IWOTA Chemnitz 2017 Operational calculus for groups with fjnite - PowerPoint PPT Presentation

IWOTA Chemnitz 2017 Operational calculus for groups with fjnite propagation speed Gordon Blower, and Ian Doust Lancaster University and University of New South Wales; research partially supported by a Scheme 2 grant from the London Mathematical Society August 10, 2017

2 Abstract Let A be the generator of a strongly continuous cosine family operational calculus for integral transforms and functions of A using the generalized harmonic analysis associated to certain hypergroups. It is shown that characters of hypergroups which have Laplace representations give rise to bounded operators on E . Examples include the Mehler–Fock transform. The paper uses functional calculus for the cosine family main results include an operational calculus theorem for Sturm–Liouville hypergroups with Laplace representation as well as analogues to the Kunze–Stein phenomenon in the hypergroup convolution setting. (cos( tA )) t ∈ R on a complex Banach space E . The paper develops an √ cos( t ∆) which is associated with waves that travel at unit speed. The

3 Cosine families wave solutions which propagate at a fjxed fjnite speed. bounded linear operators on E . Let A a closed and densely defjned linear operator in E . Formally, a cosine family on E is a strongly continuous closed densely defjned infjnitesimal generator A and one naturally writes this to defjne Let E be a separable complex Banach space and L ( E ) the algebra of family { C ( t ) } t ∈ R of bounded operators on E such that C ( s − t ) + C ( s + t ) = 2 C ( s ) C ( t ) and C ( 0 ) = I . Such a family admits a cos( tA ) for C ( t ) . Cosine families arise in describing the solutions of well-posed L 2 Cauchy problems ∂ 2 w ∂ w ∂ t 2 = − A 2 w , w ( 0 ) = u , ∂ t ( 0 ) = 0 with initial datum u ∈ L 2 . In classical situations, these systems admit Given a cosine family { cos( tA ) } t ∈ R , various authors have used this to use � ∞ f ( A ) = 1 F f ( t ) cos( tA ) dt 2 π −∞

4 Venturi regions S 0 An important idea is to work with holomorphic functions on ‘Venturi’ regions; that is, those of the form analytic functions on an open subset S of the complex plane. For ω > 0 we let Σ ω denote the strip { z ∈ C : |ℑ z | < ω } and i Σ ω the corresponding vertical strip. For 0 < θ < π , we introduce the open sector θ = { z ∈ C \ { 0 } : | arg z | < θ } and its refmection − S 0 θ = { z : − z ∈ S 0 θ } . V θ,ω = Σ ω ∪ S 0 θ ∪ ( − S 0 θ ) . Likewise, iV θ,ω will denote the corresponding Venturi region with vertical axis. As usual, H ∞ ( S ) will denote the Banach algebra of bounded

multiplicative identity element. In general, hypergroups admit an with the weak topology forms a complex vector space. When equipped X 5 Hypergroups on [ 0 , ∞ ) Let X = [ 0 , ∞ ) , and C c ( X ) the space of compactly supported continuous functions f : X → C . The set M b ( X ) of bounded Radon measures on X with a generalized convolution’ operation M b ( X ) is a convolution measure algebra called a hypergroup or ‘convo’ denoted ( X , ∗ ) . Denote the Dirac point mass at x by ε x ∈ M b ( X ) . It is a hypergroup axiom that for all x , y ∈ X , ε x ∗ ε y is a compactly supported probability measure (generally with infjnite support). The action of ∗ in a hypergroup is in fact completely determined by the convolutions ε x ∗ ε y . When the base space is X = [ 0 , ∞ ) , the convolution ∗ is necessarily commutative, ε 0 is a involution map x �→ x − . For x ∈ X , the left translation operator Λ x is defjned, initially on C c ( X ) by � Λ x f ( y ) = f ( t ) ( ε x ∗ ε y )( dt ) ( x , y ∈ X ) .

6 Convolution and invariant measure X exists an essentially unique Haar measure on X ; that is, a nontrivial X X commutative Banach algebra. One often writes the convolution X It is traditional and useful to write Λ x f ( y ) as f ( x ∗ y ) (although this is not in fact defjning an operation on X ). Since ∗ is commutative, there positive invariant measure m on [ 0 , ∞ ) satisfying � � Λ x f ( y ) m ( dy ) = f ( y ) m ( dy ) ( x ∈ X ) for all f ∈ C c ( X ) . This allows us to defjne a (commutative) convolution between two functions f , g ∈ C c ( X ) by � � ( f ∗ g )( x ) = f ( y ) Λ x g ( y ) m ( dy ) = f ( y ) g ( x ∗ y ) m ( dy ) . This map extends to L 1 ( m ) = L 1 ( X , m ) and makes ( L 1 ( m ) , ∗ ) into a operation as Λ f g = f ∗ g for f , g ∈ L 1 ( m ) .

7 Multiplicative functions and characters 2 A character on the hypergroup X is a bounded and multiplicative X is the set of all characters on X . and this simplifjes some of the defjnitions below. The set of bounded and 1 A continuous function φ : X → C is said to be multiplicative if φ ( x ∗ y ) = φ ( x ) φ ( y ) for all x , y ∈ X and φ ( z ) � = 0 for some z ∈ X . function φ such that φ ( x − ) = φ ( x ) and φ ( 0 ) = 1. The character space ˆ When X = [ 0 , ∞ ) the involution is always the identity x − = x , and the condition that φ ( x − ) = φ ( x ) is equivalent to the condition that φ ( x ) ∈ R multiplicative functions φ λ can be naturally parametrized by a domain S X ⊆ C . This occurs, in particular, for Sturm–Liouville hypergroups, in which case λ is a spectral parameter.

8 The Fourier transform X is always suffjciently large in our context to enable one to do harmonic analysis. We can defjne the Fourier transform of X The character space ˆ f ∈ L 1 ( X ; m ) by setting � ˆ ( φ ∈ ˆ f ( φ ) = f ( x ) φ ( x ) m ( dx ) , X ) . In the case that ˆ X ⊆ { φ λ : λ ∈ S X } we shall write ˆ f ( λ ) rather than ˆ f ( φ λ ) and we can extend ˆ f to be a function of the complex variable λ .

9 The Plancherel measure Theorem be difgerent from the trivial character I . (i) (Levitan) There exists a unique Plancherel measure π 0 supported on a X such that f �→ ˆ closed subset S of ˆ f for f ∈ L 2 ( m ) ∩ L 1 ( m ) extends to a unitary isomorphism L 2 ( m ) → L 2 ( π 0 ) . (ii) (Voigt) There exists a unique positive character φ 0 ∈ S , and φ 0 can

10 Laplace representation for characters Defjnition A hypergroup ( X , ∗ ) is said to have a Laplace representation if ( a , b ) ⊆ S for some 0 < a < b , and for every x ≥ 0, there exists a positive Radon measure τ x on [ − x , x ] such that τ x ([ − x , x ]) = φ 0 ( x ) and for every character φ λ in S � x φ λ ( x ) = cos( λ t ) τ x ( dt ) . − x The integral is taken over [ − x , x ] , and includes any point masses at ± x .

11 Extension of the Fourier transform is bounded and multiplicative; X ; Lemma Suppose that there exist M 0 , ω 0 > 0 such that � x cosh( ω 0 t ) τ x ( dt ) ≤ M 0 ( x ≥ 0 ) . − x 1 Then for all λ ∈ Σ ω 0 the function φ λ : X → C , � x φ λ ( x ) = cos( λ t ) τ x ( dt ) ( x ≥ 0 ) − x 2 for all x ∈ X , the map h x : λ �→ φ λ ( x ) is in H ∞ (Σ ω 0 ) ; 3 R ∪ [ − i ω 0 , i ω 0 ] is contained in ˆ 4 the Fourier transform f �→ ˆ f is bounded L 1 ( m ) → H ∞ (Σ ω 0 ) .

12 Laplace representation and cosine families In this section we shall suppose that the operator A generates a strongly as given in Defjnition 10. In this setting we defjne the family of bounded linear operators Indeed, this enables us to deal with unbounded cosine families, as in Proposition 13 below. continuous cosine family (cos( tA )) t ∈ R on E , and that ( X , ∗ ) is a hypergroup which admits a Laplace representation for its characters φ λ { φ A ( x ) } x ≥ 0 on E by the strong operator convergent integrals � x φ A ( x ) = cos( At ) τ x ( dt ) ( x ≥ 0 ) . − x

13 Operational calculus for hypergroups with Laplace representation Proposition generates a strongly continuous cosine family on E satisfying 0 and defjnes a bounded linear operator on E ; Let ( X , ∗ ) have a Laplace representation satisfying and suppose that A � cos( tA ) � L ( E ) ≤ κ cosh( t ω 0 ) ( t ≥ 0 ) . 1 Then ( φ A ( x )) x > 0 is a uniformly bounded family of operators; 2 for all f ∈ L 1 ( m ) , the following integral converges strongly � ∞ T A ( f ) = f ( x ) φ A ( x ) m ( dx ) 3 for f , g ∈ L 1 ( m ) , T A ( f ∗ g ) = T A ( f ) T A ( g ) , and so the map T A : L 1 ( m ) → L ( E ) is an algebra homomorphism.

14 Kunze Stein phenomenon For a locally compact group G , the space L 1 ( G ) acts boundedly on L 2 ( G ) by left-convolution. That is, if f ∈ L 1 ( G ) then Λ f : g �→ f ∗ g is a bounded operator on L 2 ( G ) . In general, this result does not extend to f ∈ L p ( G ) for p > 1. The Kunze–Stein phenomenon refers to the fact that for certain Lie groups, most classically for G = SL ( 2 , C ) , for 1 ≤ p < 2 one does obtain a bound of the form � f ∗ g � L 2 ( G ) ≤ C p � f � L p ( G ) � g � L 2 ( G ) . Thus the representation Λ : ( L 1 ( G ) , ∗ ) → L ( L 2 ( G )) : f �→ Λ f extends to a bounded linear map Λ : L p ( G ) → L ( L 2 ( G )) . Indeed the classical case is G = SL ( 2 , C ) has a maximal compact subgroup K = SU ( 2 , C ) such that K × K acts upon G via ( h , k ) : g �→ h − 1 gk for h , k ∈ K and g ∈ G , producing a space of orbits G // K = { KgK : g ∈ G } . The double coset space G // K inherits the structure of a commutative hypergroup modelled on X = [ 0 , ∞ ) .