Polynomials with small norm on compact Riemannian homogeneous manifolds
aa r X i v : . [ m a t h . F A ] A p r Polynomials with small norm on compactRiemannian homogeneous manifolds
A. KushpelDepartment of Mathematics,University of LeicesterOctober 9, 2018
Abstract
Let H k , k ≥ ≤ θ ≤ θ ≤ · · · ≤ θ n ≤ · · · of the Laplace-Beltrami op-erator ∆ on a compact Riemannian homogeneous manifold M d with thenormalized invariant measure ν and T n = ⊕ nk =0 H k . We consider theproblem of existence of polynomials t n ∈ T n with small norm. Namely,we show that for any ǫ ∈ (0 ,
1) and any subspace L m ⊂ T n , dim L m ≥ ǫn ,there exists such t n ∈ L m that k t n k L p ( M d ,ν ) ≤ C p,q k t n k L q ( M d ,ν ) , where C p,q depends just on p and q , 1 < q < p < ∞ . In the case p = ∞ or q = 1 an extra logarithmic factor appears. This range of problemshas been extensively studied by many authors in the case M d = T , theunit circle (or compact Abelian group G ), i.e., when the characters of G are bounded by 1. In general, on compact Riemannian homogeneousmanifolds, the eigenfunctions of the Laplace-Beltrami operator are notuniformly bounded that creates difficulties of a fundamental nature inapplications of known methods and results. The method, we develop, isbased on a geometric inequality between norms induced by two convexbodies in R n . MSC:
Keywords:
Homogeneous manifold, volume, L´evy mean, flat polynomial.
The range of problems we consider in this article has been traditionally studiedin the context of random Fourier series and has been initiated in the classicalworks of Paley and Zygmund. In many situations it is difficult or impossible togive explicitly an example of a certain object having a required property andfrequently one gets by with Lebesgue measure [11], [12]. Problems regarding flatpolynomials with coefficients ± L ( T )1orm has attracted a lot of attention [4], [2]. It was shown in [22] that for any N ∈ N there is a sequence ǫ n = ±
1, 1 ≤ n ≤ N such that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N X k =1 ǫ n e inθ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < N / . This topic has been developed in [13]. It was shown that for all | z | = 1 there isa sequence of polynomials P ( z ) = P m =1 a m,n z m , | a m,n | = 1 such that(1 − ǫ N ) N / ≤ | P N ( z ) | ≤ (1 + ǫ N ) N / , where ǫ N ≤ CN − / (log N ) / as N → ∞ . The expected L p ( T ) norm ofrandom trigonometric polynomials q N ( θ ) = P Nk =0 X k e ikθ , where X k , k ≥ E ( k q N k pp ) N p/ → Γ (cid:16) p (cid:17) , N → ∞ . The problem of existence of trigonometric polynomials with special propertiesof degree ≤ ( M + 1)(1 + ǫ ), ǫ >
0, in any subspace of L ( T ) of codimension M was considered in [14]. It was shown that for any ǫ > c ǫ M / .In this article we consider the problem of existence of polynomials on acompact homogeneous Riemannian manifold M d whose L p ( M d ) norm is closeto their L ( M d ) norm for any 1 ≤ p ≤ ∞ (see Theorem 3).The method’s possibilities are not confined to the theorem proved in theSection 4 but can be used in studying more general problems. The results wederive are apparently new even in the one dimensional case. First we give a general definition of function spaces that we consider and thenpresent various important examples.
Definition 1.
Given a measure space (Ω , ν ) . Let Ξ = { ξ k } k ∈ N be aset of orthonormal functions in L (Ω , ν ) . Suppose that there exists a sequence { k j } j ∈ N , k = 1 , such that for any j ∈ N and some C > k j +1 − X k = k j | ξ k ( x ) | ≤ Cd j a.e. on Ω , where d j = k j +1 − k j . Then we say that (Ω , ν, Ξ , { k j } j ∈ N ) ∈ K . L p = L p (Ω , ν ) be the usual set of p -integrable functions on Ω. Supposethat (Ω , ν, Ξ , { k j } j ∈ N ) ∈ K . Since all the functions ξ k are a.e. bounded on Ω,then for an arbitrary function φ ∈ L p , 1 ≤ p ≤ ∞ we can construct the sequence { c k ( φ ) } k ∈ N , where c k ( φ ) = R Ω φξ k dν and consider the formal series φ ∼ ∞ X l =1 k l +1 − X k l c k ( φ ) ξ k . The family K is sufficiently large. We consider compact, connected, orientable, d -dimensional C ∞ Riemannian manifold M d , with C ∞ metric. Let g its metrictensor, ν its normalized volume element and ∆ its Laplace-Beltrami operator.In local coordinates x l , 1 ≤ l ≤ d ,∆ = − ( g ) − / X k ∂∂x k X j g jk ( g ) / ∂∂x j . Here, g ij = g ( ∂/∂x i , ∂/∂x j ), g = | det( g ij ) | and ( g ij ) = ( g ij ) − . It is well-knownthat ∆ is an elliptic, self adjoint, invariant under isometries, second-order op-erator. The eigenvalues θ k , k ≥ ≤ θ ≤ θ ≤ · · · ≤ θ n ≤ · · · with + ∞ the only accu-mulation point. Corresponding eigenspaces H k , k ≥ d k = dim H k < ∞ , k ≥
0, orthogonal and L ( M d , ν ) = ⊕ ∞ k =0 H k . Let { Y km } d k m =1 be an orthonormal basis of H k , H k = lin { Y km } d k m =1 .Recall that a Riemannian manifold M d is called homogeneous is its groupof isometries G acts transitively on it. Let H j , j ≥ d j = dim H j , f , · · · f d j any orthonormal basis of H j , then d j X s =1 | f s ( x ) | = d j , for any x ∈ M d (see, e.g., [8]). Hence, any compact, connected, orientable, d -dimensional C ∞ , homogeneous Riemannian manifold M d , with C ∞ metric hasthe property K . Here we give several important examples of such manifolds: A Grassmannian (Grassmann manifold), G m,n ( R ) is the space of all m-dimensional subspaces of R n . Grassmann manifold also appear as coset space G m,n ( R ) = O( n ) / O( n − m ) × O( m ); A complex Grassmannian manifold G m,n ( C ) is the space of all m -dimensionalcomplex subspaces in C n ; An n -torus, T d is defined as a product of n circles: T d = S × · · · S . The n -torus can be described as a quotient of R n under shifts in any coordinate.That is, the n -torus is R n modulo the action of the integer lattice Z n (with theaction being taken as vector addition); The Stiefel manifold, denoted V k ( R d ) or V k,d , is the set of all orthonormal k -frames in R d . That is, it is the set of ordered k -tuples of orthonormal vectors3n R d . When k = 1, the manifold V ,d is just the set of unit vectors in R d ; thatis, V ,d is diffeomorphic to the d − S d − . At the other extreme, when k = d , the Stiefel manifold V d,d is the set of all ordered orthonormal bases for R d . V d,d is a principal homogeneous space for O( d ) and therefore diffeomorphicto it. In general, the orthogonal group O( d ) acts transitively on V k,d with sta-bilizer subgroup isomorphic to O( d − k ). Therefore V k,d can be viewed as thehomogeneous space V k,d = O( d ) / O( d − k ) . The unit complex sphere in C d is defined as S d C = { z ∈ C d | h z, z i = 1 } , where h z, w i = z w + · · · z d w d , z, w ∈ C d . A Riemannian manifold is two-point homogeneous if for any set of fourpoints x , y , x , y with d ( x , y ) = d ( x , y ), d being the Riemannian metricon M d , there exists φ ∈ G such that φ ( x ) = x and φ ( y ) = y . A completeclassification of the two-point homogeneous spaces was given in [23]. For infor-mation on this classification see, e.g., [6, 7, 9, 10]. They are: the spheres S d , d = 1 , , , . . . ; the real projective spaces P d ( R ), d = 2 , , , . . . ; the complex pro-jective spaces P d ( C ), l = d/ d = 4 , , , . . . ; the quaternionic projective spaces P d ( H ), d = 8 , , . . . ; the Cayley elliptic plane P (Cay). The superscripts heredenote the dimension over the reals of the underlying manifolds M d . Let α = ( α , · · · , α n ) ∈ R n , β = ( β , · · · , β n ) ∈ R n and h α, β i = P nk =1 α k β k .Let k α k (2) = h α, α i / be the Euclidean norm on R n , S n − = { α ∈ R n : k α k (2) = 1 } be the unit sphere in R n , B n (2) = { α ∈ R n : k α k (2) ≤ } bethe unit ball in R n and Vol n be the standard n -dimensional volume of subsetsin R n . Let us fix a norm k · k on R n and denote by E the Banach space E = ( R n , k · k ) with the ball B E = V . The L´evy mean M V is defined by M = M ( R n , k · k ) = Z S n − k α k dµ ( α ) . For a convex centrally symmetric body V ⊂ R n we define the polar body V o of V as V o = ( α ∈ R n : sup β ∈ V |h α, β i| ≤ ) . The dual space E o = ( R n , k · k o ) is endowed with the norm k α k V o = k α k o = sup β ∈ B E |h α, β i| and B E o = V o . Theorem 1.
Let V and W be any convex symmetric bodies in R n , V ⊂ B n (2) , then for any n ∈ N and ǫ > there is such < µ ǫ < that in any subspace L m ⊂ R n , m = dim L m ≥ µ ǫ n there is such α ∗ ∈ L m that k α ∗ k V k α ∗ k W ≤ C ǫ ( M V ) ǫ M W o . here C ǫ depends just on ǫ . Proof
From the Urysohn inequality (see, e.g., [21] p. 6-7) it follows that
Vol n ( V )Vol n ( B n (2) ) ! /n ≤ C Z S n − k α k V o dµ ( α ) = C M V o ≤ C (cid:18)Z S n − k α k V o dµ (cid:19) / = C M V o or Vol n ( V o ) ≤ C n ( M V ) n Vol n ( B n (2) ) . Comparing the last estimate with the Bourgain-Milman inequality [1] p. 320,
Vol n ( V ) · Vol n ( V o ) (cid:16) Vol n (cid:16) B n (2) (cid:17)(cid:17) /n ≥ C , we get Vol n ( V ) ≥ (cid:18) C C M V (cid:19) n Vol n ( B n (2) ) . (1)Let W be any convex symmetric body in R n . Using isoperimetric inequalityon S n − it is possible to show [1] that for every 0 < λ < L m ⊂ R n with m = dim L m ≥ λn such that for any α ∈ L m we have k α k (2) ≤ C M W o (1 − λ ) k α k W . (2)Let L m ⊂ R n be any m -dimensional subspace. Assume that m + m > n , sothat L m ∩ L m = ∅ and m := dim( L m ∩ L m ) ≥ m + m − n. Let ( L m ∩ L m ) ⊥ be the orthogonal complement of L m ∩ L m and P ( L m ∩ L m ) ⊥ ( V )be the orthogonal projection of V onto ( L m ∩ L m ) ⊥ . Assume that V ⊂ B n (2) ,then P ( L m ∩ L m ) ⊥ ( V ) ⊂ P ( L m ∩ L m ) ⊥ (cid:16) B m (2) (cid:17) and Vol m (cid:16) P ( L m ∩ L m ) ⊥ ( V ) (cid:17) ≤ Vol m (cid:16) P ( L m ∩ L m ) ⊥ (cid:16) B m (2) (cid:17)(cid:17) . Hence,Vol n ( V ) = Z V dx = Z P ( Lm ∩ Lm ⊥ ( V ) Vol m ( V ∩ ( y + L m ∩ L m )) dy. Thus, for any y ∈ P ( L m ∩ L m ) ⊥ ( V ) by the Brunn-Minkowski theoremVol m ( V ∩ ( y + L m ∩ L m )) ≤ Vol m ( V ∩ ( L m ∩ L m )) . n ( V ) ≤ Vol m ( V ∩ ( L m ∩ L m )) · Vol n − m (cid:16) P ( L m ∩ L m ) ⊥ ( V ) (cid:17) ≤ Vol m ( V ∩ ( L m ∩ L m )) · Vol n − m ( B n − m (2) ) (3)Comparing (1) and (3) we find that for any convex symmetric body V ⊂ B n (2) and any m -dimensional subspace L m ∩ L m ⊂ R n ,Vol m ( V ∩ ( L m ∩ L m )) ≥ (cid:18) C C M V (cid:19) n Vol n ( B n (2) )Vol n − m ( B n − m (2) ) . (4)Applying the Santalo inequality (see, e.g. [5])Vol m ( V ∩ ( L m ∩ L m )) · Vol m (( V ∩ ( L m ∩ L m )) o )(Vol m ( B m (2) )) ≤ m (( V ∩ ( L m ∩ L m )) o ) ≤ (Vol m ( B m (2) )) Vol m ( V ∩ ( L m ∩ L m )) . Combining this result with the Bieberbach inequality (see, e.g., [5])2 m Vol m ( B m (2) )(diam( V ∩ ( L m ∩ L m ))) − m ≤ Vol m (( V ∩ ( L m ∩ L m )) o ) , we get the lower bound for the diameter of the set V ∩ ( L m ∩ L m ),diam( V ∩ ( L m ∩ L m )) ≥ Vol m ( B m (2) )Vol m (( V ∩ ( L m ∩ L m )) o ) ! /m ≥ Vol m ( V ∩ ( L m ∩ L m ))Vol n ( B n (2) ) ! /m . (5)Comparing (4) and (5) we finddiam( V ∩ ( L m ∩ L m )) ≥ (cid:18) C C M V (cid:19) n/m ω n,m , (6)where ω n,m := Vol n ( B n (2) )Vol n − m ( B n − m (2) ) · Vol m ( B m (2) ) ! /m . Recall that Vol n ( B n (2) ) = 2 π n/ Γ (cid:16) n (cid:17) .
6t means that ω n,m can be expressed as ω n,m = (cid:18) Γ( n/ n − m ) / m / (cid:19) /m . It is well-known thatΓ( z ) = z z − / e − z (2 π ) / ǫ n , lim n →∞ ǫ n = 1 , so that for any 1 ≤ m ≤ n , n → ∞ we have ω n,m = ( n/ n/ / eǫ n (( n − m ) / ( n − m ) / / ( m / m / / (2 π ) / ǫ n − m ǫ m ! /m = (cid:18) ( n + 2) n/ / ( n − m + 2) ( n − m ) / / ( m + 2) m / / (cid:19) /m (cid:18) / eǫ n π / ǫ n − m ǫ m (cid:19) /m = n / m / / (2 m )3 × (1 + 2 /n ) n/ (2 m )+1 / (2 m ) (1 − m /n + 2 /n ) n/ (2 m ) − / / (2 m ) (1 + 2 /m ) / / (2 m ) × (cid:18) / eǫ n π / ǫ n − m ǫ m (cid:19) /m ≍ (cid:18) nm (cid:19) / n / (2 m ) . (7)Remark that if m = λn , 0 < λ <
1, then ω n,m ∼ ( λe ) − / as n → ∞ .From (6) it follows that for any L m ⊂ R n there is such α ∗ ∈ L m that k α ∗ k (2) ≥ (cid:18) C C M V (cid:19) n/m ω n,m k α ∗ k V . (8)Recall that m = dim( L m ∩ L m ). Since α ∗ ∈ L m then from (2) we get k α ∗ k (2) ≤ C M W o (cid:18) nn − m (cid:19) k α ∗ k W . (9)Finally, comparing (8) and (9) we find k α ∗ k V ≤ (cid:18) C M V C (cid:19) n/m C M W o ω n,m (cid:18) nn − m (cid:19) k α ∗ k W . (10)In particular, let m = µ n and m = µ n for some fixed µ > µ > < µ + µ <
2, then from (7) and (10) it follows that k α ∗ k V ≤ C ( M V ) / ( µ + µ − M W o k α ∗ k W , where C >
Flat Polynomials on M d Let Ω be a compact space with a normalized measure ν , Fix an orthonormal sys-tem Ξ = { ξ k } k ∈ N ⊂ L (Ω , ν ) and a sequence { k j } j ∈ N such that (Ω , ν, Ξ , { k j } j ∈ N ) ∈K . LetΞ j := lin { ξ k } k j +1 − k = k j , Ω m := { j , · · · , j m } , Ξ(Ω m ) := lin { Ξ j s } ms =1 . Put n := dim Ξ(Ω m ) = P ms =1 k j s +1 − k j s = P ms =1 d j s , where d j s := dim Ξ j s .Consider the coordinate isomorphism J : R n → Ξ(Ω m )that assigns to α = ( α · · · , α n ) ∈ R n the function Jα = ξ α = P nl =1 α l ξ j l ∈ Ξ(Ω m ). Let X be a Banach space such that Ξ(Ω m ) ⊂ X for any Ω m ⊂ N . Put X n = Ξ(Ω m ) ∩ X . The definition k α k ( X n ) = k ξ α k X induces a norm on R n . Put B n ( X n ) := { α | α ∈ R n , k α k ( X n ) ≤ } , then B nX n := JB n ( X n ) .A Banach lattice X is q -concave, q < ∞ if the there is a constant C q > n X i =1 k x i ( · ) k qX ! /q ≤ C q ( X ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 | x i ( · ) | q ! /q (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X for any n ∈ N and any sequence { x i ( · ) } ni =1 ⊂ X (see, e.g., [20], p. 46).We will need the following statement. Lemma 1.
For any ξ ∈ Ξ(Ω m ) , m ∈ N we have k ξ k L p (Ω ,ν ) ≤ Cn (1 /p − /q ) + k ξ k L q (Ω ,ν ) , where ≤ p, q ≤ ∞ and n := dim Ξ(Ω m ) . Proof
Consider the function K n ( x, y ) := X ξ k ∈ Ξ(Ω m ) ξ k ( x ) ξ k ( y ) . Clearly, K n ( x, y ) = Z Ω K n ( x, z ) K n ( z, y ) dν ( z )and K n ( x, y ) = K n ( y, x ). Hence, k K n ( · , · ) k L ∞ (Ω ,ν ) ≤ k K n ( y, · ) k L (Ω ,ν ) k K n ( x, · ) k L (Ω ,ν ) for any x, y ∈ Ω and k K n ( x, · ) k L (Ω ,ν ) ≤ Cn / , since (Ω , ν, Ξ , { k j } j ∈ N ) ∈ K . Itmeans that k K n ( · , · ) k L ∞ (Ω ,ν ) ≤ Cn. (11)8et ξ ∈ Ξ(Ω m ), then applying H¨older inequality and (11) we get k ξ k L ∞ (Ω ,ν ) ≤ k K n ( · , · ) k L ∞ (Ω ,ν ) k ξ k L (Ω ,ν ) , or k I k L (Ω ,ν ) ∩ Ξ(Ω m ) → L ∞ (Ω ,ν ) ∩ Ξ(Ω m ) ≤ Cn, where I is an embedding operator. Trivially, k I k L p (Ω ,ν ) ∩ Ξ(Ω m ) → L p (Ω , ν ) ∩ Ξ(Ω m ) = 1, 1 ≤ p ≤ ∞ . Hence, using Riesz-Thorin interpolation Theorem andan embedding arguments for any ξ ∈ Ξ(Ω m ) we obtain k ξ k L p (Ω ,ν ) ≤ Cn (1 /p − /q ) + k ξ k L q (Ω ,ν ) , ≤ p, q ≤ ∞ , where ( a ) + := (cid:26) a, a ≥ , , a < . In the case Ω m = { , · · · , m } the estimates of respective L´evy means havebeen obtained in [19]. Using Lemma 1 we can generalize our result to an arbi-trary index set Ω m . Theorem 2.
Let (Ω , ν, Ξ , { k l } ) ∈ K and X is a -concave, then for anarbitrary Ω m , M ( R n , k · k ( X n ) ) ≤ C X , X n = X ∩ Ξ(Ω m ) (12) where n := dimΞ(Ω m ) and C X > is independent on n ∈ N . In particular, M ( R n , k · k ( L p (Ω ,ν ) ∩ Ξ(Ω m )) ) ≤ C (cid:26) p / , p < ∞ , (log n ) / , p = ∞ , (13) where C > is an absolute constant. Remark that different estimates of L´evy means have been obtained in [15] -[18]. We are prepared now to prove main result of this article.
Theorem 3.
Assume that max { M j − ◦ ( X ∩ Ξ n ) , M j − ◦ ( Y ∩ Ξ n ) o } < C for any n ∈ N and some absolute constant C > and B X ⊂ B L ( M d ) . Then in anysubspace J − ◦ L s ⊂ Ξ(Ω m ) there exists such polynomial t ∗ n that k t ∗ n k X ≤ C X,Y k t ∗ n k Y . (14) In particular, the inequality (14) is valid if (Ω , ν, Ξ , { k j } j ∈ N ) ∈ K , X is -concave and k · k ( B Y ∩ Ξ n ) o ≤ k · k J − ◦ ( Y ∩ Ξ n ) for some 2-concave Y and any n ∈ N . Let (Ω , ν, Ξ , { k j } j ∈ N ) ∈ K , X = L p (Ω , ν ) , Y = L q (Ω , ν ) , ≤ p, q ≤ ∞ ,then for an arbitrary spectrum Ω m , n = dim Ξ(Ω m ) and any subspace J − ◦ L s ⊂ Ξ(Ω m ) there exists a polynomial t ∗ n ∈ J − ◦ L s such that k t ∗ n k L p (Ω ,ν ) ≤ C̺ n k t ∗ n k L q (Ω ,ν ) , where ̺ n = , < q, p < ∞ , (log n ) / , ≤ q ≤ p < ∞ , (log n ) / , < q ≤ p ≤ ∞ , log n, ≤ q ≤ p ≤ ∞ nd C > is an absolute constant. Proof
Applying Theorem 2 and Theorem 1 for a fixed ǫ ∈ (0 ,
1) and theinequality k · k ( B Y ∩ Ξ n ) o ≤ k · k Y ∩ Ξ n , where Y is a 2-concave, we get k α ∗ k V k α ∗ k W ≤ CM ǫV M W o ≤ C X,Y Hence, using (12), for any L s ⊂ R n , s = dim L s ≥ ǫn , ǫ ∈ (0 , α ∗ ∈ L s that k α ∗ k V ≤ C X,Y k α ∗ k W . It means that in any subspace J ◦ L s there exists such t ∗ n ∈ J ◦ L s ⊂ Ξ(Ω m ) that k t ∗ n k X ≤ C X,Y k t ∗ n k Y . In the case X = L p (Ω , ν ) and X = L q (Ω , ν ) we use (13) to get a similar estimate k t ∗ n k L p (Ω ,ν ) ≤ C̺ n k t ∗ n k L q (Ω ,ν ) , where C >
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