Boundedness of Hausdorff operators on Hardy spaces over homogeneous spaces of Lie groups
aa r X i v : . [ m a t h . F A ] F e b Boundedness of Hausdorff operators onHardy spaces over homogeneous spaces of Liegroups
A. R. [email protected]. The aim of this note is to give the boundedness conditions forHausdorff operators on Hardy spaces H with the norm defined via (1 , q )atoms over homogeneous spaces of Lie groups with doubling property and toapply results we obtain to generalized Delsarte operators and to Hausdorffoperators over multidimensional spheres.Key words. Hausdorff operator, Lie group, homogeneous space, Hardyspace, generalized shift operator of Delsarte.MSC classes: 43A85, 47G10, 22E30 One-dimensional Hausdorff operators were introduced by Hardy [1, Section11.18] as a transformations of functions of a continuous variable analogousto the regular Hausdorff transformations for sequences and series. Althoughoccasionally one-dimensional Hausdorff operators appeared before 2000 (see[2] and [3]), the modern development of this theory begins with the work ofLiflyand and M´oricz [4] where Hausdorff operators on one-dimensional Hardyspace were considered. The multidimensional case was studied in [5]. Formore details of the development of the theory of Hausdorff operators up to2014 see [6], and [7].Hausdorff operators on the Hardy space H over homogeneous spaces oflocally compact groups were first introduced by the author in [8] for the caseof doubling measures, and in [9] for the case of locally doubling measures.The case of locally compact groups was considered earlier in [10]. The aimof this note is to improve and generalize results from [8] to the case of Hardyspaces H ( G/K ) with the norm defined via (1 , q ) atoms when G is a Liegroup and to apply results we obtain to generalized Delsarte operators andto Hausdorff operators over multidimensional spheres.1 The main result
Let G be a locally compact metrizable group with left invariant metric ρ andleft Haar measure ν . We assume that the following doubling condition in asense of [11] holds:there exists a constant C such that ν ( B ( x, r )) ≤ Cν ( B ( x, r ))for each x ∈ G and r >
0; here B ( x, r ) denotes the ball of radius r around x. The doubling constant is the smallest constant C ≥ C ν . Then for each x ∈ G, k ≥ r > ν ( B ( x, kr )) ≤ C ν k d ν ( B ( x, r )) , ( D )where d = log C ν (see, e.g., [12, p. 76]). The number d takes the role of a”dimension” for a doubling metric measure space G .Homogeneous group in a sense of Folland and Stein [13] (i.e., a connectedsimply connected Lie group G whose Lie algebra is equipped with dilations)enjoys the doubling condition and C ν = 2 Q , where Q stands for the homoge-neous dimension of G [14, Lemma 3.2.12]. A complete Riemannian manifold,with Ricci curvature nonnegative outside a compact subset of the manifold,satisfies the doubling condition, as demonstrated in [15, Lemma 1.3]. Com-pact Lie groups endowed with Riemann metric and Haar measure satisfy thedoubling condition, too [11, p. 588, Example (7)]. For complete noncompactmanifolds with nonnegative Ricci curvature, the doubling property for thevolume measure follows from the volume comparison inequality of Bishopand Gromov [16, Theorem 10.6.6].We denote by Aut( G ) the space of all topological automorphisms of G endowed with its natural topology T β [17, Ch. X, §
3, n 5], L ( Y ) denotes thespace of linear bounded operators on a normed space Y .Let K be a compact subgroup of G with normalized Haar measure β .Consider the quotient space G/K of left cosets ˙ x := xK = π K ( x ) ( x ∈ G )where π K : G → G/K stands for a natural projection. We shall assume thatthe measure ν is normalized in such a way that (generalized) Weil’s formula Z G g ( x ) dx = Z G/K (cid:18)Z K g ( xk ) dk (cid:19) dλ ( ˙ x ) (1)holds for all g ∈ L ( G ), where λ denotes some left- G -invariant measure on G/K (see [18, Chapter VII, §
2, No. 5, Theorem 2 ] and especially remarkc) after this theorem or [19, Proposition 10.4.12]). Here G -left invariance of2 means that λ ( xE ) = λ ( E ) for every Borel subset E of G/K and for every x ∈ G . This measure is unique up to constant multiplier.Henceforth we write dx instead of dν ( x ) and dk instead of dβ ( k ). Weshall write also d ˙ x instead of dλ ( ˙ x ).The function g : G → C is called right- K -invariant if g ( xk ) = g ( x ) forall x ∈ G , k ∈ K . For such a function we put ˙ g ( ˙ x ) := g ( x ). This definitionis correct and for g ∈ L ( G ) formula (1) implies that Z G g ( x ) dx = Z G/K ˙ g ( ˙ x ) d ˙ x (2)(recall that R K dk = 1).The map g ˙ g is a bijection between the set of all right- K -invariantfunctions on G (all right- K -invariant functions from L ( G )) and the set ofall functions on G/K (respectively functions from L ( G/K, λ )).Let an automorphism A ∈ Aut( G ) maps K onto itself. Since A ( ˙ x ) := A ( xK ) = { A ( x ) A ( k ) : k ∈ K } = A ( x ) K = π K ( A ( x ))we get a homeomorphism ˙ A : G/K → G/K, ˙ A ( ˙ x ) := π K ( A ( x )) . Then forevery right- K -invariant function g on G we have ˙ g ( ˙ A ( ˙ x )) = g ( A ( x )) . We put Aut K ( G ) := { ˙ A : A ∈ Aut( G ) , A ( K ) = K } . A ν -measurable function a on G is called an (1 , q )- atom ( q ∈ (1 , ∞ ]) if(i) the support of a is contained in a ball B ( x, r );(ii) k a k ∞ ≤ ν ( B ( x,r )) if q = ∞ , and k a k q ≤ ν ( B ( x, r )) q − if q ∈ (1 , ∞ ) ;(iii) R G a ( x ) dν ( x ) = 0.In case ν ( G ) < ∞ we shall assume ν ( G ) = 1; in this case the constantfunction having value 1 is also considered to be an atom.Hereafter by atom we mean an (1 , q )-atom on G . Definition 1. [8], [9]. We define the
Hardy space H ( G/K ) = H ,q ( G/K ) as a space of such functions f on G/K that admit an atomic decompositionof the form f = ∞ X j =1 λ j ˙ a j As usual, k · k q denotes the L q norm. It is known that H ,q ( G/K ) does not depend on q ∈ (1 , ∞ ] [11, Theorem A, p. 592].We write H ,q ( G/K ) instead of H ( G/K ) in order to stress the fact that we use the norm k · k H ,q ( G/K ) described below. a j are right- K -invariant (1 , q )-atoms on G and P ∞ j =1 | λ j | < ∞ . In thiscase, k f k H ,q ( G/K ) := inf ∞ X j =1 | λ j | , and infimum is taken over all decompositions above of f .In other words, f = ˙ g where g = P ∞ j =1 λ j a j , a j are right- K -invariant(1 , q )-atoms on G , and P ∞ j =1 | λ j | < ∞ . Moreover, k f k H ,q ( G/K ) = k g k H ,q ( G ) . Remark 1 . Real Hardy spaces over compact connected (not necessaryquasi-metric) Abelian groups were defined in [20].
Proposition 1 . [9].
Let G = K . Then the space H ,q ( G/K ) is nontrivialand Banach. Definition 2. [8]. Let (Ω , µ ) be a measure space, ( ˙ A ( u )) u ∈ Ω ⊂ Aut K ( G )a family of homeomorphisms of G/K , and Φ a measurable function on (Ω , µ ).For a Borel measurable function f on G/K we define a
Hausdorff operatoron
G/K as follows( H Φ , ˙ A f )( ˙ x ) := Z Ω Φ( u ) f ( ˙ A ( u )( ˙ x )) dµ ( u ) . For the proof of our main result the next two lemmas are crucial.
Lemma 1. [10].
Let (Ω , µ ) be σ -compact quasi-metric space with positiveRadon measure µ, ( X, m ) be a measure space and F ( X ) be some Banachspace of m -measurable functions on X . Assume that the convergence of asequence strongly in F ( X ) yields the convergence of some subsequence to thesame function for m -almost all x ∈ X . Let F ( u, x ) be a function such that F ( u, · ) ∈ F ( X ) for µ -almost all u ∈ Ω and the map u F ( u, · ) : Ω → F ( X ) is Bochner integrable with respect to µ . Then for m -almost all x ∈ X (cid:18) ( B ) Z Ω F ( u, · ) dµ ( u ) (cid:19) ( x ) = Z Ω F ( u, x ) dµ ( u ) . Lemma 2.
Let G be a (finite dimensional real or complex) connectedLie group with left invariant Riemann metric ρ . Then every automorphism ϕ ∈ Aut( G ) is Lipschitz with Lipschitz constant k ( dϕ ) e k . Proof. Let T a ( G ) denotes the tangent space for G at the point a ∈ G . Let L a : x ax be the left translation in G . Then the tangent map l a := ( dL a ) e : T e ( G ) → T a ( G ) is a bijection. We fix the Euclidean norm k · k in T e ( G ) andintroduce the norm in T a ( G ) by the rule k X a k := k X e k if X a = l a ( X e ), X e ∈ T e ( G ), a ∈ G .As is well known, for every p, q ∈ Gρ ( p, q ) = inf α Z k α ′ ( t ) k dt α from [0 ,
1] to G with α (0) = p, α (1) = q ( α ′ ( t ) stands, as usual, for the tangent vector to α at the point α ( t )). Since ϕ ∈ Aut( G ), the formula β = ϕ ◦ α gives the generalform of all piecewise smooth curves in G with β (0) = ϕ ( p ) and β (1) = ϕ ( q ).Thus, by the chain rule ρ ( ϕ ( p ) , ϕ ( q )) = inf α Z k ( ϕ ◦ α ) ′ ( t ) k dt = inf α Z (cid:13)(cid:13) ( dϕ ) α ( t ) α ′ ( t ) (cid:13)(cid:13) dt ≤ inf α Z (cid:13)(cid:13) ( dϕ ) α ( t ) kk α ′ ( t ) (cid:13)(cid:13) dt. It is known (see, e.g., [21]) that for every left invariant vector field X on G (this means that X a = l a ( X e ) for all a ∈ G ) the vector field ( dϕ )( X ) isleft invariant, too. In other wards, ( dϕ ) a ( X a ) = l a ( dϕ ) e ( X e ), i.e., ( dϕ ) a = l a (( dϕ ) e ) l − a and therefore k ( dϕ ) a k = k ( dϕ ) e k for all a ∈ G . The resultfollows.Now we are in a position to prove the next Theorem 1.
Let G be a (finite dimensional real or complex) connectedLie group with left invariant Riemann metric ρ and left Haar measure ν suchthat the space ( G, ρ, ν ) is doubling. Let (Ω , µ ) be σ -compact quasi-metricspace with positive Radon measure µ , and let q ∈ (1 , ∞ ] . If k Φ k A,q := Z Ω | Φ( u ) | (mod A ( u )) − q k ( u ) (1 − q ) d dµ ( u ) < ∞ where k ( u ) := k ( d ( A ( u ) − )) e k , then the operator H Φ , ˙ A is bounded on thespace H ,q ( G/K ) and kH Φ , ˙ A k L ( H ,q ( G/K )) ≤ C − q ν k Φ k A,q . Proof. If we set X = G/K and m = λ the pair ( X, m ) satisfies theconditions of Lemma 1 with H ,q ( G/K ) in place of F ( X ). Indeed, let f n =˙ g n ∈ H ,q ( G/K ), f = ˙ g ∈ H ,q ( G/K ), and k f n − f k H ,q ( G/K ) → n → ∞ ).Since k f n − f k L ( G/K ) = Z G/K | π K ( g n − g ) | dλ = Z G | g n ( x ) − g ( x ) | dx ≤ k g n − g k H ,q ( G ) = k f n − f k H ,q ( G/K ) → k a k ≤ a ), there is a subsequence of f n that converges to f λ -a.e. In [21] the map dϕ is denoted by L ( ϕ ) f ∈ H ,q ( G/K ) that H Φ , ˙ A f = Z Ω Φ( u ) f ◦ ˙ A ( u ) dµ ( u ) , the Bochner integral (recall that H ,q ( G/K ) is a subspace of L ( G/K, λ ) [11,p. 592], and thus we identify functions that equal λ -a.e.).Therefore (below f = ˙ g ) kH Φ , ˙ A f k H ,q ( G/K ) ≤ Z Ω | Φ( u ) |k f ◦ ˙ A ( u ) k H ,q ( G/K ) dµ ( u )= Z Ω | Φ( u ) |k g ◦ A ( u ) k H ,q ( G ) dµ ( u ) . If g = P ∞ j =1 λ j a j then g ◦ A ( u ) = ∞ X j =1 λ j a j ◦ A ( u ) . (3)We claim that b j,u := C q − ν (mod( A ( u )) q k ( u ) ( q − s a j ◦ A ( u )is an atom, too. Indeed, Lemma 2 implies that A ( u ) − ( B ( x, r )) ⊆ B ( x ′ , k ( u ) r ) , where x ′ = A ( u ) − ( x ). If a j is supported in B ( x j , r j ) then b j,u is supportedin B ( x ′ j , k ( u ) r j ). So the condition (i) holds for b j,u .Next, by the property (D) we have ν ( B ( x j , k ( u ) r j )) ≤ C ν k ( u ) d ν ( B ( x j , r j )) . This estimate yields in view of (ii) and the left invariance of ρ and ν that k a j ◦ A ( u ) k q = (cid:18)Z G | a j ◦ A ( u ) | dν (cid:19) q = (mod( A ( u )) − q k a j k q ≤ (mod( A ( u )) − q ( ν ( B ( x j , r j ))) q − ≤ (mod( A ( u )) − q (cid:18) ν ( B ( x j , k ( u ) r j )) C ν k ( u ) d (cid:19) q − = (cid:18) C q − ν (mod( A ( u )) q k ( u ) ( q − d (cid:19) − ( ν ( B ( x ′ j , k ( u ) r j ))) q − . b j,u , too. Finally, the validity of (iii) followsfrom [18, VII.1.4, formula (31)].Since formula (3) can be rewritten in the form g ◦ A ( u ) = ∞ X j =1 (cid:18) λ j C − q ν (mod( A ( u )) − q k ( u ) (1 − q ) d (cid:19) b j,u , we have k g ◦ A ( u ) k H ,q ( G ) ≤ C − q ν (mod( A ( u )) − q k ( u ) (1 − q ) d ∞ X j =1 | λ j | . It follows that (recall that f = ˙ g ) k g ◦ A ( u ) k H ,q ( G ) ≤ C − q ν (mod( A ( u )) − q k ( u ) (1 − q ) d k g k H ,q ( G ) = C − q ν (mod( A ( u )) − q k ( u ) (1 − q ) d k f k H ,q ( G/K ) . Therefore kH Φ , ˙ A k L ( H ,q ( G/K )) ≤ C − q ν Z Ω | Φ( u ) | (mod A ( u )) − q k ( u ) d (1 − q ) dµ ( u )and the proof is complete.Setting in Theorem 1 Ω = Z + with counting measure µ we have the nextresult for discrete Hausdorff operators. Corollary 1.
Let ( G, ρ, ν ) and K be as in the Theorem 1, ( ˙ A ( n )) n ∈ Z + ⊂ Aut K ( G ) , and q ∈ (1 , ∞ ] . If Φ : Z + → C be such that k Φ k A,q := ∞ X n =0 | Φ( n ) | (mod A ( n )) − q k ( n ) (1 − q ) d < ∞ , then the discrete Hausdorff operator H Φ , ˙ A f ( ˙ x ) := ∞ X n =0 Φ( n ) f ( ˙ A ( n )( ˙ x )) is bounded on the space H ,q ( G/K ) and kH Φ , ˙ A k L ( H ,q ( G/K )) ≤ C − q ν k Φ k A,q . As a special case of Theorem 1 for K = { e } ( e denotes the unit of G ) onehas the 7 orollary 2. Let Let ( G, ρ, ν ) and (Ω , µ ) be as in the Theorem 1, ( A ( u )) u ∈ Ω ⊂ Aut( G ) , and q ∈ (1 , ∞ ] . If k Φ k A,q < ∞ then the operator H Φ ,A is bounded on H ,q ( G ) and kH Φ ,A k L ( H ,q ( G )) ≤ C − q ν k Φ k A,q . Remark 2 . The condition k Φ k A,q < ∞ is not necessary for boundednessof H Φ ,A in Hardy space as the following simple example shows . Considerthe Hausdorff operator( H f )( x ) := Z Ω f ( u x , . . . , u n x n ) du in H ( R n ). Here G = R n , Ω = { u ∈ R n : u j = 0 for j = 1 , . . . , n } , µ and ν are Lebesgue measures on Ω and R n respectively, K = { } , A ( u )( x ) = A u x ,where A u = diag { u , . . . , u n } ( x ∈ R n a column vector, u ∈ Ω), Φ = 1, d = n .The necessary moment condition R R n f ( u ) du = 0 for functions from H ( R n )yields that H f = 0 for all f ∈ H ( R n ). On the other hand, here mod A ( u ) = | det A u | = | u . . . u n | [23, Subsection VII.1.10, Corollary 1], ( dA ( u ) − ) X = A − u X ( X ∈ R n ), k ( u ) = k A − u k = ( P nj =1 u − j ) / ≥ n / | u . . . u n | − /n . Then k Φ k A,q = Z Ω (mod A ( u )) − q k ( u ) (1 − q ) d du ≥ n (1 − q ) n Z Ω du | u . . . u n | = ∞ . Let G be as above and A a compact subgroup of Aut( G ) with normalizedHaar measure m . Recall that the generalized shift operator of Delsarte [24],[25, Ch. I, §
2] (also the terms “generalized translation operator of Delsarte”,or “generalized displacement operator of Delsarte” are used) is defined to be T x f ( h ) = Z A f ( ha ( x )) dm ( a ) ( x, h ∈ G ) . Since the group G acts on G/K , one can define a generalization of thisoperator to
G/K as follows. Let Ω := { u ∈ A : u ( K ) = K } . Then Ω is a The sufficient boundedness conditions from [5] and [22] are also not met in this exam-ple. A . We denote by µ the normalized Haar measure of Ωand put for a Borel measurable function f on G/KT ˙ x f ( h ) := Z Ω f ( h ˙ u ( ˙ x )) dµ ( u ) ( ˙ x ∈ G/K, h ∈ G ) . Let h be fixed and L h f ( ˙ x ) := T ˙ x f ( h ). Then L h = H τ h , where H f ( ˙ x ) := Z Ω f ( ˙ u ( ˙ x )) dµ ( u )is a Hausdorff operator on G/K with Φ( u ) = 1 and A ( u ) = u , and τ h f ( ˙ x ) := f ( h ˙ x ). Note that mod is a continuous homomorphism from Aut( G ) to themultiplicative group (0 , ∞ ). Since Ω is a compact group, it follows thatmod(Ω) = { } . Assume that the doubling conditions for the Lie group G holds and Ω is quasi-metric. Then the operator H is bounded on H ,q ( G/K )by Theorem 1 and kH k ≤ C − q ν Z Ω k ( u ) (1 − q ) s dµ ( u )where k ( u ) = k ( d ( u − )) e k . Since τ h is an isometry of H ,q ( G/K ), we concludethat the operator L h is bounded on H ,q ( G/K ) and k L h k ≤ C − q ν Z Ω k ( u ) (1 − q ) d dµ ( u ) . R n Consider the unit sphere S n − ⊂ R n (the case n = 3 was considered in [9]).The compact group G = SO ( n ) acts on S n − transitively by restriction ofthe natural action of GL ( n, R ) on R n . It is known that the isotropy subgroup K of the point e n := (0 , . . . , ∈ S n − consists of all elements in SO ( n ) ofthe form e a := (cid:18) a ⊤ (cid:19) , where = (0 , . . . , ∈ R n − , a ∈ SO ( n − S n − with thehomogeneous space SO ( n ) /K . Let s ∈ S n − . If a matrix x ( s ) ∈ SO ( n − s = x ( s ) e ⊤ n we can identify the point s with the coset ˙ x ( s ) := x ( s ) K .Consider the set of automorphisms of G = SO ( n ) of the form A ( u )( x ) = e u − x e u, u ∈ O ( n − . x u − xu with u ∈ O ( n −
1) maps SO ( n −
1) ontoitself (being a connected component of unit in O ( n −
1) the group SO ( n −
1) isa normal subgroup of O ( n − A ( u ) where u ∈ O ( n −
1) map K onto K . Then by definition the coset˙ A ( u )( ˙ x ( s )) = π K ( e u − x ( s ) e u )can be identified with the point e u − x ( s ) e ue ⊤ n = e u − x ( s ) e ⊤ n = e u − s = ( u − s ′ , s n )( s ′ := ( s , . . . , s n − )) of S n − .Thus, Definition 2 takes the form (we put x = x ( s ) in this definition andidentify the coset ˙ x ( s ) with a column vector s ∈ S n − )( H Φ ,µ f )( s ) = Z O ( n − Φ( u ) f ( u − s ′ , s n ) dµ ( u ) (4)where µ stands for a (regular Borel) measure on O ( n −
1) and f is a Borelmeasurable function on S n − .Note that the point ( u − s ′ , s n ) runs over the cross-section of S n − bythe hyperplane { x = s n } ⊂ R n (which contains s ) orthogonal to the lastcoordinate axis when u runs over O ( n − S n − and the function H Φ ,µ f depends on s n ∈ [ − , k ( u ) = 1 for u ∈ O ( n − k ( u ) = k d ( A ( u − )) n k (here 1 n stands for the unit n × n matrix). It is easyto verify that for every X ∈ so ( n ), the Lie algebra of SO ( n ) d ( A ( u − )) n X = e uX e u − . On the other hand, e u ∈ O ( n ) for u ∈ O ( n − k d ( A ( u − )) n k = max k X k =1 , k Y k =1 |h d ( A ( u − )) n X, d ( A ( u − )) n Y i| =max k X k =1 , k Y k =1 |h e uX e u − , e uY e u − i| = max k X k =1 , k Y k =1 |h X, Y i| = 1(here h· , ·i stands for the Euclidean inner product). Since SO ( n ) is compact,it is doubling [11]. Next, since SO ( n ) is unimodular, we get that mod A = 1for all A ∈ Aut( SO ( n )). So if Φ ∈ L ( O ( n − , µ ) [8, Theorem 1] yields thatthe operator (4) is bounded on L p ( S n − ) and kH Φ ,µ k L ( L p ( S n − ) ≤ k Φ k L ( µ ) .Moreover, Theorem 1 yields that kH Φ ,µ k L ( H ,q ( S n − )) ≤ C − q ν k Φ k L ( µ ) C ν is the doubling constant for SO ( n ).In closing let us consider the following special case. Let Φ = 1 and m be aHaar measure of the (compact) group O ( n − f ∈ H ( S n − )the function ( H ,m f )( s ) = Z O ( n − f ( u − s ′ , s n ) dm ( u )belongs to H ( S n − ). On the other hand, this function depends on s n only.Indeed, ( s ′ , s n ) ∈ S n − if and only if s ′ belongs to the sphere S n − r centeredat 0 ∈ R n − of radius r = p − s n . Fix s ′ ∈ S n − r . Since SO ( n −
1) actstransitively on S n − r , for every s ′ ∈ S n − r there is such v ∈ SO ( n −
1) that vs ′ = s ′ . Taking into account that O ( n −
1) is unimodular, we get( H ,m f )( s ) = Z O ( n − f ( uvs ′ , s n ) dm ( u ) = Z O ( n − f ( us ′ , s n ) dm ( u )which completes the proof. Acknowledgments . This work was supported by the State Program ofScientific Research of the Republic of Belarus.
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