Restriction theorems for Hankel operators
aa r X i v : . [ m a t h . F A ] O c t RESTRICTION THEOREMS FOR HANKEL OPERATORS
NAZAR MIHEISI AND ALEXANDER PUSHNITSKI
Abstract.
We consider a class of maps from integral Hankel operators to Han-kel matrices, which we call restriction maps. In the simplest case, such a mapis simply a restriction of the integral kernel onto integers. More generally, it isgiven by an averaging of the kernel with a sufficiently regular weight function.We study the boundedness of restriction maps with respect to the operator normand the Schatten norms. Introduction
Hankel operators.
Let α = { α ( j ) } j ≥ be a sequence of complex numbers.The Hankel matrix H ( α ) is the “infinite matrix” { α ( j + k ) } j,k ≥ , considered as alinear operator on ℓ ( Z + ), Z + = { , , , . . . } , so that( H ( α ) x )( k ) = X j ≥ α ( j + k ) x ( j ) , k ≥ , x = { x ( j ) } j ≥ ∈ ℓ ( Z + ) . Similarly, for a kernel function a ∈ L (0 , ∞ ), the integral Hankel operator on L (0 , ∞ ) is defined by the formula( H ( a ) f )( t ) = Z ∞ a ( t + s ) f ( s ) ds, t > , f ∈ L (0 , ∞ ) . In order to distinguish between these two classes of operators, we use boldface fontfor objects associated with integral Hankel operators.For general background on Hankel operators, see [6, 8]. In what follows, we willonly consider bounded Hankel matrices and bounded integral Hankel operators.1.2.
Restrictions.
The purpose of this paper is to examine the linear map, whichwe call the restriction map , between the set of integral Hankel operators and theset of Hankel matrices. To set the scene, let us consider the pointwise restriction of integral kernels to integers. For a given kernel function a , define the sequence α ( j ) := a ( j + 1) , j ≥ . (1.1)Of course, for this operation to make sense, the kernel function a has to be contin-uous. Here is our first result; we denote by S p , 0 < p < ∞ , the standard Schattenclass of compact operators (see Section 2). Date : 1 October 2018.2010
Mathematics Subject Classification.
Key words and phrases.
Hankel operator, integral kernel, Schatten class.
Theorem 1.1.
Let H ( a ) ∈ S p for some < p ≤ . Then the kernel function a ( t ) is continuous in t > , so the restriction (1.1) is well defined. The operator H ( α ) is in S p and we have the estimate k H ( α ) k S p ≤ C p k H ( a ) k S p . (1.2)The continuity of the kernel function a for trace class integral Hankel operatorsis well known (see e.g. [7, Corollary 7.10]); the main point here is the estimate(1.2). In Section 3 we give a slightly more precise version of Theorem 1.1 and showthat it does not extend to p >
1. Further, we show that if we restrict the map H ( a ) H ( α ) to non-negative integral Hankel operators, then it is bounded in S p norm for all 0 < p < ∞ (and also in the operator norm).Further, along with the pointwise restriction (1.1), we consider the following restrictions by averaging . For a suitably regular function ϕ on R and for a kernelfunction a , we define the restriction R ϕ a to be the sequence R ϕ a ( j ) = Z ∞ a ( t ) ϕ ( t − j ) dt, j ≥ . In particular, formally taking ϕ ( t ) = δ ( t − δ is the Dirac δ -function, werecover the pointwise restrictions (1.1). In Section 5 we prove that, under suitableregularity conditions on ϕ , the map H ( a ) H ( R ϕ a )is bounded in S p norm for all 0 < p < ∞ (and also in the operator norm). We alsorelate this result to the well known unitary equivalence between Hankel matricesand integral Hankel operators.This paper appeared as an attempt to consider one of the technical ingredientsof [5] on a more systematic basis. Theorem 1.1 and its proof is based on the sameset of ideas as [5, Theorem 3.2].The results of this paper seem to parallel some restriction theorems for Fouriermultipliers; see e.g. [2, 4, 1]. However, this connection is not completely under-stood (at least by the authors).We note in passing that one can consider a converse operation, an extension ofa Hankel matrix to an integral kernel. For a suitably regular function ϕ and asequence α = { α ( j ) } j ≥ , one can define the extension E ϕ α to be the function E ϕ α ( t ) = X k ≥ α ( k ) ϕ ( t − k ) , t > , and one can consider the map H ( α ) H ( E ϕ α ) . Although some Schatten norm boundedness results for this map are not difficultto prove, we have not succeeded in finding a coherent set of estimates for it andtherefore we do not discuss extensions here.
ESTRICTION THEOREMS FOR HANKEL OPERATORS 3
Symbols.
For a bounded Hankel matrix H ( α ), its analytic symbol is thefunction q α ( z ) = X m ≥ α m z m , | z | < . Similarly, for a bounded integral Hankel operator H ( a ), its analytic symbol is thefunction q a ( ξ ) = Z ∞ a ( t ) e πitξ dt, Im ξ > . It is instructive to view restriction maps on Hankel operators in terms of thesymbols. If α = R ϕ a , then for the symbols we have q α ( z ) = Z R q a ( ξ + i q ϕ ( − ξ + i − ze − πiξ dξ, | z | < . (1.3)In particular, for the pointwise restriction (1.1) we have q α ( e πiξ ) = e − πiξ X j ∈ Z q a ( ξ − j ) , Im ξ > . (1.4)Since Schatten norms of Hankel operators correspond to Besov norms of the sym-bols (see Section 2), one can view the topic of this paper as the study of the mapinduced by (1.3) between Besov classes. We prefer to use an operator theoreticviewpoint whenever possible, although sometimes we have to resort to proofs interms of Besov classes. 2. Preliminaries
Throughout this paper, the symbol ‘ C ’ with a (possibly empty) set of subscriptswill denote a positive constant, depending only on the subscripts, whose precisevalue may change with each occurrence. Moreover, we write X ≍ Y for twoexpressions X and Y if X ≤ CY and Y ≤ CX .2.1. Operator theory, Schatten classes.
For a bounded linear operator A ina Hilbert space, we denote by k A k B the operator norm of A .For a compact operator A in a Hilbert space, let { s n ( A ) } ∞ n =1 be the sequenceof singular values of A , enumerated with multiplicities taken into account. For0 < p < ∞ , the standard Schatten class S p of compact operators is defined by thecondition A ∈ S p ⇔ k A k p S p := X n ≥ s n ( A ) p < ∞ . k·k S p is a norm on S p for p ≥ < p < NAZAR MIHEISI AND ALEXANDER PUSHNITSKI
Characterisation of Schatten class Hankel operators.
Let T deonte theunit circle. We consider the Fourier transform F as the unitary map from L ( T )to ℓ ( Z ), ( F f )( j ) = b f ( j ) = Z f ( e πit ) e − πijt dt, j ∈ Z . We also use its inverse F − : ℓ ( Z ) → L ( T ) and denote q α = F − α . Similarly, weuse the Fourier integral transform F in L ( R ) and its inverse( F − f )( ξ ) = q f ( ξ ) = Z R f ( t ) e πiξt dt, ξ ∈ R . Let w ∈ C ∞ ( R ) be a non-negative function such that supp w ⊂ [1 / ,
2] and X m ∈ Z w ( t/ m ) = 1 , t > . We set w m ( t ) = w ( t/ m ). For m ≥
0, we denote by w m the restriction of thefunction w m onto Z + , i.e. w m ( j ) = w m ( j ), j ≥ Proposition 2.1. [8, Theorem 6.7.4]
Let < p < ∞ . (i) For a bounded Hankel matrix H ( α ) , one has k H ( α ) k p S p ≍ | α (0) | p + X m ≥ m kF − ( αw m ) k pL p ( T ) . (ii) For a bounded integral Hankel operator H ( a ) one has k H ( a ) k p S p ≍ X m ∈ Z m k F − ( aw m ) k pL p ( R ) . The expressions in the right side here are exactly the norms of the symbols inthe Besov class B /pp .2.3. Periodization operator.
Here we discuss the map induced by (1.4). For acompactly supported function f ∈ C ( R ), we define the periodization of f as thefunction on the unit circle given by P f ( e πit ) = X j ∈ Z f ( t − j ) , e πit ∈ T . (2.1)We call P the periodization operator . Applying the “triangle inequality” | a + b | p ≤| a | p + | b | p for 0 < p ≤ t we see that kP f k L p ( T ) ≤ k f k L p ( R ) , < p ≤ . This allows one to extend P to a map from L ( R ) to L ( T ). For f ∈ L ( R ) it isstraightforward to see that c P f ( j ) = b f ( j ) , j ∈ Z . ESTRICTION THEOREMS FOR HANKEL OPERATORS 5
Thus we have the estimate (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X j ∈ Z b f ( j ) z j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ( T ) ≤ k f k L p ( R ) , < p ≤ . (2.2)3. Pointwise restrictions
Pointwise restrictions for operators of class S p . For λ >
0, let δ λ ( t ) = δ ( t − λ ), where δ ( t ) is the Dirac delta function, so that if a ∈ C (0 , ∞ ), then( R δ λ a )( j ) = a ( j + λ ) , j ≥ . If a is the kernel function of an integral Hankel operator of class S , then a isalmost everywhere equal to a continuous function on (0 , ∞ ) [7, Corollary 7.10],and the estimate | a ( t ) | ≤ C k H ( a ) k S /t, t > , (3.1)holds true with some absolute constant C . Thus, the definition of R δ λ a makessense without any further restriction on a .The aim of this section is to prove the following. Theorem 3.1.
Let < p ≤ , λ > . If H ( a ) ∈ S p then H ( R δ λ a ) ∈ S p and k H ( R δ λ a ) k S p ≤ C p (1 + 1 /λ ) k H ( a ) k S p . (3.2)The main component in the proof of Theorem 3.1 is the estimate (2.2). Proof.
Denote b ( t ) = a ( t + λ ). By Proposition 2.1(i), we have k H ( R δ λ a ) k p S p ≤ C p | b (0) | p + C p X m ≥ m (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X j ≥ b ( j ) w m ( j ) z j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) pL p ( T ) . (3.3)Let us first estimate the series in the right hand side of (3.3). Applying (2.2) to f = F − ( bw m ), we obtain (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X j ≥ b ( j ) w m ( j ) z j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) pL p ( T ) ≤ k F − ( bw m ) k pL p ( R ) for every m ≥
0. By Proposition 2.1(ii), this yields X m ≥ m (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X j ≥ b ( j ) w m ( j ) z j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) pL p ( T ) ≤ X m ∈ Z m k F − ( bw m ) k pL p ( R ) ≤ C p k H ( b ) k p S p . Let us relate the norm of H ( b ) to the norm of H ( a ). Writing Z ∞ Z ∞ b ( t + s ) f ( t ) g ( s ) dt ds = Z ∞ λ/ Z ∞ λ/ a ( t + s ) f ( t − λ/ g ( s − λ/ dt ds, NAZAR MIHEISI AND ALEXANDER PUSHNITSKI we see that H ( b ) is unitarily equivalent to the restriction of H ( a ) onto the subspace L ( λ/ , ∞ ) ⊂ L (0 , ∞ ). It follows that k H ( b ) k S p ≤ k H ( a ) k S p (3.4)for all p >
0. Finally, consider the first term in the right hand side of (3.3). By(3.1) we have | b (0) | = | a ( λ ) | ≤ C k H ( a ) k S /λ ≤ C k H ( a ) k S p /λ. Combining the above estimates, we arrive at the required statement. (cid:3)
Remark.
One can also consider restrictions of a to the scaled lattice { γj + λ } j ≥ for some γ >
0. For γ >
0, let a γ ( t ) = a ( γt ) and let V γ : L (0 , ∞ ) → L (0 , ∞ ) bethe unitary operator V γ f ( t ) = √ γ f ( γt ) , t > . Then γ H ( a γ ) = V γ H ( a ) V ∗ γ and so γ k H ( a γ ) k S p = k H ( a ) k S p for all 0 < p < ∞ .It follows from this and Theorem 3.1 that if 0 < p ≤
1, then γ k H ( R δ λ a γ ) k S p ≤ C p k H ( a ) k S p , and thus sup γ> γ k H ( R δ λ a γ ) k S p ≤ C p k H ( a ) k S p . (3.5)3.2. Counterexample for p > . For p >
1, is it no longer the case that thekernel of an integral Hankel operator of class S p is necessarily continuous. How-ever, even if we restrict to operators with continuous kernels, the conclusions ofTheorem 3.1 still fail and thus the condition 0 < p ≤ a with supp a ⊂ [1 / ,
2] and a (1) = 1 and let a ( N ) ( t ) = a (1 + N ( t − N ∈ N . Then for each N we have R δ a ( N ) (0) = a ( N ) (1) = 1 , R δ a ( N ) ( j ) = a ( N ) (1 + j ) = 0 , j ≥ . It follows that k H ( R δ a ( N ) ) k S p = 1for all p ≥ N ∈ N . On the other hand, it is not difficult to show that k H ( a ( N ) ) k p S p ≤ CN − p which tends to zero as N → ∞ whenever p >
1. Indeed, by the assumption onthe support of a we have a ( N ) = a ( N ) w − + a ( N ) w + a ( N ) w for all N , where w m are defined in Section 2.2. It is easy to conclude that X m = − , , m k F − ( a ( N ) w m ) k pL p ( R ) ≤ C p k F − ( a ( N ) ) k pL p ( R ) = CN − p . ESTRICTION THEOREMS FOR HANKEL OPERATORS 7
Partial converse of Theorem 3.1.
It is clear that one cannot bound H ( a )by H ( R δ λ a ) in any norm. However, one can achieve a partial converse if we varyour restriction operators in an appropriate sense and take a supremum over allrestrictions in the right side. Here we briefly sketch a sample argument of thisnature. Fix 0 < p ≤
1; we use “continuous” counterparts of the expressions inProposition 2.1; see e.g. [15, Section 2.3.3, p.99]: k H ( α ) k p S p ≍ | α (0) | p + Z kF − ( αw τ ) k pL p ( T ) dττ , (3.6) k H ( a ) k p S p ≍ Z ∞ k F − ( aw τ ) k pL p ( R ) dττ , where w τ ( t ) = w ( τ t ) and w τ = { w ( τ j ) } j ≥ .Let a be a continuous function on (0 , ∞ ) and, for γ >
0, let a γ ( t ) = a ( γt ).Observe that, by a change of variable, kF − ( w τ R δ λ a γ ) k pL p ( T ) = Z / − / (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X j ≥ w ( τ j ) a ( γ ( j + λ )) e πijs (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p ds = γ − p Z / γ − / γ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) γ X j ≥ w ( τ j ) a ( γ ( j + λ )) e πijγs (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p ds. By another change of variable, it then follows from (3.6) that γ p k H ( R δ λ a γ ) k p S p ≥ C p γ Z Z / γ − / γ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) γ X j ≥ w ( τ j ) a ( γ ( j + λ )) e πijγs (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p ds dττ = C p Z /γ Z / γ − / γ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) γ X j ≥ w ( τ γj ) a ( γ ( j + λ )) e πijγs (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p ds dττ . (3.7)Since a is continuous, for each s ∈ R and τ > | F − ( aw τ )( s ) | p as γ →
0. Then by Fatou’s Lemma we see that k H ( a ) k p S p ≍ Z ∞ k F − ( aw τ ) k pL p ( R ) dττ ≤ lim γ → Z /γ Z / γ − / γ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) γ X j ≥ w ( τ γj ) a ( γ ( j + λ )) e πijγs (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p ds dττ ≤ C p lim γ → γ p k H ( R δ λ a γ ) k p S p . (3.8)This gives an analogue of Igari’s theorem for Fourier multipliers [4]. Combining(3.8) with (3.5) gives the estimate k H ( a ) k S p ≍ sup γ> γ k H ( R δ λ a γ ) k S p , < p ≤ . NAZAR MIHEISI AND ALEXANDER PUSHNITSKI Pointwise restriction for non-negative operators
Statement of the result.
Although Theorem 3.1 fails for p >
1, the esti-mate (3.2) remains valid for all 0 < p < ∞ if we restrict to the class of non-negativeoperators (in the usual quadratic form sense). Before stating this precisely we re-call (see e.g. [16, page 22]) that a bounded integral Hankel operator H ( a ) isnon-negative if and only if the kernel function a can be represented as a ( t ) = Z ∞ e − tη dµ ( η ) , (4.1)where the measure µ satisfies µ ((0 , η )) ≤ Cη, η > . In particular, it follows that the kernel function a ( t ) is continuous in t >
0, andtherefore the restriction R δ λ a is well defined for all λ > Theorem 4.1.
Let H ( a ) ≥ be a bounded integral Hankel operator and λ > .Then the following hold: (i) If H ( a ) ∈ B then H ( R δ λ a ) ∈ B and k H ( R δ λ a ) k B ≤ C (1 + 1 /λ ) k H ( a ) k B . (4.2)(ii) If H ( a ) ∈ S p for some < p ≤ ∞ , then H ( R δ λ a ) ∈ S p and k H ( R δ λ a ) k S p ≤ C p (1 + 1 /λ ) k H ( a ) k S p . (4.3) Remark 4.2. (1) Observe that by (4.1), the kernel function a is necessarilypositive, monotone decreasing and continuous on (0 , ∞ ). In fact, the proofof Theorem 4.1 depends only on these properties of a .(2) If λ ≥
2, one can slightly improve the statement of Theorem 4.1. In thiscase one gets k H ( R δ λ a ) k B ≤ k H ( a ) k B , λ ≥ , k H ( R δ λ a ) k S p ≤ k H ( a ) k S p , λ ≥ , p ∈ N , i.e. the constants in the estimates are equal to one in these cases.In the rest of this section we prove Theorem 4.1. Observe that we only needto consider the case p >
1, as for 0 < p ≤ p ∈ N or p = ∞ . The second one is a direct calculation based on Proposition 2.1which applies to all p ≥ ESTRICTION THEOREMS FOR HANKEL OPERATORS 9
Proof for p ∈ N ∪ {∞} . First we need a version of (3.1) for non-negativeoperators.
Lemma 4.3.
Let H ( a ) ≥ be a bounded integral Hankel operator and λ > .Then λ a ( λ ) ≤ k H ( a ) k B . Proof.
Take f ( t ) = e − t/λ ; then k f k L (0 , ∞ = λ/ a ( t ),( H ( a ) f , f ) = Z ∞ Z ∞ a ( t + s ) e − ( t + s ) /λ dt ds = Z ∞ a ( t ) e − t/λ tdt ≥ Z λ a ( t ) e − t/λ tdt ≥ a ( λ ) Z λ e − t/λ tdt = (1 − e − ) λ a ( λ ) . On the other hand,( H ( a ) f , f ) ≤ k H ( a ) k B k f k L (0 , ∞ ) = ( λ/ k H ( a ) k B . Combining these two estimates, we obtain λ a ( λ ) ≤ − e − ) k H ( a ) k B ≤ k H ( a ) k B , as required. (cid:3) Proof of Theorem 4.1 for p ∈ N ∪ {∞} . First let us assume that λ ≥ . Let K be the integral operator in L (0 , ∞ ) with the integral kernel K ( t, s ) = a ( λ + ⌊ t ⌋ + ⌊ s ⌋ ) , where ⌊ t ⌋ is the largest integer less than or equal to t . Since λ + ⌊ t ⌋ + ⌊ s ⌋ ≥ λ + ( t −
1) + ( s − ≥ t + s, by monotonicity of a we have K ( t, s ) ≤ a ( t + s ) . In the terminology of [12, Chapter 2], this means that K is pointwise dominated by H ( a ). By [12, Theorem 2.13], it follows that k K k ≤ k H ( a ) k B and k K k S p ≤ k H ( a ) k S p for all p ∈ N . (This implication does not extend to p N ; see e.g. [9, 13].) It isalso true (see [10, 3]) that the compactness of H ( a ) implies the compactness of K .Next, let us relate K to H ( R δ λ a ). For f ∈ L (0 , ∞ ) let us write the quadraticform of K as ( K f , f ) = X j,k ≥ a ( λ + j + k ) f j f k , f j = Z j +1 j f ( t ) dt. This means that, writing L (0 , ∞ ) = ℓ ( Z + ) ⊗ L (0 , K can berepresented as K = H ( R δ λ a ) ⊗ ( · , ) , where ( · , ) is the rank one operator in L (0 ,
1) acting as f Z f ( t ) dt. It follows that k K k B = k H ( R δ λ a ) k B and k K k S p = k H ( R δ λ a ) k S p for all p >
0. This completes the proof for λ ≥ p ∈ N ∪ {∞} .Let us consider the case 0 < λ <
2. Let P be the projection onto ℓ ( { , , . . . } )in ℓ ( Z + ). Write H ( R δ λ a ) = P H ( R δ λ a ) P + e H. The operator e H is of rank ≤
4. Inspecting the matrix elements of e H and usingLemma 4.3, it is easy to see that k e H k S ≤ C k H ( a ) k B /λ, λ > . On the other hand, the operator P H ( R δ λ a ) P is unitarily equivalent to H ( R δ λ +2 a ). Thus, applying the previous step of the proof, we obtain k P H ( R δ λ a ) P k B ≤ k H ( a ) k B and k P H ( R δ λ a ) P k S p ≤ k H ( a ) k S p for p ∈ N . Combining these estimates, we arrive at (4.2) and (4.3) for p ∈ N . (cid:3) As already mentioned, this proof does not extend to p N ; see e.g. [9, 13].Below we give a different proof which works for all 1 ≤ p < ∞ , but does not giveprecise information about the constants in the estimates.4.3. Proof of Theorem 4.1 for ≤ p < ∞ . In order to simplify our notation,we set b ( t ) = a ( t + λ ), b ( k ) = a ( k + λ ), and q b m ( z ) = X k ≥ b ( k ) w m ( k ) z k , m ∈ Z + , z ∈ T , q b m ( ξ ) = Z ∞ b ( t ) w m ( t ) e πiξt dt, m ∈ Z , ξ ∈ R . The core of the proof is the bound X m ≥ m k q b m k pL p ( T ) ≤ C p X m ∈ Z m k q b m k pL p ( R ) , (4.4)which we prove below. Throughout the proof, we use the property that b and b are positive and monotone decreasing. ESTRICTION THEOREMS FOR HANKEL OPERATORS 11
First step: upper bound for k q b m k L p ( T ) . Fix m ≥
1. First we prepare twopointwise bounds for q b m ( z ). The first one is trivial: | q b m ( z ) | ≤ X k b ( k ) w m ( k ) ≤ m +1 b (2 m − ) . (4.5)The second one is obtained through a discrete version of integration by parts (Abelsummation). We have q b m ( z ) = 1 z − X k b ( k ) w m ( k )( z k +1 − z k )= 1 z − X k (cid:0) b ( k ) w m ( k ) − b ( k + 1) w m ( k + 1) (cid:1) z k +1 = 1 z − X k (cid:0) ( b ( k ) − b ( k + 1)) w m ( k ) + b ( k + 1)( w m ( k ) − w m ( k + 1)) (cid:1) z k +1 , and therefore | q b m ( z ) | ≤ | z − | m +1 X k =2 m − ( b ( k ) − b ( k + 1))+ 1 | z − | m +1 X k =2 m − b ( k ) | w m ( k − − w m ( k ) | . Clearly, the first sum here is telescoping. For the second sum, we use the estimate | w m ( k − − w m ( k ) | ≤ C − m . Putting this together, we obtain | q b m ( z ) | ≤ | z − | ( b (2 m − ) − b (1 + 2 m +1 ))+ C | z − | − m m +1 X k =2 m − b ( k ) ≤ C | z − | b (2 m − ) , (4.6)which is our second bound for q b m ( z ).Now we can estimate the norm k q b m k L p ( T ) . We split the integral over the unitcircle into two parts and estimate them separately. Using (4.5), we obtain2 m Z | t | < − m | q b m ( e πit ) | p dt ≤ C pm b (2 m − ) p . Using (4.6), we get2 m Z | t | > − m | q b m ( e πit ) | p dt ≤ C m Z | t | > − m dt | e πit − | p b (2 m − ) p ≤ C m Z − m dtt p b (2 m − ) p ≤ C pm b (2 m − ) p . Combining the estimates for two integrals above, we obtain2 m k q b m k pL p ( T ) ≤ C pm b (2 m − ) p . Second step: lower bound for k q b m k L p ( R ) . For the derivative of b m we have q b ′ m ( ξ ) = 2 πi Z ∞ b ( t ) w m ( t ) te πitξ dt, and therefore | q b ′ m ( ξ ) | ≤ π Z ∞ b ( t ) w m ( t ) tdt ≤ m +2 π Z ∞ b ( t ) w m ( t ) dt = 2 m +2 π q b m (0) . It follows that | q b m ( ξ ) − q b m (0) | ≤ | ξ | m +2 π q b m (0) , and therefore for | ξ | < − m − we have | q b m ( ξ ) | ≥ q b m (0) / . We use this to obtain a lower bound for the integral of | q b m | p :2 m Z R | q b m ( ξ ) | p dξ ≥ m Z | ξ | < − m − | q b m ( ξ ) | p dξ ≥ − ( q b m (0) / p = C q b m (0) p . Finally, q b m (0) = Z ∞ b ( t ) w m ( t ) dt ≥ b (2 m +1 ) Z ∞ w m ( t ) dt = C m b (2 m +1 ) , and so we obtain 2 m k q b m k pL p ( R ) ≥ C mp b (2 m +1 ) p . Combining the two steps and completing the proof.
Combining theupper bound for k q b m k L p ( T ) and the lower bound for k q b m k L p ( R ) , we obtain2 m k q b m k pL p ( T ) ≤ C p ( m − b (2 m − ) p ≤ C m − k q b m − k pL p ( R ) , m ≥ . Summing over m , we obtain the bound (4.4).By Proposition 2.1(i), we have k H ( b ) k p S p ≤ C p | b (0) | p + C p X m ≥ m k q b m k pL p ( T ) . (4.7) ESTRICTION THEOREMS FOR HANKEL OPERATORS 13
By Lemma 4.3, we have | b (0) | p = | a ( λ ) | p ≤ p k H ( a ) k p B /λ p . Similarly, the m = 0 term in the series in (4.7) can be estimated as follows: k q b k pL p ( T ) = | b (1) | p = | a ( λ + 1) | p ≤ p k H ( a ) k p B / (1 + λ ) p ≤ p k H ( a ) k p B /λ p . Combining this with (4.4) and using Proposition 2.1(ii), we obtain k H ( b ) k p S p ≤ C p k H ( a ) k p B /λ p + C p X m ∈ Z m k q b m k pL p ( R ) ≤ C p k H ( a ) k p B /λ p + C p k H ( b ) k p S p . Finally, as in (3.4), we have k H ( b ) k S p ≤ k H ( a ) k S p , and we arrive at the requiredestimate (4.3). 5. Restriction by averaging
Boundedness of restrictions by averaging.
The main result of this sec-tion says that if the function ϕ is sufficiently regular, then the map H ( a ) H ( R ϕ a ) is bounded with respect to all Schatten norms. We will make use of theperiodisation operator P from Section 2.3. Theorem 5.1.
Let ϕ ∈ C ( R ) be such that supp ϕ ⊂ [0 , ∞ ) and P ( | q ϕ | ) ∈ L ∞ ( T ) .Then there exist bounded operators Φ and Φ acting from ℓ ( Z + ) to L (0 , ∞ ) suchthat Φ ∗ H ( a )Φ = H ( R ϕ a ) (5.1) and k Φ k B = k Φ k B = √ A , where A = kP ( | q ϕ | ) k L ∞ ( T ) . Consequently, we have k H ( R ϕ a ) k B ≤ A k H ( a ) k B and k H ( R ϕ a ) k S p ≤ A k H ( a ) k S p for every < p < ∞ . A close inspection of the proof of Theorem 5.1 will reveal that the condition P ( | q ϕ | ) ∈ L ∞ ( T ) is necessary, in the sense that if there exist bounded operatorsΦ , Φ : ℓ ( Z + ) → L (0 , ∞ ) such that (5.1) holds, then P ( | q ϕ | ) ∈ L ∞ ( T ).It will be convenient to separate the statement related to the boundedness ofthe maps Φ and Φ . Lemma 5.2.
Let ψ ∈ L ( R ) with P ( | q ψ | ) ∈ L ∞ ( T ) . Then the map Φ : x = { x ( j ) } j ≥ X j ≥ x ( j ) ψ ( t − j ) , t ∈ R , is bounded from ℓ ( Z + ) to L ( R ) , with k Φ k B = kP ( | q ψ | ) k L ∞ ( T ) . Proof.
Let x be a finitely supported sequence. We have, using Parseval’s theorem, k Φ x k L ( R ) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X j ≥ x ( j ) ψ ( · − j ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ( R ) = Z R (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X j ≥ x ( j ) q ψ ( ξ ) e πijξ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dξ = Z R | q x ( e πiξ ) | | q ψ ( ξ ) | dξ = X j ∈ Z Z | q x ( e πiξ ) | | q ψ ( ξ − j ) | dξ = Z | q x ( e πiξ ) | P ( | q ψ | )( ξ ) dξ ≤ kP ( | q ψ | ) k L ∞ ( T ) k q x k L ( T ) = kP ( | q ψ | ) k L ∞ ( T ) k x k ℓ ( Z + ) . It is also clear that the inequality here is sharp in the sense thatsup k q x k L T ) =1 Z R | q x ( e πiξ ) | | q ψ ( ξ ) | dξ = kP ( | q ψ | ) k L ∞ ( T ) . This proves the claim. (cid:3)
Below C + will denote the upper half-plane; H ∞ ( C + ), H ( C + ) etc. are thestandard Hardy classes. We will sometimes identify functions in these Hardyclasses with their boundary values on R . Proof of Theorem 5.1.
By assumption, we have P ( | q ϕ | ) ∈ L ∞ ( T ) ⊂ L ( T ); it fol-lows that q ϕ ∈ L ( R ). Recalling that supp ϕ ⊂ [0 , ∞ ), we obtain that q ϕ ∈ H ( C + ).Thus, we can factorise q ϕ into a product of two H ( C + )-functions. More precisely,there exist ϕ , ϕ ∈ L (0 , ∞ ) such that q ϕ ( ξ ) = q ϕ ( ξ ) q ϕ ( − ξ ) and | q ϕ ( ξ ) | = | q ϕ ( − ξ ) | , ∀ ξ ∈ R . Then ϕ = ϕ ∗ ϕ and kP ( | q ϕ | ) k L ∞ ( T ) = kP ( | q ϕ | ) k L ∞ ( T ) = kP ( | q ϕ | ) k L ∞ ( T ) . Next, for i = 1 ,
2, let us define the map Φ i : ℓ ( Z + ) → L (0 , ∞ ) byΦ i : x = { x ( j ) } j ≥ X j ≥ x ( j ) ϕ i ( · − j ) . By Lemma 5.2, both Φ and Φ are bounded with norms equal to p kP ( | q ϕ | ) k L ∞ ( T ) .In order to prove (5.1), let us first rearrange the definition of R ϕ a . For each j, k ≥ R ϕ a ( j + k ) = Z ∞ a ( t ) ϕ ( t − j − k ) dt = Z ∞ a ( t ) Z R ϕ ( t − s − j − k ) ϕ ( s ) dt ds = Z Z s + t> a ( s + t ) ϕ ( t − j ) ϕ ( s − k ) dt ds. ESTRICTION THEOREMS FOR HANKEL OPERATORS 15
Since both ϕ and ϕ are supported on (0 , ∞ ), we can rewrite this as R ϕ a ( j + k ) = Z ∞ Z ∞ a ( s + t ) ϕ ( t − j ) ϕ ( s − k ) ds dt. Now for x = { x ( j ) } j ≥ ∈ ℓ ( Z + ), let us compute the quadratic form( H ( a )Φ x, Φ x ) = X j,k ≥ Z ∞ Z ∞ a ( t + s ) x ( j ) x ( k ) ϕ ( t − j ) ϕ ( s − k ) dt ds = X j,k ≥ R ϕ a ( j + k ) x ( j ) x ( k ) , which yields (5.1). (cid:3) Unitary equivalence and restrictions associated to general convolu-tions.
Let L n = L (0) n be the n -th Laguerre polynomial (see [14, Ch. V] for thedefinition) and let u n ( t ) = − i √ πL n (4 πt ) e − πt , t > . (5.2)Then { u n } n ≥ is an orthonormal basis of L (0 , ∞ ). It is well known that thematrix of an integral Hankel operator is a Hankel matrix in the basis { u n } n ≥ andhence the classes of Hankel matrices and integral Hankel operators are unitarilyequivalent [8, Ch. 1, Thm 8.9].In this subsection we discuss how this unitary equivalence fits into our “restric-tion by averaging” framework. This requires looking at restrictions by averagingof a more general type than considered above. To a given integral Hankel operator H ( a ) we associate the Hankel matrix H ( α ) with α j = Z ∞ a ( t ) ϕ j ( t ) dt, j ≥ , where ϕ j is a certain sequence of smooth functions, a more general one than justtranslations of a single function. Our sequence ϕ j will be given by the multipleconvolution of the form ϕ j = ϕ ∗ ν ∗ ν ∗ · · · ∗ ν | {z } j terms , j ≥ , where ϕ is a sufficiently regular function supported on [0 , ∞ ), and ν is a positivefinite measure supported on [0 , ∞ ). Observe that if dν ( t ) = δ ( t − dt , then ϕ j ( t ) = ϕ ( t − j ), so we recover the definition of R ϕ .To make the multiple convolution notation more readable, we introduce the(formal) convolution with ν operator T ν f = f ∗ ν ;then ϕ j = T jν ϕ . Theorem 5.3.
Let ν be a positive measure on [0 , ∞ ) with ν ([0 , ∞ )) ≤ , and let ϕ ∈ C ( R ) satisfy supp ϕ ⊂ [0 , ∞ ) and | q ϕ ( ξ ) | ≤ C ξ , ξ ∈ R . (5.3) For j ≥ , set ϕ j = T jν ϕ and consider the map a ( t ) α = { α ( j ) } ∞ j =0 , α ( j ) = Z ∞ a ( t ) ϕ j ( t ) dt. Then there exist bounded operators Φ and Φ acting from ℓ ( Z + ) to L (0 , ∞ ) suchthat Φ ∗ H ( a )Φ = H ( α ) . (5.4) Consequently, k H ( α ) k ≤ A k H ( a ) k and k H ( α ) k S p ≤ A k H ( a ) k S p for all < p < ∞ , where A = k Φ kk Φ k . It will again be convenient to separate the boundedness of Φ and Φ into alemma. Lemma 5.4.
Let ω ∈ H ∞ ( C + ) with k ω k H ∞ ≤ . Then the map x = { x ( j ) } ∞ j =0 X j ≥ x ( j ) ω ( ξ ) j ξ + i , ξ ∈ C + , (5.5) is bounded from ℓ ( Z + ) to H ( C + ) .Proof. Consider the conformal map D ∋ ζ ξ = i ζ − ζ ∈ C + and the corresponding unitary operator U : H ( C + ) → H ( D ),( U f )( ζ ) = 2 √ π − ζ f (cid:18) i ζ − ζ (cid:19) . We have U : ω ( ξ ) j ξ + i
7→ − i √ πψ ( ζ ) j , ψ ( ζ ) = ω (cid:18) i ζ − ζ (cid:19) . It follows that U maps the right hand side of (5.5) to the function − i √ π X j ≥ x ( j ) ψ ( ζ ) j . Since | ψ ( ζ ) | ≤
1, by the Littlewood subordination theorem [11, Chap. 1.3], wehave (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X j ≥ x ( j ) ψ ( ζ ) j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) H ( D ) ≤ C (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X j ≥ x ( j ) ζ j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) H ( D ) = C k x k ℓ . ESTRICTION THEOREMS FOR HANKEL OPERATORS 17
Putting this together, we obtain the required statement. (cid:3)
Proof of Theorem 5.3.
Let us write q ϕ ( ξ ) = q ϕ ( ξ ) q ϕ ( − ξ ) , q ϕ ( ξ ) = q ϕ ( ξ )( ξ + i ) , q ϕ ( ξ ) = − ξ + i , so that ϕ = ϕ ∗ ϕ . By (5.3) combined with the condition on the support of ϕ ,we have q ϕ , q ϕ ∈ H ( C + ) and so ϕ , ϕ ∈ L (0 , ∞ ).For i = 1 ,
2, let Φ i : ℓ ( Z + ) → L (0 , ∞ ) be the mapΦ i : x = { x ( j ) } j ≥ X j ≥ x ( j ) T jν ϕ i . Observe that ~ T jν ϕ i ( ξ ) = q ν ( ξ ) j q ϕ i ( ξ ) . Further, since by hypothesis ν ([0 , ∞ )) ≤
1, we have that the inverse Fourier trans-form q ν is in H ∞ ( C + ) with k q ν k H ∞ ( C + ) ≤ is bounded. By applying the inverse Fourier transform,it suffices to check that the map x = { x ( j ) } j ≥ X j ≥ x ( j ) q ν ( ξ ) j ! q ϕ ( ξ )is bounded from ℓ to H ( C + ). Recalling the definition of q ϕ , we see that thisimmediately follows from Lemma 5.4.To prove that Φ is bounded, we write q ϕ ( ξ ) = h ( ξ ) ξ + i , h ( ξ ) = q ϕ ( ξ )( ξ + i ) . By (5.3), we have h ∈ H ∞ ( C + ), and so the boundedness of Φ again follows by anapplication of Lemma 5.4.It remains to check formula (5.4). This is the same argument as the one in theproof of Theorem 5.1. Indeed, we have T j + kν ϕ = T j + kν ( ϕ ∗ ϕ ) = ( T jν ϕ ) ∗ ( T kν ϕ ) , and therefore α ( j + k ) = Z ∞ a ( t ) Z R ( T jν ϕ )( t − s )( T kν ϕ )( s ) ds dt. Since supp T jν ϕ i ⊂ [0 , ∞ ), by a change of variable this can be rewritten as α ( j + k ) = Z ∞ Z ∞ a ( t + s )( T jν ϕ )( t )( T kν ϕ )( s ) ds dt. Now we see that( H ( a )Φ x, Φ x ) = X j,k ≥ x ( j ) x ( k ) Z ∞ Z ∞ a ( t + s )( T jν ϕ )( t )( T kν ϕ )( s ) ds dt = ( H ( α ) x, x ) . (cid:3) Example 5.5.
For t ≥
0, let ϕ ( t ) = − πte − πt and ν ( t ) = δ ( t ) − πe − πt . Then q ϕ ( ξ ) = 1 π ( ξ + i ) and q ν ( ξ ) = ξ − iξ + i ∈ L ∞ ( R ) . Hence the conclusions of Theorem 5.3 hold. However, we can say more in thiscase. We also have that ϕ = ψ ∗ ψ , with ψ ( t ) = − i √ πe − πt , t ≥
0, and so wecan take Φ x = Φ x = X j ≥ x ( j ) T jν ψ in the proof of Theorem 5.3. It can be shown that T jν ψ = u j , j ≥
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Department of Mathematics, King’s College London, Strand, London WC2R2LS, United Kingdom
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