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Entropy of Sobolev's classes on Compact Homogeneous Riemannian Manifolds

We develop a general method to calculate entropy numbers of standard Sobolev's classes on an arbitrary compact homogeneous Riemannian manifold. Our method is essentially based on a detailed study of geometric characteristics of norms induced by subspaces of harmonics. The method's possibilities are not confined to the statements proved but can be applied in studying more general problems such as entropy of multiplier operators. As an application, we establish sharp orders of entropy of Sobolev's classes and respective n-widths.


Introduction
Let (Ω, ν) be a measure space and {ξ k } k∈N be a sequence of orthonormal, functions on Ω. Let X be a Banach space with the norm · X and {ξ k } k∈N ⊂ X. Clearly, Ξ n (X) := lin{ξ 1 , · · · , ξ n } ⊂ X, ∀n ∈ N is a sequence of closed subspaces of X with the norm induced by X. Consider the coordinate isomorphism J defined as J : R n −→ Ξ n (X) α = (α 1 , · · · , α n ) −→ n k=1 α k ξ k . Hence, the definition α J −1 Ξn(X) = Jα X induces the norm on R n which appears to be useful in various applications. Of course, not much can be said regarding such kind of norms even in lower dimensions. To be able to apply methods of geometry of Banach spaces to various open problems in different spaces of functions on Ω we will need to calculate an expectation of the function ρ n (α) := α J −1 Ξn(X) on the unit sphere S n−1 ⊂ R n with respect to the invariant probabilistic measure dµ n , i.e., to find the Lévy mean M ( · J −1 Ξn(X) ) = S n−1 α J −1 Ξn(X) dµ n (α).
Observe that the sequence of Lévy means M ( · J −1 Ξn(X) ) contain more information then the sequence of volumes Vol n B J −1 Ξn(X) , n ∈ N, where B J −1 Ξn(X) := {α| α ∈ R n , α J −1 Ξn(X) ≤ 1} is the unit ball induced by the norm · J −1 Ξn(X) and therefore is more useful in various applications.
As a motivating example consider the case Ω = M d , where M d is a compact homogeneous Riemannian manifold, ν its normalized volume element, {ξ k } k∈N is a sequence of orthonormal harmonics on M d and X = L p = L p (M d , ν), p ≥ 2. In general, the sequence {ξ k } k∈N is not uniformly bounded on M d . Hence, the method of estimating of Lévy means developed in [10] - [12] can not give sharp order result. Various modifications of this method presented in [14] - [16] give an extra (log n) 1/2 factor even if p < ∞. Our general result concentrated in Lemma 3.2 gives sharp order estimates for the Lévy means which correspond to the norm induced on R n by the subspace ⊕ m s=1 H ks ∩ L p , dim ⊕ m s=1 H ks := n with an arbitrary index set (k 1 , · · · , k m ), where H ks are the eigenspaces of the Laplace-Beltrami operator for M d defined by (2.2). To show the boundness of the respective Lévy means as n → ∞ we impose a technical condition (2.1) which holds in particular for any compact homogeneous Riemannian manifold because of the addition formula (2.3) and employ the equality 1] is the sequence of Rademacher functions [19], [18]. To extend our estimates to the case p = ∞ we apply Lemma 3.1 which gives a useful inequality between 1 ≤ p, q ≤ ∞ norms of polynomials on M d with an arbitrary spectrum. It seems that the factor (log n) 1/2 obtained in Lemma 3.2 is essential because of the lower bound for the Lévy means found in [9] in the case of trigonometric system. This fact explains a logarithmic slot in our estimates of entropy numbers presented in Theorem 3.12. Section 3 deals with estimates of entropy numbers and n-widths. Theorem 3.3 establishes general lower bounds for entropy numbers in terms of Lévy means and is of independent interest. We derive lower bounds for the entropy numbers of Sobolev's classes (3.9) using Theorem 3.3 and estimates of Lévy means given by Lemma 3.2. At this point we apply Lemma 2.2 to get the dependence between eigenvalues and dimensions of eigenspaces of the Laplace-Beltrami operator. The proof of Lemma 2.2 is based on Weyl's formula (see [21]) where V (M d ) is the volume of M d and n(a) is the number of eigenvalues (each counted with its multiplicity) smaller than a. To get upper bounds for entropy numbers contained in Theorem 3.12 we apply estimates of Lévy means established in Lemma 3.2 and make use of the Pajor-Tomczak-Jaegermann inequality [22] which states in our notations that for any λ ∈ (0, 1) there exists a subspace X s ⊂ J −1 Ξ n (X), dim X s = s > λn and a universal constant C > 0 such that where | · | is the Euclidean norm on R n and · o J −1 Ξn(X) is the dual norm with respect to · J −1 Ξn(X) . Remark that (1.2) is essentially based on a technical result due to Gluskin [8]. Hawever, for our applications is sufficient to apply a less sharp result established by Bourgain and Milman [3] which is based on averaging arguments and isoperimetric inequality.
The paper ends with estimates of different n-widths and their applications in calculation of entropy which extend previous results [1], [2], [16].
In this article there are several universal constants which enter into the estimates. These positive constants are mostly denoted by C, C 1 , .... We will only distinguish between the different constants where confusion is likely to arise, but we have not attempted to obtain good estimates for them. For ease of notation we will write a n ≪ b n for two sequences, if a n ≤ Cb n , ∀n ∈ N and a n ≍ b n , if C 1 b n ≤ a n ≤ C 2 b n , ∀n ∈ N and some constants C, C 1 and C 2 . Also, we shall put (a) + := max{a, 0}.
Though the main purpose of this paper is to present new results, we have tried to make the text selfcontained by presenting well known definitions and elementary properties of entropy numbers and n-widths.
Let X and Y be Banach spaces with the closed unit balls B X and B Y respectively. Let v : X → Y be a compact operator. Then the nth entropy number e n (v) = e n (v : X → Y ) is the infimum of all positive ǫ such that there exist y 1 , · · · , Similarly, for a compact set A ⊂ Y we define the entropy number e n (A, Y ) as the infimum of all positive ǫ such that there exist Suppose that A is a convex, compact, centrally symmetric subset of a Banach space X with unit ball B X . The Kolmogorov n-width of A in X is defined by d n (A, X) := d n (A, B X ) := inf Xn sup f ∈A inf g∈Xn f − g X , where X n runs over all subspaces of X of dimension n. The Gel'fand n-width of A in X is defined by d n (A, X) := d n (A, B X ) := inf L n sup x∈L n ∩A x X , where L n runs over all subspaces of X of codimension n. The Bernstein n-width of A in X is defined by b n (A, X) := b n (A, B X ) := sup Xn+1 sup{ǫ > 0 : ǫB ∩ X n+1 ⊂ A}, where X n+1 is any (n + 1)-dimensional subspace of X. For a compact operator v : X → Y we define Kolmogorov's numbers and Gelfand's numbers Proposition 1.1. This proposition records some simple properties of n-widths and entropy numbers.
(c) Kolmogorov and Gel'fand n-widths are dual. Let X and Y be Banach , [20], p.408). (d) Later we will wish to restrict estimation of entropy numbers over infinitedimensional sets to finite-dimensional sets. In order to do this let i be any linear isometry, i : Y →Ỹ (here we will think of Y as finite dimensional and i as the imbedding into the infinite dimensional space). Then ([24,

Elements of Harmonic Analysis on Compact Riemannian Homogeneous Manifolds
Definition 2.1. Let (Ω, ν) be a measure space for some compact set Ω ∈ R s , s ∈ N. Let Ξ = {ξ k } k∈N be a set of orthonormal functions in L 2 (Ω, ν). Suppose that there exists a sequence κ = {k j } j∈N , k 1 = 1, such that for any j ∈ N and some C > 0 Consider the set of p-integrable functions on (Ω, ν), L p = L p (Ω, ν). It follows from (2.1) that the functions ξ k are a.e. bounded for every n ∈ N. Hence, for an arbitrary φ ∈ L p , 1 ≤ p ≤ ∞ it is possible to construct the Fourier coefficients and consider the formal Fourier series Let U p := {φ| φ p ≤ 1} be the unit ball in L p , and Λ = {λ l } l∈N be a fixed sequence of complex numbers. We shall say that the multiplier operator Λ is of type (κ, p, q) with the norm Λ κ p,q := sup φ∈Up Λφ q , , if for any Let us present here several important examples of measure spaces (Ω, ν, Ξ, κ) ∈ K. Consider a compact, connected, d-dimensional C ∞ Riemannian manifold M d with C ∞ metric. Let g its metric tensor, ν its normalized volume element and ∆ its Laplace-Beltrami operator. In local coordinates x l , 1 ≤ l ≤ d, where g jk := g(∂/x j , ∂/x k ), g := |det(g jk )|, and (g jk ) := (g jk ) −1 . It is wellknown that ∆ is an elliptic, self adjoint, invariant under isometry, second order operator. The eigenvalues θ k , k ≥ 0, of ∆ are discrete, nonnegative and form an increasing sequence 0 ≤ θ 0 ≤ θ 1 ≤ · · · ≤ θ n ≤ · · · with +∞ the only accumulation point. The corresponding eigenspaces The Sobolev classes W γ p are defined as sets of functions with formal Fourier expansions (??) where φ p ≤ 1 and M d φdν = 0. In this article we assume c 0 , c = 0 to guarantee compactness of the set W γ p in L q . We recall that a Riemannian manifold M d is called homogeneous if its group of isometries G acts transitively on it, i.e. for every x, y ∈ M d , there is a g ∈ G such that gx = y. For a compact homogeneous Riemannian manifold M d the following addition formula is known [7] is an arbitrary orthonormal basis of H k , k ≥ 0. Hence, any such manifold possesses the property K, and these include real and complex Grassmannians, the n-torus, the Stiefel manifold, two point homogeneous spaces (spheres, the real, complex and quaternionic projective spaces and the Cayley elliptic plane), and the complex sphere.
Proof. Applying Weyl's formula (1.1) for a = θ N we get and it follows that Since both quotients in the last equation are positive we have using the second quotient in (2.7), and (2.4).
In what follows, a * will be used to denote norms and balls in Euclidean space, and lack of a * will indicate the same quantities in function spaces. Let us define the norms α * Xn = ξ α Xn := ξ α X . Put B * Xn := {α ∈ R n , α * Xn ≤ 1}, and B Xn := JB * Xn .
Proof. Let be the reproducing kernel for Ξ(Ω m ). Clearly, and K n (x, y) = K n (y, x). Hence, using the Cauchy-Schwarz inequality, for any x, y ∈ M d . Since (Ω, ν, Ξ, κ) ∈ K, from (2.1), we have K n (x, ·) 2 ≤ Cn 1/2 . Therefore, Let ξ ∈ Ξ(Ω m ). Then applying Hölder inequality and (3.1) we get Let us fix a norm · on R n and let E = (R n , · ) be a Banach space with the unit ball B E . The dual space E o = (R n , · o ) is endowed with the norm ξ o = sup σ∈BE | ξ, σ | and has the unit ball B E o . In these notations the Lévy mean M BE is where dµ denotes the normalized invariant measure on S n−1 , the unit sphere in R n . We are interested in the case where · = α * In,Lp . In the case Ω m = {1, · · · , m} the estimates of the associated Lévy means were obtained in [18]. Using Lemma 3.1 we can straightforwardly generalize this result to be valid for an arbitrary index set Ω m . This we state as We can now give lower bounds for entropy in terms of Lévy means. In the following the reader should be identifying the spaces X and Y with L p and L q respectively for some 1 ≤ p, q ≤ ∞. However, we wished to state the result in greater generality, and then apply the previous result to extract particular results in these cases. Theorem 3.3. Viewing Ξ(Ω m ) as a subspace X n ⊂ X and Y n ⊂ Y , we have where k, n ∈ N are arbitrary.
Proof. First, we use Proposition 1.1 (d) to obtain the estimate , using the appropriate norms in X n and Y n . Let ϑ 1 , · · · , ϑ N (ǫ) be a minimal ǫ-net for Λ n (B * Xn ) in B * Yn , R n . Then, By comparing volumes we get Yn . If we put N (ǫ) = 2 k−1 , then from the last inequality and the definition of entropy numbers we obtain Also, a direct calculation shows that Remark 3.4. Assume that |λ 1 | ≥ · · · ≥ |λ n |. Then | det Λ n | 1/n ≥ |λ n |, and for k = n we have Remark 3.5. Let (Ω, ν, Ξ, κ) ∈ K, X = L p and Y = L q , 1 ≤ q ≤ 2 ≤ p ≤ ∞. Then using Hölder's inequality we get where 1/q + 1/q ′ = 1. Comparing Lemma 3.2 with (3.6) and (3.7) we find Remark 3.6. Remind that τ N = dim T N . Put n = τ N andñ = τ N +1 in (3.8). Then in the case of Sobolev's classes λ n = θ From (3.8) it follows that where the last step follows from Lemma 2.2. Since the sequence of entropy numbers is not increasing, then where we have used (2.6) in the third inequality. From the last inequality we get the following lower bounds for the entropy of Sobolev's classes W γ p in L q , Remark 3.7. From [18,Theorem 4] it follows that Let φ ∈ L p , 2 ≤ p ≤ ∞ and 1 ≤ q ≤ 2. Then I γ φ q ≤ I γ φ 2 ≤ C φ 2 ≤ C φ p , i.e., I γ ∈ L(L p , L q ) for any γ > 0. It is easy to check that I γ L p is dense in L q since and L 2 is dense in L q . Also, L p is reflexive if 2 ≤ p < ∞. Hence, for any N ∈ N and 1 < p, q < ∞, from the duality of Kolmogorov and Gel'fand n-widths given by Proposition 1.1 (c), By Lemma 2.2 lim N →∞ θ N +1 /θ N = 1. Thus, θ N +1 /θ N ≤ 2 for some N 0 ∈ N and any N ≥ N 0 . The last estimate can be rewritten as Set m = dim T M . By [22,Theorem 1] there exists a subspace X s ⊂ {R m , · q ′ }, 1 ≤ q ≤ 2, 1/q + 1/q ′ = 1, dimX s = s > λl, 0 < λ < 1, such that Since, by Lemma 3.2, M B * Xs ≤ C 2 , 2 < q ′ < ∞, we have Therefore X s ∩ B * q ⊂ C 4 X s ∩ B * 2 and since the spaces R m and JR m = T M are isometrically isomorphic we get z 2 ≤ C 3 z q , ∀z ∈ JX s ⊂ T M , s ≥ [m/2].
Hence, denoting an arbitrary s-dimensional subspace of T M by Y s , Finally, applying the same line of arguments as in Remark 3.7 we get b n (W γ 2 , L q ) ≥ C −1 3 n −γ/d q > 1.
We now turn to estimates for the upper bounds on entropy and n-widths.
Theorem 3.9. Let 2 ≤ p, q ≤ ∞ and γ > d/2. Then Proof. It is sufficient to consider the case p = 2, since the case p ≥ 2 follows by imbedding. For a given N ∈ N let θ N be the corresponding eigenvalue of the Laplace-Beltrami operator for M d . Put N −1 := 1, N 0 := N and for any k ≥ 0 let N k+1 be such that θ N k+1 −1 ≤ 2 2/γ θ N k ≤ θ N k+1 . This is always possible to do since the sequence of eigenvalues forms an increasing sequence with +∞ as the only accumulation point. By Lemma 2.2, lim k→∞ θ N k+1 /θ N k+1 −1 = 1. Then, a simple argument shows that, lim k→∞ θ N k+1 /θ N k = 2 2/γ .
From here we conclude that there is a δ(k), with δ → 0 as k → ∞, and constants C 1 , C 2 > 0 such that l=N k Ξ l , and dim T N k ,N k+1 = l k . Using (2.6) we get It is easy to check that Thus, by Lemma 3.1 and (3.12), Clearly, P L2→L2∩TN k ,N k+1 = 1, where P is the orthogonal projection. Hence, by (??), Let ǫ > 0 be a fixed parameter which will be specified later Using Proposition 1.1 (a) and (b), (3.13) and (3.14) we find where using the fact that d 0 (U q , L q ) = 1, Using (3.11), Since δ > 0, and for sufficiently large N we can choose δ as small as we please, then the last series converges if γ/d > 1/2 − 1/q. In this case To complete the proof we need to get upper bounds for σ 1 . From (3.10), there exists a subspace L l s ⊂ R q l = {R l , · * q }, dimL l s = s > λl, 0 < λ < 1, such that for any α ∈ L l s . Put m := l − s, then for any z ∈ JL l s . By duality of Kolmogorov's and Gel'fand's n-widths, recalling the definition of m k , and letting X l k m k ⊂ T N k ,N k+1 be an arbitrary subspace of codimension m k , we get where s k = l k − m k , since JL l k s k is a specific subspace of codimension m k . Thus, using Lemma 2.2 and (3.12), from (3.11). Simplifying this last expression, it follows from Lemma 3.2 that Then, using estimate (3.15) and (3.11) again, we get The last sum is bounded for some δ > 0 if γ > d/2, and 0 < ǫ < 2 − d/γ. Thus we must choose ǫ less than the aforementioned and the bound given in (3.16). In this case, Finally, comparing (3.17) and (3.18) we get Remark 3.10. Comparing the above theorem with Remark 3.7, and applying an embedding arguments we get Remark 3.11. From Lemma 3.1 and (3.14) we see that the set W γ 2 is bounded in L q , q ≥ 2 if γ > d(1/2 − 1/q). If γ > d/2 then lim n→∞ d n (W γ 2 , L q ) = 0, by Theorem 3.9. Hence, W γ 2 is relatively compact in L q (see, e.g., [20, p. 402]), or the corresponding operator I γ : L 2 → L q and its conjugate I γ : L q ′ → L 2 are compact if γ > d/2, 1/q + 1/q ′ = 1.
We are prepared now to prove the main result of this article.