The theory of pseudo-differential operators on the noncommutative n ...

The Theory of Pseudo-differential Operators on the Noncommutative n-Torus

arXiv:1704.02507v1 [math.OA] 8 Apr 2017

Jim Tao April 11, 2017

1

Introduction

The methods of spectral geometry are useful for investigating the metric aspects of noncommutative geometry [1, 3–5] and in these contexts require extensive use of pseudo-differential operators. In the foundational paper [2], Connes showed that, by direct analogy with the theory [12–14] of pseudo-differential operators on Rn , one may derive a similar pseudo-differential calculus on noncommutative n tori Tnθ . With the development of this calculus came many results concerning the local differential geometry of noncommutative tori for n = 2, 4, as shown in the groundbreaking paper [7] in which the Gauss–Bonnet theorem on T2θ is proved and later papers [6, 8–11]. In these papers, the flat geometry of Tnθ which was studied in [2] is conformally perturbed using a Weyl factor given by a positive invertible smooth element in C ∞ (Tnθ ). Connes’ pseudodifferential calculus is critically used to apply heat kernel techniques to geometric operators on Tnθ to derive small time heat kernel expansions that encode local geometric information such as scalar curvature. As discovered in [6,9,11], a purely noncommutative feature that appears in the computations and in the final formula for the curvature is the modular automorphism of the state implementing the conformal perturbation of the metric. Certain details of the proofs in the derivation of the calculus in [2] were omitted, such as the evaluation of oscillatory integrals, so we make it the objective of this paper to fill in all the details. After reproving in more detail the formula for the symbol of the adjoint of a pseudo-differential operator and the formula for the symbol of a product of two pseudo-differential operators, we define the corresponding analog of Sobolev spaces for which we prove the Sobolev and Rellich lemmas. We then extend these results to finitely generated projective right modules over the noncommutative n torus. We list these results below. Theorem 1.1. Suppose P is a pseudodifferential operator with symbol σ(P ) = ρ = ρ(ξ) of order M . Then the symbol of the adjoint P ∗ is of order M and satisfies σ(P ∗ ) ∼

X ∂ ℓ δ ℓ [(ρ(ξ))∗ ] . ℓ1 !ℓ2 ! n

ℓ∈Z≥0

Theorem 1.2. Suppose that P is a pseudodifferential operator with symbol σ(P ) = ρ = ρ(ξ) of order M1 , and Q is a pseudodifferential operator with symbol σ(Q) = φ = φ(ξ) of order M2 . Then the symbol of the product QP is of order M1 + M2 and satisfies σ(QP ) ∼ where ∂ ℓ :=

Q

j

ℓ

∂j j and δ ℓ :=

Q

X

ℓ∈Zn ≥0

1 ∂ ℓ φ(ξ)δ ℓ ρ(ξ), ℓ1 !ℓ2 !

ℓ

j

δj j .

Theorem 1.3. For s > k + 1, H s ⊆ Akθ . Theorem 1.4. Let {aN } ∈ A∞ θ be a sequence. Suppose that there is a constant C so that ||aN ||s ≤ C for all N . Let s > t. Then there is a subsequence {aNj } that converges in H t . 1

Theorem 1.5. (a) For a pseudodifferential operator P with r × r matrix valued symbol σ(P ) = ρ = ρ(ξ), the symbol of the adjoint P ∗ satisfies σ(P ∗ ) ∼

X

(ℓ1 ,...,ℓn )∈(Z≥0 )n

∂1ℓ1 · · · ∂nℓn δ1ℓ1 · · · δnℓn (ρ(ξ))∗ . ℓ1 ! · · · ℓn !

(b) If Q is a pseudodifferential operator with r × r matrix valued symbol σ(Q) = ρ′ = ρ′ (ξ), then the product P Q is also a pseudodifferential operator and has symbol σ(P Q) ∼

X

(ℓ1 ,...,ℓn )∈(Z≥0 )n

∂1ℓ1 · · · ∂nℓn (ρ(ξ))δ1ℓ1 · · · δnℓn (ρ′ (ξ)) . ℓ1 ! · · · ℓn !

Theorem 1.6. For s > k + 1, H s ⊆ (Akθ )r e.

r aN ||s ≤ C Theorem 1.7. Let {~aN } ∈ (A∞ θ ) e be a sequence. Suppose that there is a constant C so that ||~ for all N . Let s > t. Then there is a subsequence {~aNj } that converges in H t .

2

Preliminaries

Fix some skew symmetric n × n matrix θ with upper triangular entries in R\Q that are linearly independent over Q. Consider the irrational rotation C ∗ -algebra Aθ with n unitary generators U1 , . . . , Un which satisfy Uk Uj = e2πiθj,k Uj Uk and Uj∗ = Uj−1 . Let {αs }s∈Rn be a n-parameter group of automorphisms given by Q m Q mj 7→ eis·m j Uj j . We define the subalgebra Akθ of C k elements of Aθ to be those a ∈ Aθ such that j Uj the mapping Rn → Aθ given by s 7→ αs (a) is C k , and we define the subalgebra A∞ θ of smooth elements of Aθ to be those a ∈ Aθ such that the mapping Rn → Aθ given by s 7→ αs (a) is smooth. An alternative definition of theP subalgebraQ A∞ θ of smooth elements is the elements in Aθ that can be expressed by an expansion of the m form m∈Zn am j Uj j , where the sequence {am }m∈Zn is in the Schwartz space S(Zn ) in the sense that, for all α ∈ Zn , Y sup ( |mj |αj |am |) < ∞. m∈Zn

j

Q m Define the trace τ : Aθ → C by τ ( j Uj j ) = 0 for mj not all zero and τ (1) = 1 and define an inner product h·, ·i : Aθ × Aθ → C by ha, bi = τ (b∗ a) with induced norm || · || : Aθ → R≥0 . Let Dj = −i∂j and define Q ℓ derivations δj by the relations δj (Uj ) = Uj and δj (Uk ) = 0 for j 6= k. For convenience, denote ∂ ℓ := j ∂j j , Q Q ℓ ℓ δ ℓ := j δj j , and Dℓ := j Dj j . We define a map ψ : ρ 7→ Pρ assigning a pseudo-differential operator on ∞ ∞ n Aθ to a symbol ρ ∈ C (R , A∞ θ ).

∞ Definition 2.1. For ρ ∈ C ∞ (Rn , A∞ θ ), let Pρ be the pseudo-differential operator sending arbitrary a ∈ Aθ to Z Z 1 Pρ (a) := e−is·ξ ρ(ξ)αs (a) ds dξ. (2π)n Rn Rn

The integral above does not converge absolutely; it is an oscillatory integral. We define oscillatory integrals below as in [13]. Definition 2.2. Let q be a nondegenerate real quadratic form on Rn , a be a C ∞ complex-valued function defined on Rn such that the functions (1 + |x|2 )−m/2 ∂ α a(x) are bounded on Rn for all α ∈ Zn≥0 , and ϕ be a Schwartz function, i.e. the functions xα ∂ β ϕ(x) are bounded on Rn for all pairs α, β ∈ Zn≥0 . Suppose further that ϕ(0) = 1. Then the limit Z eiq(x) a(x)ϕ(ǫx) dx R exists, is independent of ϕ (as long as = 1), and is equal to eiq(x) a(x) dx when a ∈ L1 . When a 6∈ L1 , R ϕ(0) we continue to denote this limit by eiq(x) a(x) dx, and have an estimate Z iq(x) e ≤ Cq,m a(x) dx max inf{U ∈ R : |(1 + |x|2 )−m/2 ∂ α a| ≤ U almost everywhere} lim

ǫ→0

|α|≤m+n+1

2

where Cq,m depends only on the quadratic form q and the order m. As shown in [13], oscillatory integrals behave essentially like absolutely convergent integrals in that one can still make changes of variables, integrate by parts, differentiate under the integral sign, and interverse integral signs. Given certain conditions on ρ, Pρ (a) satisfies the conditions of the oscillatory integral, and we can evaluate Pρ (a). Definition 2.3. An element ρ = ρ(ξ) = ρ(ξ1 , . . . , ξn ) of C ∞ (Rn , A∞ θ ) is a symbol of order m if and only if for all non-negative integers i1 , . . . , in , j1 , . . . , jn ||δ1i1 · · · δnin (∂1j1 · · · ∂njn ρ)(ξ)|| ≤ Cρ (1 + |ξ|)m−|j| . Example 2.6(i) of [13] gives a convenient formula for evaluating the oscillatory integrals that appear in our calculation of Pρ (a). Proposition 2.4. Suppose that, for some m, a is a C ∞ complex-valued function defined on Rn such that the functions (1 + |x|2 )−m/2 ∂ α a(x) are bounded on Rn for all α ∈ Zn≥0 . Then 1 (2π)n

Z

Z

1 (2π)n

e−iy·η a(y) dy dη =

Rn Rn

Z

Z

e−iy·η a(η) dy dη = a(0).

Rn Rn

We apply Proposition 2.4 to get a basic result. Q m P ∞ n ∞ Lemma 2.5. Let a = m∈Zn am j Uj j be an arbitrary element of A∞ θ and let ρ ∈ C (R , Aθ ) be a symbol of order M . Then n X Y m Uj j . Pρ (a) = ρ(m)am m∈Zn

Proof. First consider the case a = 

Pρ 

n Y

j=1

Q

mj j Uj .



m Uj j  =

= = =

j=1

We get 

n Y



1 (2π)n

Z

Z

1 (2π)n

Z

Z

1 (2π)n

Z

Z

e−is·(ξ−m) ρ(ξ) ds dξ

1 (2π)n

Z

Z

e−is·η ρ(η + m) ds dη

= ρ(m)

Rn

Rn

Rn

e−is·ξ ρ(ξ)αs 

j=1

e−is·ξ ρ(ξ)eis·m

Rn

n Y

m Uj j  ds dξ

n Y

ds dξ

j=1

Rn Rn

Rn

mj

Uj

Rn

n Y

mj

Uj

j=1 n Y

mj

Uj

j=1

m

Uj j ,

j=1

as desired, Q m η = ξ − m and applied the result of Proposition 2.4. Now consider the general Phaving substituted case a = m∈Zn am j Uj j . Since αs is an automorphism on Aθ , we get Pρ (a) =

X

ρ(m)am

m∈Zn

n Y

j=1

and we are done.

3

m

Uj j ,

3

Asymptotic formula for the symbol of the adjoint of a pseudodifferential operator

Here we prove the formula for the symbol of the adjoint for the noncommutative n torus, adapting the proof of Lemma 1.2.3 of [12] to the noncommutative n torus. Theorem 3.1. Suppose P is a pseudodifferential operator with symbol σ(P ) = ρ = ρ(ξ) of order M . Then the symbol of the adjoint P ∗ is of order M and satisfies σ(P ∗ ) ∼

X ∂ ℓ δ ℓ [(ρ(ξ))∗ ] . ℓ1 !ℓ2 ! n

ℓ∈Z≥0

Proof. Let a, b ∈ A∞ θ . We have  hPρ (a), bi = τ b∗

1 (2π)n Z Z

Z

Z

e

−is·ξ

ρ(ξ)αs (a) ds dξ

Rn Rn



1 e−is·ξ τ (b∗ ρ(ξ)αs (a)) ds dξ (2π)n Rn Rn Z Z 1 e−is·ξ τ (α−s (ρ(ξ)∗ b)∗ a) ds dξ = (2π)n Rn Rn ∗   Z Z 1 +is·ξ ∗ e α (ρ(ξ) b) ds dξ a =τ −s (2π)n Rn Rn =

= ha, Pρ∗ (b)i

where Z Z 1 e+is·ξ α−s (ρ(ξ)∗ b) ds dξ (2π)n Rn Rn Z Z 1 e+is·ξ α−s (ρ(ξ)∗ )α−s (b) ds dξ = (2π)n Rn Rn X 1 Z Z = e+is·ξ e+is·m ρm (ξ)∗ e−is·k bk ds dξ Un−mn · · · U1−m1 U1k1 · · · Unkn (2π)n Rn Rn m,k X 1 Z Z e−is·η ρm ((k − m) − η)∗ bk ds dη Un−mn · · · U1−m1 U1k1 · · · Unkn = (2π)n Rn Rn m,k X = ρm (k − m)∗ bk Un−mn · · · U1−m1 U1k1 · · · Unkn

Pρ∗ (b) =

m,k

=

X m,k

(ρm (k − m)U1m1 · · · Unmn )∗ (bk U1k1 · · · Unkn ),

so ∗  n X Y m Uj j  σ(Pρ∗ )(ξ) =  ρm (ξ − m) m

j=1

 ∗  Z Z n X Y 1 m Uj j  dx dy  e−ix·y ρm (ξ − y)αx  = (2π)n Rn Rn m j=1 ∗  Z Z 1 e−ix·y αx (ρ(ξ − y)) dx dy = (2π)n Rn Rn 

4

We have X (−y)ℓ (∂ ℓ ρ)(ξ) + RN1 (ξ, y) ℓ! |ℓ| 0 is given, we choose r so that 2C 2 (1 + r2 )t−s < ǫ. The remaining part of√the sum is over |k1 |2 + · · · + |kn |2 ≤ r2 and can be bounded above by ǫ′ := ǫ − 2C 2 (1 + r2 )t−s if i, j ≥ M ( n ǫ′ /(2r + 1)) because a ball of radius r centered at the origin is contained in a cube of side length 2r that has (2r + 1)n lattice points. Then the total sum is bounded above by ǫ, and we are done.

6

The pseudodifferential calculus on finitely generated projective modules over the noncommutative n torus

We can generalize these results to arbitrary finitely generated projective right modules over the noncommutative n torus following p. 553 of [4], which considers finitely generated projective modules over an arbitrary unital ∗-algebra. Let E be a finitely generated projective right A∞ θ -module. Since E is a finitely generated r ∞ r projective right A∞ -module, we can write E as a direct summand E = (A∞ θ θ ) e of a free module (Aθ ) ∞ r ∞ with direct complement F = (Aθ ) (id − e), where the idempotent e ∈ Mr (Aθ ) is self-adjoint. Consider an d r × r matrix valued symbol ρ = (ρj,k ) where ρj,k : Rn → A∞ θ are scalar symbols and ρj,k ∈ S . Define the operator Pρ : E → E as follows: Z Z −n Pρ (~a) := (2π) e−is·ξ ρ(ξ)αs (~a) ds dξ. Rn Rn

Define the inner product h~a, ~bi : E × E → C sending (~a, ~b) 7→ τ (~b∗~a). Since −n

Pρ (~a)j = (2π)

Z

Z

e−is·ξ

Rn Rn

r X

ρj,k (ξ)αs (ak ) ds dξ,

k=1

Lemma 2.5 generalizes to E as follows after applying it to each component: Pρ (~a) =

X

ρ(m)~am

m

n Y

m

Uj j .

j=1

Theorems 3.1 and 4.1 generalize as follows. Theorem 6.1. (a) For a pseudodifferential operator P with r × r matrix valued symbol σ(P ) = ρ = ρ(ξ), the symbol of the adjoint P ∗ satisfies σ(P ∗ ) ∼

X

(ℓ1 ,...,ℓn )∈(Z≥0 )n

∂1ℓ1 · · · ∂nℓn δ1ℓ1 · · · δnℓn (ρ(ξ))∗ . ℓ1 ! · · · ℓn !

(b) If Q is a pseudodifferential operator with r × r matrix valued symbol σ(Q) = ρ′ = ρ′ (ξ), then the product P Q is also a pseudodifferential operator and has symbol σ(P Q) ∼

X

(ℓ1 ,...,ℓn )∈(Z≥0 )n

∂1ℓ1 · · · ∂nℓn (ρ(ξ))δ1ℓ1 · · · δnℓn (ρ′ (ξ)) . ℓ1 ! · · · ℓn ! 13

Proof. First let’s prove part (a). Let ρ be an r × r matrix valued symbol of order M and ~a, ~b ∈ E. We have   Z Z 1 ∗ −is·ξ ~ ~ hPρ (~a), bi = τ b e ρ(ξ)αs (~a) ds dξ (2π)n Rn Rn ∗   Z Z 1 +is·ξ ∗~ e α−s (ρ(ξ) b) ds dξ ~a =τ (2π)n Rn Rn = h~a, P ∗ (~b)i ρ

where Pρ∗ (~b)j =

1 (2π)n

Z

Z

e+is·ξ

=

1 (2π)n

Z

Z

e+is·ξ

=

X m,p

Rn Rn

Rn

Rn

(ρm (p − m)

r X

k=1 r X

α−s (ρ(ξ)∗j,k bk ) ds dξ α−s (ρ(ξ)∗j,k )α−s (bk ) ds dξ

k=1 n Y

Uhmh )∗ (bp

h=1

n Y

Uhph )

h=1

so σ(Pρ∗ )(ξ) =

" X m

ρm (ξ − m)

n Y

Uhmh

h=1

#∗

n X Y 1 −ix·y e ρ (ξ − y)α Uhmh = m x (2π)n Rn Rn m h=1 ∗  Z Z 1 e−ix·y αx (ρ(ξ − y)) dx dy . = (2π)n Rn Rn

"

Z

Z

!

dx dy

#∗

The rest of the proof reduces to the r = 1 case, applying it to each entry in ρ = (ρj,k ). We proceed to part (b). Let ρ be an r × r matrix valued symbol of order m1 and φ be an r × r matrix valued symbol of order m2 . Let {ϕk } be a partition of unity and define φk (ξ) := φ(ξ)ϕk (ξ). Let ~a ∈ E. We have Z Z 1 Pφk (Pρ (~a)) = e−is·ξ φk (ξ)αs (Pρ (~a)) ds dξ (2π)n Rn Rn   Z Z Z Z 1 1 −is·ξ −it·η = e φk (ξ)αs e ρ(η)αt (~a) dt dη ds dξ (2π)n Rn Rn (2π)n Rn Rn  Z Z Z Z 1 −is·ξ−it·η e φk (ξ)αs (ρ(η))αs+t (~a) dt dη ds dξ = (2π)2n Rn Rn Rn Rn  Z Z Z Z 1 −ix·ξ−i(y−x)·η = e φk (ξ)αx (ρ(η))αy (~a) dy dη dx dξ (2π)2n Rn Rn Rn Rn  Z Z Z Z 1 −ix·(ξ−η)−iy·η e φk (ξ)αx (ρ(η))αy (~a) dy dη dx dξ = (2π)2n Rn Rn Rn Rn  Z Z Z Z 1 −ix·σ−iy·η = e φk (σ + τ )αx (ρ(τ ))αy (~a) dy dτ dx dσ (2π)2n Rn Rn Rn Rn Z Z  Z Z 1 −iy·τ −ix·σ e e φk (σ + τ )αx (ρ(τ )) dx dσ αy (~a) dy dτ = (2π)2n Rn Rn Rn Rn Z Z 1 e−iy·τ λk (τ )αy (~a) dy dτ = (2π)n Rn Rn where λk (τ ) =

1 (2π)n

Z

Rn

Z

e−ix·σ φk (σ + τ )αx (ρ(τ )) dx dσ

Rn

14

so

1 Pφ (Pρ (~a)) = (2π)n

where λ(τ ) = Let

P∞

k=0

Z

Z

e−iy·τ λ(τ )αy (~a) dy dτ

Rn Rn

λk (τ ). λk (ξ) :=

1 (2π)n

Z

Z

Z

Z

e−ix·y

1 (2π)n

Z

Z

Since λk (ξ)α,γ

1 = (2π)n =

r X

β=1

Rn

e−ix·y φk (ξ + y)αx (ρ(ξ)) dx dy.

Rn

Rn Rn

Rn

r X

φk (ξ + y)α,β αx (ρ(ξ)β,γ ) dx dy

β=1

e−ix·y φk (ξ + y)α,β αx (ρ(ξ)β,γ ) dx dy,

Rn

the rest of the proof reduces to the r = 1 case, applying it to each summand in the above sum. Let λ(ξ) = (1 + ξ12 + · · · + ξn2 )1/2 idE . Consider the following inner product on E. Definition 6.2. Define the Sobolev inner product h·, ·is : E × E → C by X (1 + |m1 |2 + · · · + |mn |2 )s bj,m aj,m . h~a, ~bis := hPλs (~a), Pλs (~b)i = j,m

Note that for s = 0 this agrees with h·, ·i. This inner product induces the following norm. Definition 6.3. Define the Sobolev norm || · ||s : E → R≥0 by X (1 + |m1 |2 + · · · + |mn |2 )s |aj,m |2 . ||~a||2s := hPλs (~a), Pλs (~a)i = j,m

Using this norm, we can define the analog of Sobolev spaces on E. Definition 6.4. Define the Sobolev space H s to be the completion of E with respect to || · ||s . We can prove that a pseudo-differential operator of order d ∈ R continuously maps H s into H s−d . However we must first prove the case where s = d. Theorem 6.5. Suppose ρ is a matrix valued symbol of order d. Then, for any ~a ∈ E, ||Pρ (~a)||0 ≤ C||~a||d for some constant C > 0 and Pρ defines a bounded operator Pρ : H d → H 0 . Proof. Q Let F := {fj : 1 ≤ j ≤ r} be an orthogonal eigenbasis of e normalized with respect to h·, ·i. Note m that { g Ug g fj : m ∈ Zn , 1 ≤ j ≤ r} is an orthogonal basis of E considered as a C-vector space, with Q m Q m respect to h·, ·is . We have || g Ug g fj ||20 = 1, ||ρm (ξ)h,j g Ug g fj ||20 = |ρm (ξ)h,j |2 , and ||ρ(ξ)h,j fj ||20 = P 2 d 2 m |ρm (ξ)h,j | . Since ρh,j is of order d, we have ||ρ(ξ)h,j ||0 ≤ Cρ (1 + |ξ|) , and since (1 − |ξ|) ≥ 0 gives us 2 2 (1 + |ξ|) ≤ 2(1 + |ξ| ), we have ||ρ(ξ)h,j ||20 ≤ Cρ2 (1 + |ξ|)2d ≤ Cρ2 2d (1 + |ξ|2 )d . Let kρ := Cρ2 2d . Then we have X m

|ρm (ξ)h,j |2 ≤ kρ (1 + |ξ|2 )d .

Q mg and Es := {es,m | n, m ∈ Z}. By definition we have Let es,m := (1 + |m1 | + · · · + |mn | ) g Ug Es F orthonormal with respect to h·, ·is . It suffices to prove this theorem for the case ~a = ed,m fj by the orthonormality of Ed F since X |aj,m |2 ||Pρ (ed,m fj )||20 ||Pρ (~a)||20 = 2

2 −s/2

j,m

15

and ||~a||2d = Since

||ed,m fj ||2d

= 1, it suffices to show that

X j,m

|aj,m |2 ||ed,m fj ||2d .

||Pρ (ed,m fj )||20 ≤ K

for some constant K > 0. We have ||Pρ (ed,m fj )||20 = ||ρ(m)ed,m fj ||20

= ||ρ(m)(1 + |m1 |2 + · · · + |mn |2 )−d/2

Y g

Ugmg fj ||20

2 X Y Y ρk (m) Ugkg (1 + |m1 |2 + · · · + |mn |2 )−d/2 Ugmg fj = g g k 0 2 X Y Y ρk (m) Ugmg fj Ugkg = (1 + |m1 |2 + · · · + |mn |2 )−d g g k 0 2 X Y = (1 + |m1 |2 + · · · + |mn |2 )−d ρk (m)w(m, k) Ugkg +mg fj g k 0 2 X Y 2 2 −d kg ρk−m (m)w(m, k − m) = (1 + |m1 | + · · · + |mn | ) Ug fj g k 0 2 X Y ρk−m (m)h,j w(m, k − m) = (1 + |m1 |2 + · · · + |mn |2 )−d Ugkg fj h,k g 0 X 2 2 −d 2 = (1 + |m1 | + · · · + |mn | ) |ρk−m (m)h,j | h,k

2

2 −d

2

2 −d

= (1 + |m1 | + · · · + |mn | )

h,k

≤ (1 + |m1 | + · · · + |mn | ) where w(k, m) :=

n Y

mj

Uj

j=1

so our desired constant is K = rkρ =

rCρ2 2d

n Y

j=1

X

k

|ρk (m)h,j |2

rkρ (1 + |m1 |2 + · · · + |mn |2 )d = rkρ



Uj j 

n Y

mj +kj 

Uj

j=1

and we are done.



∈ S1 ⊂ C

For the general case s 6= d we need to prove a lemma saying that || · ||s = || · ||s−t ◦ Pλt .

Lemma 6.6. For any ~a ∈ E and s, t ∈ R, ~a ∈ H s if and only if Pλt (~a) ∈ H s−t with ||~a||s = ||Pλt (~a)||s−t . Proof. Suppose that ~a ∈ H s or Pλt (~a) ∈ H s−t . Then X (1 + |m1 |2 + · · · + |mn |2 )s−t λ2t (m)|aj,m |2 ||Pλt (~a)||2s−t = j,m

=

X j,m

=

X j,m

(1 + |m1 |2 + · · · + |mn |2 )s−t (1 + |m1 |2 + · · · + |mn |2 )t |aj,m |2 (1 + |m1 |2 + · · · + |mn |2 )s |aj,m |2

= ||~a||s so we know that ~a ∈ H s and Pλt (~a) ∈ H s−t . 16

Then the general case follows quite easily. Corollary 6.7. Suppose ρ is a matrix valued symbol of order d. Then ||Pρ (~a)||s−d ≤ C||~a||s for some constant C > 0 and Pρ defines a bounded operator Pρ : H s → H s−d . Proof. By Lemma 6.6, we have ||Pρ (~a)||s−d = ||Pλs−d (Pρ (~a))||0 . By Proposition 6.1(b), the matrix valued symbol σ(Pλs−d ◦ Pρ ) is of order d + (s − d) = s, so Theorem 6.5 gives us ||Pλs−d (Pρ (~a))||0 ≤ C||~a||s for some constant C > 0. We can also define an analog of the C k norm on E. Definition 6.8. Define the C k norm || · ||∞,k : E → R≥0 as follows: X ||δ ℓ (~a)||C ∗ ||~a||∞,k := |ℓ|≤k

where the C ∗ norm || · ||C ∗ is given by ||~a||2C ∗ := sup{|λ| : ~a∗~a − λ · 1 not invertible}. Q m P Since, for arbitrary ~a = j,m aj,m g Ug g fj , Dsℓ αs (~a) = (−i∂s )ℓ

X

eis·m aj,m

X

mℓ eis·m aj,m

X

δ ℓ eis·m aj,m

j,m

=

Ugmg fj

g=1

j,m

=

n Y

n Y

Ugmg fj

g=1 n Y

Ugmg fj

g=1

j,m ℓ

= δ αs (~a)

we have (Akθ )r e = C k . We can easily prove an analog of the Sobolev lemma on E as follows. Theorem 6.9. For s > k + 1, H s ⊆ (Akθ )r e. Proof. First consider the case k = 0. Note that || · ||∞,0 = || · ||C ∗ so for arbitrary aj,m ||aj,m and for arbitrary ~a =

n Y

g=1

P

j,m

Qn

g=1

Ugmg fj ||2∞,0 = sup{|λ| : |aj,m |2 − λ · 1 not invertible} = |aj,m |2 aj,m

Qn

m

g=1

Ug g fj we have

X

||aj,m

~a =

X

||~a||2∞,0 ≤

j,m

n Y

g=1

Ugmg fj ||2∞,0 =

X j,m

|aj,m |2 = ||~a||20

by the triangle inequality. We have aj,m λs (m)λ−s (m)

n Y

Ugmg fj

g=1

j,m

so by the Cauchy-Schwarz inequality we get ||~a||20 ≤ ||~a||2s

X m

(1 + |m1 |2 + · · · + |mn |2 )−s . 17

m

Ug g fj we have

Since 2s > 2, (1 + |m1 |2 + · · · + |mn |2 )−s is summable over m ∈ Zn and j ∈ {1, . . . , r} so ||~a||20 ≤ C||~a||s . Thus we get ||~a||∞,0 ≤ ||~a||0 ≤ C||~a||s and H s ⊆ (A0θ )r e. Now suppose k > 0. Using what we’ve proven for the previous case, we have ||δ ℓ (~a)||∞,0 ≤ C||δ ℓ (~a)||s−|ℓ| n Y X mℓ aj,m Ugmg fj ||s−|ℓ| = C|| j,m

< C||

X j,m

g=1

(1 + |m1 |2 + · · · + |mn |2 )|ℓ| aj,m

= C||Pλ|ℓ| (~a)||s−|ℓ|

n Y

g=1

Ugmg fj ||s−|ℓ|

= C||~a||s for |ℓ| ≤ k since s − |ℓ| ≥ s − k > 1. Therefore, X X C||~a||s ≤ C||~a||s (k + 1)(k + 2)/2 ||δ ℓ (~a)||∞,0 ≤ ||~a||∞,k = |ℓ|≤k

|ℓ|≤k

and we get H s ⊆ (Akθ )r e. We get the following corollary. \ r Corollary 6.10. H s = (A∞ θ ) e. s∈R

T Proof. Suppose a ∈ s∈R H s . Then for any k ∈ Z≥0 , ~a ∈ H k+2 , so by the theorem we just proved, T s ∞ r r ~a ∈ (Akθ )r e. Consequently ~a ∈ (A∞ θ ) e, so s∈R H ⊆ (Aθ ) e. ∞ r s ∞ r ∞ r s Suppose a ∈ (Aθ ) e. Then T sinces H is the completion of (Aθ ) e with respect to || · ||s , (Aθ ) e ⊆ H ∞ r for all s ∈ R, and (Aθ ) e ⊆ s∈R H . We can also prove an analog of the Rellich lemma on E.

r Theorem 6.11. Let {~aN } ∈ (A∞ aN ||s ≤ C θ ) e be a sequence. Suppose that there is a constant C so that ||~ for all N . Let s > t. Then there is a subsequence {~aNj } that converges in H t .

Proof. Let F := {fj : 1 ≤ j ≤ r} be a set of eigenvectors of e normalized with respect h·, ·i, where fj has Q to m and Es := {es,m | corresponding eigenvalue λj for 1 ≤ j ≤ r. Let es,m := (1+|m1 |2 +· · ·+|mn |2 )−s/2 g Ug gP n, m ∈ Z}. Es F is an orthonormal basis with respect to h·, ·is , so we can write ~aN := h,k aN,h,k es,k fh . Then X |aN,h,k |2 ≤ |aN,h,k |2 ≤ C 2 h,k

and |aN,h,k | ≤ C. Applying the Arzela-Ascoli theorem to {aN,h,k } for some fixed (h, k), we can get a subsequence {aNj ,h,k } of {aN,h,k } such that for any ǫ > 0 there exists M (ǫ) ∈ N such that |aNi ,h,k −aNj ,h,k | < ǫ whenever i, j ≥ M (ǫ). Do this for all 1 ≤ h ≤ r and |k1 |2 + · · · + |kn |2 ≤ R, replacing {aN } with {aNj } each time. Then we get a subsequence {aNj } of {aN } such that for any ǫ > 0 there exists M (ǫ) ∈ N such that, for all 1 ≤ h ≤ r and |k1 |2 + · · · + |kn |2 ≤ R, |aNi ,h,k − aNj ,h,k | < ǫ whenever i, j ≥ M (ǫ). Now consider the sum X ||aNi − aNj ||2t = |aNi ,h,k − aNj ,h,k |2 (1 + |k1 |2 + · · · + |kn |2 )t−s . h,k

Decompose it into two parts: one where |k1 |2 + · · · + |kn |2 > R2 and one where |k1 |2 + · · · + |kn |2 ≤ R2 . On |k1 |2 + · · · + |kn |2 > R2 we estimate (1 + |k1 |2 + · · · + |kn |2 )t−s < (1 + R2 )t−s

18

so that X

X

h |k1 |2 +···+|kn |2 ≥R2

|aNi ,h,k − aNj ,h,k |2 (1 + |k1 |2 + · · · + |kn |2 )t−s < (1 + R2 )t−s 2

X h,k

|aNi ,h,k − aNj ,h,k |2

2 t−s

≤ 2rC (1 + R )

.

If ǫ > 0 is given, we choose R so that 2rC 2 (1 + R2 )t−s < ǫ. The remaining part of the√ sum is over |k1 |2 + · · · + |kn |2 ≤ R2 and can be bounded above by ǫ′ := ǫ − 2rC 2 (1 + R2 )t−s if i, j ≥ M ( n ǫ′ /(2R + 1)) because a ball of radius R centered at the origin is contained in a cube of side length 2R that has (2R + 1)2 lattice points. Then the total sum is bounded above by ǫ, and we are done.

7

Acknowledgement

I would like to thank Farzad Fathizadeh for suggesting the problem of filling in the details in the pseudodifferential calculus on the noncommutative n torus. I would also like thank Matilde Marcolli for helping me make plans to take my candidacy exam. I would like to thank Vlad Markovic and Eric Rains for agreeing to be on my candidacy exam committee.

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[13] Xavier S. Raymond. Elementary Introduction to the Theory of Pseudodifferential Operators. CRC Press, Boston, 1991. [14] M. W. Wong. An introduction to pseudo-differential operators, volume 6 of Series on Analysis, Applications and Computation. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, third edition, 2014.

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