≤ ∞, 1 ≤ q 0 , q 1 ≤ ∞ and s 0 − s 1 − d( where κ is given by (8).On the other hand if we take α such that α + 1 where the last inequality was proved in [23] with κ given by (8).Now the lemma follows from (35)-(38). 2