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

1

Notation, basic setting Groups, representations and Fourier transform Sequence spaces on the unitary dual Operators, kernels and symbols

2

Symbol classes and operators Difference operators Symbol classes of type (ρ, δ) Ellipticity and hypoellipticity Lp-multiplier theorems

3

Examples Symbols and difference operators on the sphere S3 The sub-Laplacian on S3 Subelliptic first order operators Further examples

Notation, basic setting Symbol classes and operators Examples Groups, representations and Fourier transform Sequence spaces on the unitary dual Operators, kernels and symbols

Notation / basic setting: G compact Lie group (e.g., Tn, S3 ⊂ H, SU(n), SO(n), etc.)

• G the unitary dual to G , consisting of equivalence classes of

irreducible representations ξ : G → U(dξ), i.e., ξ(xy) = ξ(x)ξ(y) for all x, y ∈ G , span{ξ(x)v : x ∈ G } = Cdξ for all v ∈ Cdξ \ {0}. Fourier transform for φ ∈ L2(G ) F[φ](ξ) = φ(ξ) =

• G

φ(x)ξ∗(x)dx ∈ Cdξ×dξ φ(x) =

• [ξ]∈

G dξ Tr

• ξ(x)

φ(ξ)

• Plancherel identity

φ2

L2 =

• [ξ]∈

G dξ

φ(ξ)2

HS

Notation, basic setting Symbol classes and operators Examples Groups, representations and Fourier transform Sequence spaces on the unitary dual 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) ∈ Cdξ×dξ}. We denote ξ = max{1, λξ} for Laplace eigenvalues Lξ = −λ2

ξξ.

Rapid decay and moderate growth s( G ) = {σ ∈ Σ( G ) : ∀M σ(ξ) ξ−M} s′( G ) = {σ ∈ Σ( G ) : ∃M σ(ξ) ξM}

Notation, basic setting Symbol classes and operators Examples Groups, representations and Fourier transform Sequence spaces on the unitary dual Operators, kernels and symbols

ℓp spaces: ℓp( G ) = {σ ∈ Σ( G ) : σℓp < ∞} where σp

ℓp =

• [ξ]∈

G dξσ(ξ)p Ip,

σℓ∞ = supξ σ(ξ)op Then the following mapping properties follow: F : L2(G ) → ℓ2( G ) is unitary F : L1(G ) → ℓ∞( G ) and F −1 : ℓ1( G ) → L∞(G ) are contractions. Remark: ℓp( G ) spaces form an interpolating scale with respect to complex interpolation.

Notation, basic setting Symbol classes and operators Examples Groups, representations and Fourier transform Sequence spaces on the unitary dual Operators, kernels and symbols

L2-based Sobolev spaces can be characterised by Fourier transform. f ∈ Hs(G ) ⇐ ⇒ ξ → ξs f (ξ) ∈ ℓ2( G ) f ∈ C∞(G ) ⇐ ⇒

• f ∈ s(

G ) f ∈ D′(G ) ⇐ ⇒

• f ∈ s′(

G )

Notation, basic setting Symbol classes and operators Examples Groups, representations and Fourier transform Sequence spaces on the unitary dual Operators, kernels and symbols

Operators and symbols: Let A : C ∞(G ) → C ∞(G ). Then we associate to A a symbol σA(x, ξ) = ξ∗(x)(Aξ)(x) such that Aφ(x) =

• [ξ]∈

G dξ Tr

• ξ(x)σA(x, ξ)

φ(ξ)

• .

Questions / Problems under consideration characterisation of operator classes in terms of symbols mapping properties symbolic calculus

Notation, basic setting Symbol classes and operators Examples Groups, representations and Fourier transform Sequence spaces on the unitary dual Operators, kernels and symbols

Lemma

The symbol σA is the Fourier transform of the right convolution kernel RA

• f 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, Af (x) =

• G

KA(x, y)f (y)dy =

• G

RA(x, y−1x)f (y)dy, and thus σA(x, ξ) =

• G

RA(x, y)ξ∗(y)dy.

Notation, basic setting Symbol classes and operators Examples Difference operators Symbol classes of type (ρ, δ) Ellipticity and hypoellipticity Lp-multiplier theorems

Difference operators Q ∈ diffk( G ) of order k acting on s′( G ): Qσ = F[q(x)F −1σ] are defined in terms of functions q ∈ C ∞(G ) with ∂α

x q(e) = 0, |α| < k.

△1, . . . , △n ∈ diff1( G ) are admissible if the functions q1, . . . qn satisfy dqj(e) = 0, rank{dq1(e), . . . , dqn(e)} = dim G for e ∈ G the identity element; strongly admissible if in addition {e} = ∩j{qj(x) = 0}. We use multi-index notation △α

ξ = △α1 1 · · · △αn n , α ∈ Nn

and define associated differential operators ∂(α)

x

∈ Diff|α|(G ) s.th. f (x) −

• |α|<N

qα(x) α! ∂(α)

x

f (e) vanishes to order N in e ∈ G .

Notation, basic setting Symbol classes and operators Examples Difference operators Symbol classes of type (ρ, δ) Ellipticity and hypoellipticity Lp-multiplier theorems

We define the symbol class S m

ρ,δ(G ) with symbolic estimates

∂β

x △α ξ σA(x, ξ)op ≤ Cα,βξm−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 ∈ Ψm(G ) if and only if σA ∈ S m

1,0(G ).

1H¨

• rmander class of pseudo-differential operators on compact Lie groups and global

hypoellipticity, arXiv:1004.4396

Notation, basic setting Symbol classes and operators Examples Difference operators Symbol classes of type (ρ, δ) Ellipticity and hypoellipticity Lp-multiplier theorems

The following theorem gives the ’usual’ symbolic calculus.

Theorem (R.-T. 2009)2

Assume A, B : C ∞(G ) → C ∞(G ) have symbols σA ∈ S m1

ρ,δ (G ) and

σB ∈ S m2

ρ,δ (G ) with ρ > δ. Then the concatenation A ◦ B belongs to

Op S m1+m2

ρ,δ

(G ) and satisfies σA◦B = σA♯σB with σA♯σB ∼

• α

1 α!

• △α

ξ σA

• ∂(α)

x

σB

• .

2Pseudo-differential operators and symmetries, Birkh¨

auser, Basel, 2010.

Notation, basic setting Symbol classes and operators Examples Difference operators Symbol classes of type (ρ, δ) Ellipticity and hypoellipticity Lp-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

ηDij ∈ diff(

G ) corresponding to qij(x) = ηij(x) − δij.

Lemma1

Let σ, τ ∈ s′( G ). Then we obtain the quadratic Leibniz rule

ηDij(στ) = (ηDijσ)τ + σ(ηDijτ) + dη

• k=1

(ηDikσ)(ηDkjτ).

Notation, basic setting Symbol classes and operators Examples Difference operators Symbol classes of type (ρ, δ) Ellipticity and hypoellipticity Lp-multiplier theorems

From σσ−1 = I, we obtain 0 = (ηDijσ)σ−1 + σ(ηDijσ−1) +

dη

• k=1

(ηDikσ)(ηDkjσ−1). which allows to determine ηDijσ−1 and prove estimates for it. Choosing enough representations η, we obtain a strongly admissible selection.

Theorem (R.-T.-W. 2010)1

A ∈ Ψm(G ) is elliptic if and only if its symbol σA ∈ S m

1,0(G ) is invertible

for all x and all but finitely many ξ and its inverse satisfies σA(x, ξ)−1op ≤ Cξ−m.

Notation, basic setting Symbol classes and operators Examples Difference operators Symbol classes of type (ρ, δ) Ellipticity and hypoellipticity Lp-multiplier theorems

Similarly, we can construct parametrices for certain hypoelliptic operators.

Theorem (R.-T.-W. 2010)1

Assume that σA ∈ 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 σA(x, ξ)−1op ≤ Cξ−m0 for some m0 ≤ m and σA(x, ξ)−1 △α

ξ ∂β x σA(x, ξ)

• p ≤ Cα,βξ−ρ|α|+δ|β|

if m0 < m. Then there exists a symbol σB ∈ S −m0

ρ,δ

(G ) with σA♯σB = σB♯σA = I modulo S −∞(G ).

Notation, basic setting Symbol classes and operators Examples Difference operators Symbol classes of type (ρ, δ) Ellipticity and hypoellipticity Lp-multiplier theorems

We make a short excursion to Fourier multipliers Tσφ(x) =

• [ξ]∈

G dξ Tr

• ξ(x)σ(ξ)

φ(ξ)

• ,

σ ∈ ℓ∞( G ). Then as operators on L2(G ) Tσ : L2(G ) → L2(G ), Tσ2→2 = σℓ∞, Tσ ∈ Ip(L2) if and only if σ ∈ ℓp( G ). The situation on Lp(G ) is more delicate, in the following we ask for a H¨

• rmander-Mikhlin type multiplier theorem.

Notation, basic setting Symbol classes and operators Examples Difference operators Symbol classes of type (ρ, δ) Ellipticity and hypoellipticity Lp-multiplier theorems

Let for the following Z(G ) = {e}, ∆0 be the set of roots of G and ρ(x) = dim G − Tr Ad(x) =

• η∈∆0(dη − Tr η(x))

with associated difference △ ∗ ∈ diff2( G ). Let further be the s.a. selection

• f differences be given by ηDij, η ∈ ∆0 and κ > 1

2 dim G even.

Theorem (R.-W. 2011)4

Assume for |α| ≤ κ − 1 △α

ξ σ(ξ)op ≤ Cξ−|α|,

△ ∗ κ/2σ(ξ)op ≤ Cξ−κ. Then Tσ is weak type (1,1) and Lp-bounded for 1 < p < ∞.

4Lp-multipliers on compact Lie groups, Funct. Anal. Appl., to appear.

Notation, basic setting Symbol classes and operators Examples Difference operators Symbol classes of type (ρ, δ) Ellipticity and hypoellipticity Lp-multiplier theorems

Corollary4

Assume σA(ξ) ∈ S 0

ρ,0(G ) for ρ ∈ [0, 1]. Then

A : Wp,rp(G ) → Lp(G ) for all 1 < p < ∞ and with rp = κ(1 − ρ)|1/p − 1/2|.

Corollary4

Let X be a left-invariant vector field on G . Then there exists a discrete exceptional set C ⊂ iR such that for all c ∈ C the operator X + c is invertible with inverse in Op S 0

0,0(G ) and consequently

f Lp(G ) ≤ Cp(X + c)f Wp,rp (G ) holds true for all 1 < p < ∞ and with rp = κ|1/p − 1/2|.

Notation, basic setting Symbol classes and operators Examples Symbols and difference operators on the sphere S3 The sub-Laplacian on S3 Subelliptic first order operators Further examples

In the remaining part we consider the case G = S3. G = S3 ∼ = SU(2)

• G ≃ 1

2N0 with dℓ = 2ℓ + 1,

we use (half-) integers as m, n = −ℓ, −ℓ + 1, . . . , ℓ as indices in matrices. Orthonormal basis of invariant vector fields: D1, D2, D3 Corresponding symbols σD1(ℓ)m,n = i 2(

• (ℓ − n)(ℓ + n + 1)δℓ

m,n+1

+

• (ℓ + m + 1)(ℓ − m)δℓ

m+1,n)

σD2(ℓ)m,n = 1 2(

• (ℓ − n)(ℓ + n + 1)δℓ

m,n+1

−

• (ℓ + m + 1)(ℓ − m)δℓ

m+1,n)

σD3(ℓ)m,n = −imδℓ

mn

Notation, basic setting Symbol classes and operators Examples Symbols and difference operators on the sphere S3 The sub-Laplacian on S3 Subelliptic first order operators Further examples

First order operators (creation, annihilation and neutral operator) ∂± = iD1 ∓ D2, and ∂0 = iD3 with (simpler) symbols σ∂+(ℓ)m,n = −

• (ℓ − n)(ℓ + n + 1)δℓ

m,n+1

σ∂−(ℓ)m,n = −

• (ℓ + m + 1)(ℓ − m)δℓ

m+1,n

σ∂0(ℓ)m,n = mδℓ

mn

Laplacian L = D2

1 + D2 2 + D2 3 = −∂2 0 − 1

2(∂−∂+ + ∂+∂−) with symbol σL(ℓ) = −ℓ(ℓ + 1)δℓ

m,n

Notation, basic setting Symbol classes and operators Examples Symbols and difference operators on the sphere S3 The sub-Laplacian on S3 Subelliptic first order operators Further examples

Difference operators Dij in terms of the representation with ℓ = 1/2 σ∂0 σ∂+ σ∂− σL D11

1 2I

−σ∂0 + 1

4I

D12 I −σ∂− D21 I −σ∂+ D22 − 1

2I

σ∂0 + 1

4I

Notation, basic setting Symbol classes and operators Examples Symbols and difference operators on the sphere S3 The sub-Laplacian on S3 Subelliptic first order operators Further examples

Example 1: Sub-Laplacian The Sub-Laplacian Ls = D2

1 + D2 2 has a parametrix L−♯ s

∈ Op S −1

1/2,0(G ).

Therefore, the following sub-elliptic estimate φWp,s ≤ CpLsφWp,s+rp holds true with rp = |1/p − 1/2| − 1 and for 1 < p < ∞, s ∈ R.

Notation, basic setting Symbol classes and operators Examples Symbols and difference operators on the sphere S3 The sub-Laplacian on S3 Subelliptic first order operators Further examples

Example 2: Perturbations by lower order terms The operator H = D3 − D2

1 − D2 2 = D3 − Ls is also hypoelliptic and has a

parametrix H♯ ∈ Op(S −1

1/2,0(G )). This can be shown in analogy to the

case of the sub-Laplacian. Therefore, the following sub-elliptic estimates φWp,s ≤ CpHφWp,s+rp are valid for rp = |1/p − 1/2| − 1, 1 < p < ∞ and s ∈ R.

Notation, basic setting Symbol classes and operators Examples Symbols and difference operators on the sphere S3 The sub-Laplacian on S3 Subelliptic first order operators Further examples

Example 3: Resolvents of vector fields The operator Rζ = (D3 + ζ)−1 for ζ ∈ i

2Z belongs to Op S 0 0,0(G ).

Therefore, the following sub-elliptic estimates φWp,s ≤ Cp(D3 + ζ)φWp,s+rp are valid for rp = 2|1/p − 1/2|, 1 < p < ∞.

Notation, basic setting Symbol classes and operators Examples Symbols and difference operators on the sphere S3 The sub-Laplacian on S3 Subelliptic first order operators Further examples

Example 4: Global hypoellipticity The operator P = 2iD3

3 − L is globally hypoelliptic. Its symbol is invertible

for all ℓ = 0 and satisfies uniform bounds σP(ℓ)−1op ≤ C for all remaining quantum numbers ℓ. Example 5: A non-hypoelliptic operator The operator Q = D2

1 − D2 2 = − 1 2(∂2 − + ∂2 +) has a symbol which is only

invertible for ℓ + 1

2 ∈ 2N0. Thus the operator is not hypoelliptic nor does

it satisfy any sub-elliptic estimates. Its kernel is infinite dimensional and contains distributions of arbitrary order.

Notation, basic setting Symbol classes and operators Examples Symbols and difference operators on the sphere S3 The sub-Laplacian on S3 Subelliptic first order operators Further examples

• M. Ruzhansky and V. Turunen,

Pseudo-differential operators and symmetries, Birkh¨ auser, Basel, 2010.

• M. Ruzhansky, V. Turunen and J. Wirth,

H¨

• rmander class of pseudo-differential operators on compact Lie

groups and global hypoellipticity, preprint, arXiv:1004.4396.

• M. Ruzhansky and J. Wirth,

Lp-multipliers on compact Lie groups,

• Funct. Anal. Appl., to appear

preprint, arXiv:1102.3988.