Function spaces

In this section we provide the associated function spaces, which come from the original symbol estimate. Moreover, we are going to prove the corresponding mapping properties.

ForK ≥1definehξi_K :=^pK²+|ξ|², and setλ^m,p_K (t, x, ξ) :=hξi^p_Khσξi^m−p_K . Then it is clear, thatλ^m,p_K ∈Σe^m,pfor everyK≥1. Similar to the estimates inΣ^m,p one can derive, that

|∂_t^j∂_x^α∂_ξ^βλ^m,p_K |.hξip+2j−|β|+|α|

K hσξi^{m−p−|α|−2j}_K . (3.2)

DenoteΛ^m,p_K = Op(λ^m,p_K ). We are going to prove the following proposition:

Proposition 3.32. Given (m, p) ∈ Z×R there is a K0 ≥ 1, such that the opera-torΛ^m,p_K ∈ Op(Σe^m,p) is invertible for all K ≥ K0 with (Λ^m,p_K ) ∼ Λ^−m,−p_K modulo smoothing terms.

This is an application of the parameter-dependent pseudodifferential calculus, see for instance [Shu]. The idea is to consider operators, depending on a parameter K. Usually, when constructing parametrices, one gets remainder terms of lower order. In this case, the parameterK can be taken sufficiently large such that the norm of that remainder term is strictly less than one. Hence, the operator becomes invertible.

Proof. The symbolλ^m,p_K belongs toΣ^em,p. Similarlyλ^−m,−p_K is inΣ^e−m,−p. Define R_K :=λ^m,p_K #λ^−m,−p_K −λ^m,p_K λ^−m,−p_K =λ^m,p_K #λ^−m,−p_K −1.

For any`∈N0 let(j, α, β)∈N<sup>1+2d</sup><sub>0</sub> withj+|α|+|β| ≤`. We want to show

|R_K|_{0,0;`</sub> ≤ C<sub>`} K, where

|R_K|_0,0;`= sup

(t,x,ξ)∈[0,T]×R^2d j+|α|+|β|≤`

hξi^{2j+|β|−|α|}hσξi^|α|+2j|∂_t^j∂^α_x∂_ξ^βR_K|.

We will proveR_K−→0inΣ^0,0 forK−→ ∞. By Leibniz rule, it is

|∂_t^j∂_x^α∂_ξ^βR_K| ≤ ^X

|γ|≥1

1 γ!

X

j⁰+|α⁰|+|β⁰|≤`

c_j⁰_α⁰_β⁰|∂_t^j0∂_x^α0∂_ξ^β0+γλ^m,p_K ||∂_t^j−j0∂_x^α−α0+γ∂_ξ^β−β0λ^−m,−p_K |.

3.4 Function spaces 39

The estimates (3.2), gives that the product in the inner sum can be estimated by the function

hξi^{2j−|β|+|α|}_K hσξi−|α|−2j−|γ|

K

and so

|∂_t^j∂_x^α∂^β_ξR_K| ≤Chξi^{2j−|β|+|α|}_K hσξi^−|α|−2j_K ^X

|γ|≥1

1

γ!hσξi^−|γ|

≤C`hξi^{2j−|β|+|α|}_K hσξi^{−|α|−2j−1}_K . Thus, we obtain

|R_K|<sub>0,0;`</sub> ≤C` sup

(t,x,ξ)∈[0,T]×R^2d j+|α|+|β|≤`

hξi^{2j+|β|−|α|}hσξi^|α|+2jhξi^{2j−|β|+|α|}_K hσξi^{−|α|−2j−1}_K .

Note thathξi^|β|/hξi^|β|_K ≤1for allβ ∈N^d₀ andK ≥1, so the previous estimate can be rewritten, by sorting the powers, as

|R_K|_{0,0;`</sub> ≤C<sub>`} sup

(t,x,ξ)∈[0,T]×R^2d j+|α|+|β|≤`

hσξi²hξi²_K hσξi²_Khξi²

!^2j+|α|₂

hσξi⁻¹_K .

Since

hσξi²hξi²_K hσξi²_Khξi² ≤1 for all(t, x, ξ)∈[0, T]×R^2d, the estimate reduces to

|R_K|_{0,0;`</sub>≤C<sub>`} sup

(t,x,ξ)∈[0,T]×R^2d

hσξi⁻¹_K = C_` K.

Hence, RK −→ 0 in Σ^0,0 for K −→ ∞. This implies RK(t, x, Dx) −→ 0 in L(L²((0, T) ×R^d) for K −→ ∞. Thus Λ^−m,−p_K ◦(1 + R_K)⁻¹ is a right inverse to Λ^m,p_K for large K ≥ 1. In a similar way, a left inverse to Λ^m,p_K is contructed.

Moreover, (Λ^m,p_K )⁻¹ = Λ^−m,−p_K modulo smoothing terms, as it is seen from the construction.

We extend the Example 3.27 since we now know that the operators Λ^m,pare inverti-ble.

Example 3.33. ForΛ^m,p =λ^m,p(t, x, D_x), whereλ^m,p(t, x, D_x) = hσξi^m−p_K hξi^p_K,K sufficiently large, we have

σ_Ψ^m(Λ^m,p) =σ^m−p|ξ|^m, σ^m,p_Ψ,d(Λ^m,p) =r^m−p|ξ|^m, σ_Ψ,d^m−1,p(Λ^m,p) = 0.

Moreover it holds(Λ^m,p)⁻¹ ∈Op(Σe^−m,−p)and, by using the composition rules, σ_Ψ^−m((Λ^m,p)⁻¹) =σ^p−m|ξ|^−m, σ_Ψ,d^−m,−p((Λ^m,p)⁻¹) =r^p−m|ξ|^−m, and

σ^{−m−1,−p}_Ψ,d ((Λ^m,p)⁻¹) =im(p−m)hξ, xir^p−m−2|ξ|^−m−2.

We now consider those invertible operators and always neglect the parameterK.

Let us define the following function spaces:

Definition 3.34. Fors∈N0,δ∈Rand we define

CH^s,δ ={u: (0, T)×R^d−→C|Λ^s,δu∈C([0, T], L²(R^d))}

and

L¹H^s,δ ={u: (0, T)×R^d−→C|Λ^s,δu∈L¹((0, T), L²(R^d))}, respectively.

We are able to formulate the following mapping property result:

Proposition 3.35. Form∈N0we have

Λ^m,p ∈L(CH^s+m,δ+p,CH^s,δ).

for all(s, δ)∈N0×R.

Proof. Lets∈N0. Then we have to show thatΛ^m,pu∈CH^s,δ, which isΛ^s,δ(Λ^m,pu)∈ C([0, T], L²(R^d)), providedu ∈CH^s+m,δ+p. By the composition of two operators we get

Λ^s,δΛ^m,p= Λ^s+m,δ+p+R,

whereR∈Op(Σe^s+m−1,δ+p). By assumption, it holdsΛ^s+m,δ+pu∈C([0, T], L²(R^d)).

We now write the remainder term as

R=R(Λ^s+m,δ+p)⁻¹Λ^s+m,δ+p, since the operatorsΛ^s+m,δ+p are invertible. It turns out that

R(Λ^s+m,δ+p)⁻¹ ∈Op(Σ^e−1,0)⊆Op(Σ^e0,0).

In view of Corollary 3.14, we also have

Ru=R(Λ^s+m,δ+p)⁻¹Λ^s+m,δ+pu∈C([0, T], L²(R^d)).

3.4 Function spaces 41

This completes the proof.

Remark3.36. We get a similar result forL¹H^s,δ.

Let us now come to the main function spaces involving time derivatives.

Definition 3.37. Fors∈N0,δ∈Rand we define

X^s,δ ={u: (0, T)×R^d−→C|Λ^{s−j,δ−2j}D_t^ju∈C([0, T], L²(R^d)),0≤j≤s}

and

Y^s,δ ={u: (0, T)×R^d−→C|Λ^{s−j,δ−2j}D^j_tu∈L¹((0, T), L²(R^d)),0≤j≤s}, respectively. The corresponding norms are given by

kuk_Xs,δ = sup

0≤j≤s

sup

t∈[0,T]

k(hσD_xi^s−δ+jhD_xi^δ−2jD^j_tu)(t,·)k_L2(R^d)

and

kuk_Ys,δ =

s

X

j=0

Z T 0

k(hσD_xi^s−δ+jhD_xi^δ−2jD^j_tu)(t,·)k_L2(R^d), respectively.

In view of these definitions, we automatically have the embeddingX^s,δ ⊆Y^s,δ. We also define the trace space as

H^s,δ(R^d) =X^s,δ|_t=0.

This coincides with the2-microlocal Sobolev with respect to the LangrangianT₀^∗(R^d), see the Appendix.

Remark 3.38. Surely, another convenient definition would be the following: For parameters s ∈ N0, δ ∈ R, the space H^s,δ((0, T)×R^d) consists of all functions u: (0, T)×R^d−→Csatisfying

Λ^{s−j,δ−2j}D^j_tu∈L²((0, T)×R^d), 0≤j≤s.

For generals∈R,δ ∈R, the spaceH^s,δ is then defined by interpolation and duality.

This would be a huge advantage in contrast to the spacesX^s,δ. However, withX^s,δ we do not have to worry about the traces att= 0, because they continuously run into the above mentioned 2-microlocal spaces.

We are now going to prove the important mapping properties.

Proposition 3.39. Form∈N0we have

Λ^m,p∈L(X^s+m,δ+p, X^s,δ) ∀(s, δ)∈N0×R.

Proof. We use a similar reduction argument as in the previous proposition. Let s∈N0. Then we have to show, that

Λ^{s−j,δ−2j}D^j_t(Λ^m,pu)∈C([0, T], L²(R^d))

for0≤j≤s, providedu∈X^s+m,δ+p. By using commutators we have D^j_tΛ^m,p= Λ^m,pD_t^j+ [D^j_t,Λ^m,p],

and also

Λ^{s−j,δ−2j}D^j_t(Λ^m,pu) = Λ^{s−j,δ−2j}Λ^m,pD_t^ju+ Λ^{s−j,δ−2j}[D_t^j,Λ^m,p].

In the first term we getΛ^{s−j,δ−2j}Λ^m,p = Λs+m−j,δ+p−2j+R, where the remainder belongs toOp(Σes+m−j−1,δ+p−2j). Hence,

Λ^{s−j,δ−2j}Λ^m,pD_t^ju= Λs+m−j,δ+p−2j

D_t^ju+RD^j_tu.

The first summand belongs toC([0, T], L²(R^d))by assumption, whereas the second summand can be rewritten as

RD^j_tu=R(Λs+m−j,p+δ−2j

)⁻¹Λs+m−j,p+δ−2jD_t^ju

for all0≤j≤s. Again, the functionΛs+m−j,p+δ−2jD^j_tubelongs toC([0, T], L²(R^d)) by assumption, whereas

R(Λs+m−j,p+δ−2j)⁻¹∈Op(Σe^−1,0)⊆Op(Σe^0,0),

which are C([0, T], L²(R^d))-continuous, since Op(Σe^0,0) ⊆ Op(Σ^0,0,†) in view of Corollary 3.14.

Note thatΛ^m,p= Op(λ^m,p). By using iterated commutators, we further have [D^j_t,Λ^m,p] =

j

X

k=1

j k

!

Op(D^k_tλ)D_t^j−k, and so

Λ^{s−j,δ−2j}[D^j_t,Λ^m,p]u∼

j

X

k=1

Λs+m−j,p+δ+2k−2jD^j−k_t u.

Inductively also these terms belong toC([0, T], L²(R^d)).

3.4 Function spaces 43

This leads to the following corollary:

Corollary 3.40. Form∈N0 we have

Op(Σ^m,p,†)⊂L(X^s+m,δ+p, X^s,δ) ∀(s, δ)∈N0×R.

Let us now prove the following helpful lemma:

Lemma 3.41. For(s, δ)∈N0×Rwe haveD_t:X^s+1,δ −→X^s,δ−2 and u∈X^s+1,δ ⇐⇒Λ^1,0u∈X^s,δ and Λ^0,−2D_tu∈X^s,δ.

Proof. The first part is easily done by a simple index-shift. In fact, letu∈ X^s+1,δ, then

Λ^{s+1−j,δ−2j}D^j_tu∈C([0, T], L²(R^d))

for0≤j≤s+ 1. Definek:=j−1, then forj6= 0we have0≤k≤sand Λ^{s−k,δ−2k−2}D_t^k+1u∈C([0, T], L²(R^d)),

which isΛ^{s−k,δ−2−2k}D_t^k(Dtu)∈C([0, T], L²(R^d))and soDtu∈X^s,δ.

The second statement follows directly by the mapping properties ofD_tandΛ^m,p.

With the same arguments we finally get the following similar results:

Corollary 3.42. Let(s, δ)∈N0×R. Form∈N0 we have Op(Σe^m,p)∈L(Y^s+m,δ+p, Y^s,δ).

Moreover, ifA(t, x, D_x)∈Op(Σe^m,p), then

A(0, x, D_x)∈L(H^s+m,δ+p, H^s,δ).

Remark3.43. Compared to the results in [Bon], the spaceCH^s,δ|t= 0corresponds to the weighted SobolevSP(δ, s−δ), whileΛ^m,p(0, x, Dx)belongs toOp(Σ^p,m−p₀ ), see the equivalence in (A.1).

Next, we now want to prove partial hypoellipticity. Forr∈N0, we also set X^s,r,δ :={u|∂_t^ju∈CH^{s−j,δ−2j},0≤j≤r}

Y^s,r,δ :={f|∂_t^jf ∈L¹H^{s−j,δ−2j},0≤j≤r}.

Note thatX^s,s,δ =X^s,δ fors∈N0andX^s,0,δ =CH^s,δ. Then the following result is valid:

Theorem 3.44. Letr∈N0,u∈CH^s,δ,f ∈Y^s,r,δ andDtu=a(t, x, Dx)u+f, where a(t, x, D_x)∈Op(Σe^1,2). Thenu∈X^s,r,δ.

Proof. We prove this statement by an induction overr ∈N0. Ifr = 0, then nothing is to do. Assume that this statement is valid for a certainr∈N0andf ∈Y^s,r+1,δ. Then we have to show, that D_t^r+1u ∈ CHs−r−1,δ−2r−2. Using the differential equation leads us to

D^r+1_t u=D^r_t(Dtu) =D^r_t(a(t, x, Dx)u+f) =D^r_ta(t, x, Dx)u+D^r_tf,

so we have to compute the operatorD_t^ra(t, x, D_x). Using iterated commutators, we get again

D_t^ra(t, x, Dx) =

r

X

j=0

r j

!

(D_t^ja)(t, x, Dx)D^r−j_t . Thus,

D^r+1_t u=a(t, x, Dx)D^r_tu+

r

X

j=1

r j

!

(D^j_ta)(t, x, Dx)D_t^r−ju+D_t^rf.

First, we haveD_t^ru∈CH^{s−r,δ−2r}, and by mapping properties also a(t, x, D_x)D^r_tu∈CHs−r−1,δ−2r−2.

Since(D_t^ja)(t, x, D_x)∈Op(Σ^e1,2+2j)andD_t^r−ju∈CHs−r+j,δ−2r+2j, we also obtain (D_t^ja)(t, x, Dx)D^r−j_t u∈CHs−r−1+j,δ−2r−2

, 1≤j≤r, which is even better thanCHs−r−1,δ−2r−2. Iff ∈Y^s,r+1,δ, so

D^r+1_t f ∈L¹Hs−r−1,δ−2r−2

.

With the identity

f(x) =f(0) + Z x

0

(∂_t^rf)(s)ds

we haveD_t^rf ∈CHs−r−1,δ−2r−2. This finally yieldsD_t^r+1u∈CHs−r−1,δ−2r−2, and sou∈X^s,r+1,δ.

As a consequence we obtain:

Corollary 3.45. In the proof of the main theorem it suffices to considerr = 0.

3.4 Function spaces 45

4

Proof of the Main Theorem

In this Chapter we are going to prove the main theorem. We first define the following expression:

Definition 4.1. We define

Φ(A₀₀, A₀₁, M₀₀, M₀₁) :=M AM⁻¹+ (DtM)M⁻¹

01.

Let us now state the main theorem with the notation we got so far.

Theorem 4.2. LetA∈Op(Σe^1,2). Assume there exists aM₀ ∈Σ^(0),0with|detM₀|&1, M0A0M₀⁻¹is Hermitian, such that

I−2xM00A00M₀₀⁻¹

ξ>0,

whereM₀₀∈Σ^(0,0) is the(0,0)-bihomogeneous component of M₀.

Then for everyM₀₁there exists aδ₀ =δ₀(A₀₀, A₀₁)∈Rsuch that Im (Φ(A00, A01, M00, M01))≤ δ0

2r²

I −2xM00A00M₀₀⁻¹

ξ

, (4.1)

and aγ₀ = γ(A₀₀), with the property that for alls∈ N0,δ ≥ δ₀+sγ₀, U₀ ∈ H^s,δ, F ∈Y^s,δ, the Cauchy problem (1.1) possesses a unique solutionU ∈X^s,δ. Moreover, the a priori estimate

kUk_Xs,δ .kU₀k_Hs,δ+kFk_Ys,δ

is valid.

For the meaning ofΦ, see Definition 4.4. The relation betweenδ₀,γ₀ andA₀₀, A₀₁is quite complicated. We will provide upper bounds on bothδ0 andγ0in Section 4.3.

In fact, the operatorA is not assumed to have Hermitian principal part, but we assume the existence of a symmetrizer, see Definition 4.6. With this operator we can formulate an equivalent system, whose principal symbol is Hermitian. The second reduction is, that we can "shift" the parameterδ to 0.

47

Note thatδ depends onsand increases ass→ ∞. To prove an energy estimate for everys∈N0, we will us an inductive argument. By using the spectral shiftδ →0, we will assume, that the energy estimate holds for ans∈N0 withδ = 0. We then show, that with the steps→s+ 1, also the spectral parameterδ increases byγ0. For the next step in the induction, we again shift the newδto0.

In the sequel, we will consider operators of the formKAK⁻¹+ (DtK)K⁻¹, where A∈Op(Σe^1,2),K ∈Op(Σe^m,p)andKis invertible. Using the composition rules, see Proposition 3.30, we can formulate the following technical lemma:

Lemma 4.3. We have M AM⁻¹+ (DtM)M⁻¹

01

=M00

h

(M₀₀⁻¹M01, A00

i

(M00)⁻¹+M00A01(M00)⁻¹−i(M00)_ξ(A00)_x(M00)⁻¹

−i(M00)_t(M00)⁻¹

−iM00A00(M00)⁻¹(M00)_ξ−(M00)_ξA00−M00(A00)_ξ(M00)⁻¹(M00)_x(M00)⁻¹.

WhenM is scalar withM₀₁= 0, then this simplifies to (see last summand):

M AM⁻¹+ (D_tM)M⁻¹

01=A₀₁−i(M₀₀)_ξ(A₀₀)_x(M₀₀)⁻¹+i(M₀₀)⁻¹(A₀₀)_ξ(M₀₀)_x

−i(M00)_t(M00)⁻¹. Consider nowM = Λ^m,p. This gives us

ΛAΛ⁻¹+ (D_tΛ)Λ⁻¹

01=A₀₁−imξ

|ξ|²(A₀₀)_x+i(m−p)x

r² (A₀₀)_ξ−i(m−p) 2r² I.

If additionallym= 0then ΛAΛ⁻¹+ (DtΛ)Λ⁻¹

01=A₀₁+i(m−p)x

r² (A₀₀)_ξ−i(m−p) 2r² I.

In this case we also have

ΛAΛ⁻¹

01=A₀₁+i(m−p)x

r² (A₀₀)_ξ.

We end this short overview with giving the precise formulation forΦ.

Corollary 4.4. We have Φ(A₀₀, A₀₁, M₀₀, M₀₁)

=M₀₀^h(M₀₀⁻¹M₀₁, A₀₀ⁱ(M₀₀)⁻¹+M₀₀A₀₁(M₀₀)⁻¹−i(M₀₀)_ξ(A₀₀)_x(M₀₀)⁻¹

−i(M00)_t(M00)⁻¹

−iM00A00(M00)⁻¹(M00)_ξ−(M00)_ξA00−M00(A00)_ξ(M00)⁻¹(M00)_x(M00)⁻¹.

Note thatΦ(A₀₀, A₀₁, I,0) =A₀₁.

4.1 Basic C L

²

-energy estimate

Let us derive the basic energy estimate, which later will give us the a priori estimate inCL². For this, we do not need any special structure onAin terms of principal symbols. However, in the following theorem we see the influence of the remainder classΣ^m−2,p,†+ Σ^{m−1,p−1,†}for(m, p) = (1,2).

Theorem 4.5. LetA∈Op(Σ^1,2,†)be an operator, such thatIm(A)∈Op(Σ^0,2,†)and Im(A(t, x, ξ)). hξi

hσξi + hξi² hσξi³

! I

Then each solutionU =U(t, x) ∈CL² to the problem (1.1) with U0 ∈ L²(R^d) and F ∈L¹L² satisfies the a priori estimate

kUk_CL2 .kU₀k_L2+kFk_L1L².

Proof. Let us writeq(t, x, ξ) =hξihσξi⁻¹+hξi²hσξi⁻³. We assume, that the operator Dt−Apossesses a forward fundamental solutionX(t, t⁰) :S(R^d)−→S(R^d), i.e.







(Dt−A(t, x, Dx))X(t, t⁰) = 0, 0≤t⁰ ≤t≤T, X(t⁰, t⁰) =I, .

Then we can represent the solutionU via U(t, x) =X(t,0)U₀(x) +i

Z T 0

X(t, t⁰)F(t⁰, x)dt⁰ and so have to prove an uniform estimate of the form

kX(t, t⁰)Vk_L2(R^d).kVk_L2(R^d), (4.2)

4.1 BasicCL²-energy estimate 49

for all0≤t⁰ ≤t≤T andV ∈S(R^d). Together with the representation forU we

We are now going to show (4.2). Consider first the primitive p(t, x, ξ) =

Then we calculate, by change of variables, i.e.t(s) =hσhξii², I₁ =hξi

Here the operatorse^±p(t,x,Dx)are of order zero and invertible. Furthermore, we have by constructionY(t⁰, t⁰) =I. Then

∂_tY(t, t⁰) =−q(t, x, D_x)Y(t, t⁰) +ie^−p(t,x,Dx)e^p(t0,x,Dx)A(t, x, D_x)X(t, t⁰)

=B(t, x, D_x)Y(t, t⁰)

where

B(t, x, Dx) =iA−qI+^he^−p(t,x,Dx)e^p(t0,x,Dx), iAⁱe^{−p(t0,x,Dx)}e^p(t,x,Dx).

For B we haveB(t, x, ξ) ∈ Σ^1,2,† and Re(B) ≤ K for a constant K > 0. Thus, Garding’s inequality gives us

∂_tkY(t, t⁰)Vk²_L2(R^d)= 2 Re(∂_tY(t, t⁰)V, Y(t, t⁰)V) = 2 Re(BY(t, t⁰)V, Y(t, t⁰)V) .kY(t, t⁰)Vk²_L2(R^d).

Finally using Gronwall’s lemma, we obtain

kY(t, t⁰)Vk²_L2(R^d).kY(t⁰, t⁰)Vk²_L2(R^d).kVk²_L2(R^d).

This gives us (4.2), since the operatorse^±p(t,x,Dx)are isomorphisms onL²(R^d).

Later, this estimate basically will give us theCL²-wellposedness.

Im Dokument On the Cauchy problem for a class of degenerate hyperbolic equations (Seite 49-61)