Classes of pseudo-differential operators on compact Lie groups and - PowerPoint PPT Presentation

Notation, basic setting Symbol classes and operators Examples Classes of pseudo-differential operators on compact Lie groups and global hypoellipticity Jens Wirth Vienna, October 2012 The talk is based on joint work with Michael Ruzhansky (London) and Ville Turunen (Helsinki).

Notation, basic setting Symbol classes and operators Examples Notation, basic setting 1 Groups, representations and Fourier transform Sequence spaces on the unitary dual Operators, kernels and symbols Symbol classes and operators 2 Difference operators Symbol classes of type ( ρ, δ ) Ellipticity and hypoellipticity L p -multiplier theorems Examples 3 Symbols and difference operators on the sphere S 3 The sub-Laplacian on S 3 Subelliptic first order operators Further examples

Notation, basic setting Groups, representations and Fourier transform Symbol classes and operators Sequence spaces on the unitary dual Examples Operators, kernels and symbols Notation / basic setting: G compact Lie group (e.g., T n , S 3 ⊂ H , SU ( n ), SO ( n ), etc.) � G the unitary dual to G , consisting of equivalence classes of irreducible representations ξ : G → U ( d ξ ), i.e., ξ ( xy ) = ξ ( x ) ξ ( y ) for all x , y ∈ G , for all v ∈ C d ξ \ { 0 } . span { ξ ( x ) v : x ∈ G } = C d ξ Fourier transform for φ ∈ L 2 ( G ) � F [ φ ]( ξ ) = � φ ( x ) ξ ∗ ( x ) d x ∈ C d ξ × d ξ φ ( ξ ) = G � � � ξ ( x ) � φ ( x ) = G d ξ Tr φ ( ξ ) [ ξ ] ∈ � Plancherel identity � G d ξ � � � φ � 2 φ ( ξ ) � 2 L 2 = HS [ ξ ] ∈ �

Notation, basic setting Groups, representations and Fourier transform Symbol classes and operators Sequence spaces on the unitary dual Examples Operators, kernels and symbols Sequence spaces: We fix one representative from each class in � G and consider sequences from Σ( � G ) := { σ : � G ∋ ξ �→ σ ( ξ ) ∈ C d ξ × d ξ } . We denote � ξ � = max { 1 , λ ξ } for Laplace eigenvalues L ξ = − λ 2 ξ ξ . Rapid decay and moderate growth s ( � G ) = { σ ∈ Σ( � G ) : ∀ M � σ ( ξ ) � � � ξ � − M } s ′ ( � G ) = { σ ∈ Σ( � G ) : ∃ M � σ ( ξ ) � � � ξ � M }

Notation, basic setting Groups, representations and Fourier transform Symbol classes and operators Sequence spaces on the unitary dual Examples Operators, kernels and symbols ℓ p spaces: ℓ p ( � G ) = { σ ∈ Σ( � G ) : � σ � ℓ p < ∞} where � � σ � p G d ξ � σ ( ξ ) � p � σ � ℓ ∞ = sup ξ � σ ( ξ ) � op ℓ p = I p , [ ξ ] ∈ � Then the following mapping properties follow: F : L 2 ( G ) → ℓ 2 ( � G ) is unitary G ) and F − 1 : ℓ 1 ( � F : L 1 ( G ) → ℓ ∞ ( � G ) → L ∞ ( G ) are contractions. Remark: ℓ p ( � G ) spaces form an interpolating scale with respect to complex interpolation.

Notation, basic setting Groups, representations and Fourier transform Symbol classes and operators Sequence spaces on the unitary dual Examples Operators, kernels and symbols L 2 -based Sobolev spaces can be characterised by Fourier transform. ξ �→ � ξ � s � f ( ξ ) ∈ ℓ 2 ( � f ∈ H s ( G ) ⇐ ⇒ G ) f ∈ C ∞ ( G ) � f ∈ s ( � ⇐ ⇒ G ) f ∈ s ′ ( � f ∈ D ′ ( G ) � ⇐ ⇒ G )

Notation, basic setting Groups, representations and Fourier transform Symbol classes and operators Sequence spaces on the unitary dual Examples Operators, kernels and symbols Operators and symbols: Let A : C ∞ ( G ) → C ∞ ( G ). Then we associate to A a symbol σ A ( x , ξ ) = ξ ∗ ( x )( A ξ )( x ) such that � � � ξ ( x ) σ A ( x , ξ ) � A φ ( x ) = G d ξ Tr φ ( ξ ) . [ ξ ] ∈ � Questions / Problems under consideration characterisation of operator classes in terms of symbols mapping properties symbolic calculus

Notation, basic setting Groups, representations and Fourier transform Symbol classes and operators Sequence spaces on the unitary dual Examples Operators, kernels and symbols Lemma The symbol σ A is the Fourier transform of the right convolution kernel R A of the operator A with respect to the second variable. Indeed, by Schwartz kernel theorem the operator A has a distributional kernel und by substitution also a (right) convolution kernel, � � R A ( x , y − 1 x ) f ( y ) d y , Af ( x ) = K A ( x , y ) f ( y ) d y = G G and thus � R A ( x , y ) ξ ∗ ( y ) d y . σ A ( x , ξ ) = G

Difference operators Notation, basic setting Symbol classes of type ( ρ, δ ) Symbol classes and operators Ellipticity and hypoellipticity Examples L p -multiplier theorems Difference operators Q ∈ diff k ( � G ) of order k acting on s ′ ( � G ): Q σ = F [ q ( x ) F − 1 σ ] are defined in terms of functions q ∈ C ∞ ( G ) with ∂ α x q ( e ) = 0, | α | < k . △ 1 , . . . , △ n ∈ diff 1 ( � G ) are admissible if the functions q 1 , . . . q n satisfy d q j ( e ) � = 0 , rank { d q 1 ( e ) , . . . , d q n ( e ) } = dim G for e ∈ G the identity element; strongly admissible if in addition { e } = ∩ j { q j ( x ) = 0 } . ξ = △ α 1 We use multi-index notation △ α 1 · · · △ α n n , α ∈ N n 0 and define associated differential operators ∂ ( α ) ∈ Diff | α | ( G ) s.th. x � q α ( x ) ∂ ( α ) f ( x ) − f ( e ) vanishes to order N in e ∈ G . x α ! | α | < N

Difference operators Notation, basic setting Symbol classes of type ( ρ, δ ) Symbol classes and operators Ellipticity and hypoellipticity Examples L p -multiplier theorems We define the symbol class S m ρ,δ ( G ) with symbolic estimates � ∂ β x △ α ξ σ A ( x , ξ ) � op ≤ C α,β � ξ � m − ρ | α | + δ | β | . If ρ > δ any s.a. selection of difference operators defines the same class. Theorem (R.-T.-W. 2010) 1 A ∈ Ψ m ( G ) if and only if σ A ∈ S m 1 , 0 ( G ). 1 H¨ ormander class of pseudo-differential operators on compact Lie groups and global hypoellipticity , arXiv:1004.4396

Difference operators Notation, basic setting Symbol classes of type ( ρ, δ ) Symbol classes and operators Ellipticity and hypoellipticity Examples L p -multiplier theorems The following theorem gives the ’usual’ symbolic calculus. Theorem (R.-T. 2009) 2 Assume A , B : C ∞ ( G ) → C ∞ ( G ) have symbols σ A ∈ S m 1 ρ,δ ( G ) and σ B ∈ S m 2 ρ,δ ( G ) with ρ > δ . Then the concatenation A ◦ B belongs to Op S m 1 + m 2 ( G ) and satisfies σ A ◦ B = σ A ♯σ B with ρ,δ � � �� � 1 ∂ ( α ) △ α σ A ♯σ B ∼ ξ σ A σ B . x α ! α 2 Pseudo-differential operators and symmetries , Birkh¨ auser, Basel, 2010.

Difference operators Notation, basic setting Symbol classes of type ( ρ, δ ) Symbol classes and operators Ellipticity and hypoellipticity Examples L p -multiplier theorems To really apply the calculus theorem to prove statements, we need to choose difference operators more carefully. For fixed η ∈ � G we define η D ij ∈ diff( � G ) corresponding to q ij ( x ) = η ij ( x ) − δ ij . Lemma 1 Let σ, τ ∈ s ′ ( � G ). Then we obtain the quadratic Leibniz rule d η � η D ij ( στ ) = ( η D ij σ ) τ + σ ( η D ij τ ) + ( η D ik σ )( η D kj τ ) . k =1

Difference operators Notation, basic setting Symbol classes of type ( ρ, δ ) Symbol classes and operators Ellipticity and hypoellipticity Examples L p -multiplier theorems From σσ − 1 = I , we obtain d η � 0 = ( η D ij σ ) σ − 1 + σ ( η D ij σ − 1 ) + ( η D ik σ )( η D kj σ − 1 ) . k =1 which allows to determine η D ij σ − 1 and prove estimates for it. Choosing enough representations η , we obtain a strongly admissible selection. Theorem (R.-T.-W. 2010) 1 A ∈ Ψ m ( G ) is elliptic if and only if its symbol σ A ∈ S m 1 , 0 ( G ) is invertible for all x and all but finitely many ξ and its inverse satisfies � σ A ( x , ξ ) − 1 � op ≤ C � ξ � − m .

Difference operators Notation, basic setting Symbol classes of type ( ρ, δ ) Symbol classes and operators Ellipticity and hypoellipticity Examples L p -multiplier theorems Similarly, we can construct parametrices for certain hypoelliptic operators. Theorem (R.-T.-W. 2010) 1 Assume that σ A ∈ S m ρ,δ ( G ) for some 1 ≥ ρ > δ ≥ 0 is invertible for all x and all but finitely many ξ and its inverse satisfies � σ A ( x , ξ ) − 1 � op ≤ C � ξ � − m 0 for some m 0 ≤ m and � σ A ( x , ξ ) − 1 � � △ α ξ ∂ β � op ≤ C α,β � ξ � − ρ | α | + δ | β | x σ A ( x , ξ ) if m 0 < m . Then there exists a symbol σ B ∈ S − m 0 ( G ) with ρ,δ σ A ♯σ B = σ B ♯σ A = I modulo S −∞ ( G ).

Difference operators Notation, basic setting Symbol classes of type ( ρ, δ ) Symbol classes and operators Ellipticity and hypoellipticity Examples L p -multiplier theorems We make a short excursion to Fourier multipliers � � � ξ ( x ) σ ( ξ ) � σ ∈ ℓ ∞ ( � T σ φ ( x ) = G d ξ Tr φ ( ξ ) , G ) . [ ξ ] ∈ � Then as operators on L 2 ( G ) T σ : L 2 ( G ) → L 2 ( G ), � T σ � 2 → 2 = � σ � ℓ ∞ , T σ ∈ I p ( L 2 ) if and only if σ ∈ ℓ p ( � G ). The situation on L p ( G ) is more delicate, in the following we ask for a H¨ ormander-Mikhlin type multiplier theorem.