trace theorems for sobolev spaces of variable order of ......on lp sobolev spaces requires that ρ =...

Math. Nachr. (1996),

Trace Theorems for Sobolev Spaces of Variable Order ofDifferentiation

By Hans-Gerd Leopold of Jena, Elmar Schrohe of Potsdam

(Received December 29, 1994 )

Abstract. We prove a trace theorem and an extension theorem for Sobolev spaces of variable

order of differentiation which are defined by pseudodifferential operators.

1. Introduction

Trace and extension theorems for the classical Sobolev spaces have been knownfor well over 30 years. They go back to Aronszajn, Besov, Slobodeckij, Stein,and many others. Kalyabin obtained corresponding results for the so-called spacesof functions of generalized smoothness.

The theory of pseudodifferential operators led to yet another type of function spaces,where the smoothness varies in space. Although these spaces have been investigatedfor almost three decades, cf. Unterberger and Bokobza [UB], it seems that nogeneral trace or extension theorems have been proven. One reason might be that thestandard pseudodifferential methods fail in this situation. The present paper gives asolution for a class of such spaces.

For certain hypoelliptic pseudodifferential symbols a we define the Sobolev spaceW 1,a

2 (IRn) by

W 1,a2 (IRn) = u ∈ S ′(IRn) : u ∈ L2(IRn) andAu ∈ L2(IRn) ;

here, A = Opa. We then show that the restrictions of these functions to the hyper-plane xn = 0 belong to W 1,a′

2 (IRn−1), where a′(x′, ξ′) = 〈ξ′〉−1/2 #a(x′, 0, ξ′, 0), andthat the restriction operator is in fact surjective. Except for the required hypoelliptic-ity, the essential assumption for the trace theorem is a conormal ellipticity conditionguaranteeing a minimal growth for the quotient a(x, ξ)/a(x, ξ′, 0); for details see The-orem 3.1. Conversely, in order to obtain the extension theorem, we have to control

1991 Mathematics Subject Classification. Primary 46E35; Secondary 47G30, 35S05Keywords and phrases. Function Spaces, Pseudodifferential Operators, Trace Theorems.

2 Math. Nachr. (1996)

the quotient a(x′, 0, ξ)/a(x′, 0, ξ′, 0) from above, cf. Theorem 3.2. Both conditionsare quite natural from the point of view of B.-W. Schulze’s wedge Sobolev spaces,cf. [Sc], [ScS], although, for technical reasons, this relationship is not exploited in thepresent proof.

An example for a symbol a where all assumptions are fulfilled is a(x, ξ) = 〈ξ〉σ(x) (1+ln 〈ξ〉)t where σ is a smooth function with all derivatives bounded and inf σ(x) >1/2; t ∈ IR. This includes the classes considered by Unterberger-Bokobza andBeauzamy, where σ was the sum of a constant and a rapidly decreasing function.In addition, symbols of the form a(x, ξ) = 〈ξ〉σ(x) recently have gained interest asgenerators of Feller semi-groups and Markov processes [JL], [Ne], [KN]. The presentresults should provide a further step towards the understanding of boundary valueproblems also for operators with symbols such as these.

We reduce the proof of the trace theorem to a boundedness result for a compositionof pseudodifferential operators on standard Sobolev spaces over IRn. Although simplein spirit, this requires a careful analysis since the symbols a and a′ live on spaces ofdifferent dimension.

Vice versa, the extension operator is constructed in a straightforward way. Againwe reduce the problem to showing the boundedness of a composition of operators onSobolev spaces. This time, however, we have to show continuity from Hk

2 (IRn−1) toHk

2 (IRn) for large k, and the operators are partly pseudodifferential, partly potentialoperators in Boutet de Monvel’s calculus.

Using different methods, Pesenson [Pe] has recently proven trace and extensiontheorems for a certain class of non-isotropic Sobolev spaces on the Heisenberg group,given by vector fields satisfying Hormander’s condition.

The present proof does not carry over to the case of Besov spaces Bsp,q with p, q 6= 2.

The reason is the well-known limitation of the boundedness results for standard pseu-dodifferential operators: For p 6= 2, the continuity of operators with symbols in S0

ρ,δ

on Lp Sobolev spaces requires that ρ = 1, cf. [Fe], while here we take advantage of thefact that, for p = 2, the much weaker condition 0 ≤ δ ≤ ρ ≤ 1, δ < 1 is sufficient. Thebasic concept of reducing both the trace and the extension theorem to boundednessresults seems to be quite general. We therefore hope that, for a restricted class ofsymbols, trace and extension theorems can be obtained also in the case p 6= 2 usingmore refined continuity results.

2. Pseudodifferential Operators and Sobolev Spaces of VariableOrder of Differentiation

A function p(x, ξ) belongs to the class Smρ,δ, −∞ < m < ∞, 0 ≤ δ, ρ ≤ 1, δ < 1,

provided that for all multi-indices α, β there is a constant cαβ such that

∣∣∣p(α)(β)(x, ξ)

∣∣∣ ≤ cαβ 〈ξ〉m−ρ|α|+δ|β|(2.1)

for all x, ξ ∈ IRn. Here p(α)(β) = ∂α

ξ Dβxp(x, ξ) and 〈ξ〉 =

(1 + |ξ|2)1/2.

Leopold, Schrohe, Trace Theorems 3

The set of all functions p(x, ξ) such that (2.1) only holds for |α| ≤ I and |β| ≤ Jwill be denoted by Sm

ρ,δ(I, J), I, J ∈ IN.For p ∈ Sm

ρ,δ or p ∈ Smρ,δ(I, J) define the semi-norms

|p|(m)(l,k) = max

|α|≤l|β|≤k

supx,ξ

∣∣∣p(α)(β)(x, ξ)

∣∣∣ 〈ξ〉−m+ρ|α|−δ|β|.

The pseudodifferential operator p(x,Dx) or Opp with symbol p(x, ξ) is defined by

[p(x,Dx)u] (x) =∫eixξ p(x, ξ) u(ξ) dξ

for u in the Schwartz space S(IRn); u(ξ) =∫e−ixξu(x) dx is the Fourier transform of

u, and dξ = (2π)−ndξ. For p ∈ Smρ,δ,

p(x,Dx) : S(IRn) → S(IRn)

is continuous and can be extended to a continuous operator from S ′(IRn) to S ′(IRn).Concerning the composition we will need the following observation.

Lemma 2.1. Let p1 ∈ Sm1ρ1,δ1

, p2 ∈ Sm2ρ2,δ2

. Then p1(x,Dx)p2(x,Dx) = p(x,Dx) hasa symbol p(x, ξ) with the following expansion

p(x, ξ) =∑

|α|≤N

1α!p(α)1 (x, ξ) p2(α)(x, ξ) + rN (x, ξ),(2.2)

where

rN (x, ξ) = (N + 1)∑

|γ|=N+1

∫ 1

0

(1− θ)N

γ!rγ,θ(x, ξ)dθ

andrγ,θ(x, ξ) = (2π)−nOs−

∫ ∫e−iyη p

(γ)1 (x, ξ + θη) p2(γ)(x+ y, ξ) dy dη .(2.3)

We shall write p = p1#p2.As usual, cf. [Kg, Chapter 2, Theorem 3.1], this follows from Taylor’s formula com-

bined with integration by parts for the particular form of rγ,θ.For ρ1 = ρ2 = ρ, δ1 = δ2 = δ and δ ≤ ρ we have p ∈ Sm1+m2

ρ,δ and rN ∈S

m1+m2−(N+1)(ρ−δ)ρ,δ .

Theorem 2.2. Let p ∈ Smρ,δ, 0 ≤ δ ≤ ρ ≤ 1, δ < 1, and s ∈ IR. Then there exist

integers I, J such that

‖p(x,Dx)|L(Hs2(IRn), Hs−m

2 (IRn))‖ ≤ C|p|m(I,J) .

Hs2(IRn) is the usual Sobolev space of all distributions u with (1−∆)

s2u ∈ L2(IRn).

The theorem goes back to Calderon and Vaillancourt, cf. [Kg, Chapter 7,Theorem 1.6]; in fact it is sufficient to ask that p ∈ Sm

ρ,δ(I, J). I and J may dependon δ, but they will not increase as δ decreases.


In [Le2] and [Le3], Besov and Sobolev spaces of variable order of differentiation weredefined.

Definition 2.3. Let 0 ≤ δ < 1 and 0 < m′ ≤ m. A symbol a(x, ξ) ∈ Sm1,δ belongs

to the class S(m,m′; δ) if there is a constant R ≥ 0 with the following properties

1. for all multi-indices α, β, and all (x, ξ) ∈ IRn × IRn with |ξ| ≥ R∣∣∣a(α)

(β)(x, ξ)∣∣∣ ≤ cαβ |a(x, ξ)| 〈ξ〉−|α|+δ|β| ;(2.4)

2. there exist constants c, c′ > 0 such that for all (x, ξ) ∈ IRn × IRn with |ξ| ≥ R

c′ 〈ξ〉m′ ≤ |a(x, ξ)| ≤ c 〈ξ〉m .(2.5)

The symbols of the class S(m,m′; δ) are a good substitute for the symbol |ξ|2 of theLaplacian which is used in the definition of the usual function spaces. For examplessee [Le2], [Le3], [Kg].

Remark 2.4.(i) If a ∈ S(m,m′; δ), b ∈ S(m, m′; δ), then

ab ∈ S(m+ m,m′ + m′;max (δ, δ)).

(ii) The symbol c(x′, ξ′) = a(x′, 0, ξ′, 0) belongs to S(m,m′, δ) with respect to IRn−1.

(iii) The elements of S(m,m′, δ) are hypoelliptic. Given a ∈ S(m,m′, δ) we canconstruct a parametrix b(x,Dx) to a(x,Dx); i.e. b(x,Dx)a(x,Dx) − I and a(x,Dx)b(x,Dx) − I both have symbols in S−∞ =

⋂Sm

ρ,δ. Moreover, b will satisfy the esti-mates ∣∣∣b(α)

(β)(x, ξ)∣∣∣ ≤ cαβ |a(x, ξ)|−1 〈ξ〉−|α|+δ|β|(2.6)

for all (x, ξ) ∈ IRn × IRn, |ξ| ≥ R, cf. [Kg, Chapter 2, §5].

Definition 2.5. For j = 1, 2, . . ., and a ∈ S(m,m′; δ) let A = Opa and

W j,a2 (IRn) =

u ∈ S ′(IRn) :

∥∥∥u∣∣∣W j,a

2 (IRn)∥∥∥ <∞

,

where ∥∥∥u∣∣∣W j,a

2 (IRn)∥∥∥ =

∥∥Aju∣∣L2

∥∥ + ‖u |L2 ‖ .

The definition extends to 1 < p <∞, cf. [Le2]. It shows a convenient way to relatefunction spaces and pseudodifferential operators, cf. [Un], [VE1],[VE2] [VK], [Be]. Weshall need the following fact.

Theorem 2.6. W j,a2 (IRn) is a Hilbert space, and S(IRn) is a dense subset. For

p ∈ S01,δ we have p(x,Dx) ∈ L(W j,a

2 (IRn)). If a is an other symbol and a− a ∈ S−∞then W j,a

2 (IRn) = W j,a2 (IRn).

For a proof and further results see [Le3] and [LS1].


Lemma 2.7. W j,a2 (IRn) = W 1,aj

2 (IRn).

Proof. By Remark 2.4, aj ∈ S(jm, jm′; δ). Let B(j) = Op b(j) be a parametrix toOpaj . Then ∥∥Aju

∣∣L2

∥∥ ≤ ∥∥AjB(j)Opaju

∣∣L2

∥∥ +∥∥AjR(j)u

∣∣L2

∥∥with R(j) ∈ S−∞, so that the last term can be estimated by c‖u|L2‖. Applying (2.6),a careful analysis of a# . . .#a#b(j) gives AjB(j) ∈ S0

1,δ. By Theorem 2.6 we get

∥∥Aju∣∣L2

∥∥ ≤ c∥∥∥u

∣∣∣W 1,aj

2

∥∥∥ .

Vice versa, let B = Op b be a parametrix to A = Opa. Then Bj is a parametrix toAj , and ∥∥Opaju

∣∣L2

∥∥ ≤∥∥OpajBjAju

∣∣L2

∥∥ +∥∥OpajRju

∣∣L2

∥∥with Rj ∈ S−∞. Again, the last term can be estimated by c ‖u |L2 ‖, and just likebefore OpajBj ∈ S0

1,δ. Hence

∥∥Opaju∣∣L2

∥∥ ≤ C∥∥∥u

∣∣∣W j,a2

∥∥∥ .

2

3. The Main Results

As usual we write the variables in IRn in the form x = (x′, xn), ξ = (ξ′, ξn).

Theorem 3.1. (Trace Theorem)

Let a ∈⋂

δ>0S(m,m′; δ), m′ > 1

2 , satisfy

∣∣∣∣a(x′, xn, ξ

′, 0)a(x′, xn, ξ′, ξn)

∣∣∣∣ ≤ C

( 〈ξ′〉〈ξ〉

) 12+ε∗

(3.1)

for some ε∗ > 0. Then the restriction operator

γ0 : W 1,a2 (IRn) →W 1,a′

2 (IRn−1)

is continuous. Here, a′(x′, ξ′) = 〈ξ′〉−1/2 #a(x′, 0, ξ′, 0).

The restriction operator is in fact surjective. This follows from

Theorem 3.2. (Extension Theorem)

Let a ∈⋂

δ>0S(m,m′; δ), m′ > 1

2 , satisfy

∣∣∣∣a(x′, 0, ξ′, ξn)a(x′, 0, ξ′, 0)

∣∣∣∣ ≤ C

( 〈ξ〉〈ξ′〉

)κ

(3.2)


for some κ > 0. Then there exists a linear bounded operator

E : W 1,a′2 (IRn−1) →W 1,a

2 (IRn)

with γ0E = id. Again, a′(x′, ξ′) = 〈ξ′〉−1/2 #a(x′, 0, ξ′, 0).

Remark 3.3. In view of the fact that we may change the symbol a for small |ξ|without changing the space W 1,a

2 (IRn), cf. Theorem 2.6, it is sufficient to ask thatestimates (3.1) and (3.2) hold for |ξ| ≥ R.Likewise it is obviously sufficient to ask that estimate (3.1) holds in a small strip|xn| < ε only.

Remark 3.4. For symbols of the form

a(x, ξ) = 〈ξ〉σ(x)(3.3)

the spaces W 1,a2 (IRn) were considered by Unterberger and Bokobza [UB], Visik

and Eskin [VE1], [VE2], Beauzamy [Bea]. Here, σ(x) = s+ ψ(x) with a constant s,and ψ ∈ S(IRn).

If t ∈ IR, then the spaces W 1,a2 (IRn) associated with

a(x, ξ) = 〈ξ〉σ(x) (1 + ln 〈ξ〉)t(3.4)

were treated in Unterberger-Bokobza [UB] and Unterberger [Un].

It is easy to check that the symbols in (3.3) and (3.4) belong to⋂

δ>0S(m,m′, δ) and

satisfy (3.1) and (3.2) provided inf σ(x) > 1/2. This is also true for a generalization of(3.3) used in [KN], where σ(x) is a function belonging to B∞(IRn) ,i.e., σ and all itsderivatives are bounded on IRn. For all these examples, trace and extension theoremswere so far unknown.

4. Proof of the Trace Theorem

In the following we shall assume that a ∈⋂

δ>0S(m,m′, δ), m′ > 1

2 , satisfies inequal-ity (3.1) in Theorem 3.1.

Lemma 4.1. Let u ∈W 1,a2 (IRn). Then γ0u ∈ L2(IRn−1).

Proof. Since A is hypoelliptic, there is a parametrix B with symbol in S−m′1,δ such

that BA = I+R with R ∈ S−∞. Now u = BAu−Ru. By assumption Au ∈ L2(IRn)so that u ∈ Hm′

2 (IRn) by Theorem 2.2. The classical trace theorem now gives theassertion, since m′ > 1/2. 2

We denote by Λ and Λ′ the operators with symbols 〈ξ〉 and 〈ξ′〉, respectively. As inTheorem 3.1, A′ is the operator with the symbol 〈ξ′〉−1/2 #a(x′, 0, ξ′, 0), while A0 isthe operator with the symbol a(x′, xn, ξ

′, 0).


Lemma 4.2. Fix k = 1+ε∗2 , cf.Theorem 3.1. Without loss of generality we may and

will assume that k < m′. Let B, R be as in the proof of Lemma 4.1. For the proof ofTheorem 3.1 it is sufficient to show that

Λ′−k A0B Λk ∈ L(Hk2 (IRn)).(4.1)

Proof. We have γ0 u ∈ L2(IRn−1) by Lemma 4.1. It remains to show that A′ γ0 u ∈L2(IRn−1). For u ∈W 1,a

2 (IRn) we have

A′ γ0 u = Λ′k−1/2γ0

(Λ′−kA0B Λk

)Λ−k Au−A′ γ0Ru,(4.2)

since γ0 Λ′−k A0 = Λ′−k+1/2A′ γ0. The second summand on the right hand side of (4.2)certainly is a function in L2(IRn−1), since R ∈ S−∞ and A′ ∈ S(m− 1/2,m′− 1/2; δ).By assumption, Au ∈ L2(IRn), hence Λ−kAu ∈ Hk

2 (IRn). If (4.1) holds, the traceof

(Λ′−k A0B Λk

)Λ−k Au is in Hk−1/2

2 (IRn−1). So the first summand also belongs toL2(IRn−1). 2

Lemma 4.3. Let b denote the symbol of B and

q(x, ξ) = 〈ξ′〉−k #a(x′, xn, ξ′, 0)#b(x, ξ)# 〈ξ〉k(4.3)

with the fixed k of Lemma 4.2. In order to prove the boundedness in Lemma 4.2, itis sufficient to show that q = q1 + q2, where q2 induces an Hk

2 (IRn)-bounded operator,while for every δ > 0, q1 satisfies the estimates

∣∣∣q (α)1 (β)(x, ξ)

∣∣∣ ≤ Cαβ 〈ξ〉2[m]δ− ε∗2 +δ|β| 〈ξ′〉−2[m]δ+ ε∗

2 −|α|(4.4)

for all α, β with |α| ≤ I, |β| ≤ J . Here I and J denote the number of derivativesrequired in Theorem 2.2 for s = k.

Proof. The right hand side of (4.4) equals

Cαβ 〈ξ〉−ρ|α|+δ|β| 〈ξ′〉|α|(ρ−1)( 〈ξ′〉〈ξ〉

) ε∗2 −2[m]δ−ρ|α|

.(4.5)

By assumption, a ∈ S(m,m′; δ) for all δ > 0. Choose ρ = min(

ε∗4I , 1

)and δ <

min(ρ, ε∗

8[m]+1

). Then (4.5) implies that q1 ∈ S0

ρ,δ(I, J) and Theorem 2.2 shows the

boundedness in Hk2 (IRn). 2

Lemma 4.4. The symbol q in (4.3) has a decomposition q = q1+q2 as in Lemma 4.3.

Proof. The symbol of b(x, ξ)# 〈ξ〉k is b(x, ξ) 〈ξ〉k. Choose M = [m]. The composi-tion formula for pseudodifferential symbols gives

a(x, ξ′, 0)#b(x, ξ)# 〈ξ〉k =∑

|µ|≤M

1µ!a(µ)(x, ξ′, 0)b(µ)(x, ξ) 〈ξ〉k + rM (x, ξ),(4.6)


where the precise form of the remainder term rM (x, ξ) is given in Lemma 2.1. Let usfirst look at the terms under the summation. For arbitrary α and β,

[a(µ)(x, ξ′, 0)b(µ)(x, ξ) 〈ξ〉k

](α)

(β)≤ Cαβ 〈ξ′〉−|µ|−|α| 〈ξ〉|µ|δ+|β|δ+k |a(x, ξ′, 0)| |a(x, ξ)|−1.

Here we have used the estimates (2.4) and (2.6) of Section 2. Summing for all |µ| ≤M ,we obtain the estimate

[a(µ)(x, ξ′, 0)b(µ)(x, ξ) 〈ξ〉k

](α)

(β)≤ C 〈ξ′〉−|α| 〈ξ〉|β|δ+k 〈ξ′〉−Mδ 〈ξ〉Mδ ·

· |a(x, ξ′, 0)| |a(x, ξ)|−1 max|µ|≤M

(〈ξ′〉Mδ−|µ| 〈ξ〉−Mδ+|µ|δ

)(4.7)

where the maximum is less than or equal to 1. In order to control the remainder termwe consider

rµ,θ(x, ξ) = Os−∫ ∫

e−iyηa(µ)(x, ξ′ + ηθ)b(µ)(x+ y, ξ) 〈ξ〉k dydη

for |µ| = M + 1, and we identify ξ′ and (ξ′, 0).The estimate is quite technical. It will therefore be given in Lemma 4.5, below. Wewill show that, for all multi-indices α, β, with |α| ≤ I and |β| ≤ J , with the numbersI and J of Lemma 4.3, there exists an L ∈ IN, independent of δ, such that

∣∣∣r(α)µ,θ(β)(x, ξ)

∣∣∣ ≤ C 〈ξ〉k−m′+|µ|δ+2Lδ+|β|δ 〈ξ′〉−|α| .(4.8)

Since δ can be chosen arbitrarily small, we may assume that

0 ≤ ([m] + 1 + 2L+ J)δ < m′ − k.

Then (4.8) implies that rM ∈ S00,0(I, J), so that Op rM ∈ L(Hk

2 (IRn)) by Theorem 2.2.By cM (x, ξ) denote the sum on the right hand side of (4.6). Consider

〈ξ′〉−k #a(x, ξ′, 0)#b(x, ξ)# 〈ξ〉k = 〈ξ′〉−k #(cM (x, ξ) + rM (x, ξ)).

Clearly, 〈ξ′〉−k #rM (x, ξ) induces a bounded operator on Hk2 (IRn). Using the compo-

sition formula for N = [m]

〈ξ′〉−k #cM (x, ξ) =∑

|ν|≤N

1ν!

(〈ξ′〉−k

)(ν)

cM(ν)(x, ξ) + rN (x, ξ).(4.9)

Proceeding as before and using (4.7) and (4.1)

∑

|ν|≤N

[(〈ξ′〉−k

)(ν)

cM(ν)(x, ξ)](α)

(β)

≤ cαβ 〈ξ′〉−k−|α| 〈ξ〉|β|δ+k 〈ξ′〉−(M+N)δ 〈ξ〉(M+N)δ |a(x, ξ′, 0)| |a(x, ξ)|−1

≤ c′αβ 〈ξ′〉−k−|α|+1/2+ε∗−(M+N)δ 〈ξ〉k+|β|δ+(M+N)δ−1/2−ε∗.


Since k ≤ 1+ε∗2 , M = N = [m], the summation on the right hand side of (4.9) satisfies

precisely the estimates required for the function q1 in (4.4). We may therefore setq2(x, ξ) = 〈ξ′〉−k #rM (x, ξ)+ rN (x, ξ), and we only have to show that rN (x, ξ) inducesa bounded operator on Hk

2 (IRn). Like before, we introduce rγ,θ and show in Lemma4.5, below, that for all α, β with |α| ≤ I, |β| ≤ J , there exists an L independent of δsuch that ∣∣∣r(α)

γ,θ(β)(x, ξ)∣∣∣ ≤ C 〈ξ〉k−m′−|µ|δ+Mδ+2Lδ+|β|δ 〈ξ′〉−|α| .(4.10)

We may decrease δ again and achieve that the first exponent in (4.10) is negative forall |β| ≤ J . This will imply that Op rN ∈ L(Hk

2 (IRn)) and complete the proof of thelemma. 2

Lemma 4.5. Let 0 ≤ δ < 1 and let p, q be two symbols for which the followingestimates hold ∣∣∣p(α)

(β)(x, ξ)∣∣∣ ≤ cpαβ 〈ξ〉s+|β|δ 〈ξ′〉

s′−|α| ;(4.11)∣∣∣q(α)

(β)(x, ξ)∣∣∣ ≤ cqαβ 〈ξ〉t+|β|δ 〈ξ′〉

t′−|α|.(4.12)

For |θ| ≤ 1 and a multi-index γ define

rγ,θ(x, ξ) = Os−∫ ∫

e−iyηp(γ)(x, ξ′ + θη)q(γ)(x+ y, ξ)dydη.(4.13)

Here we write ξ′ instead of (ξ′, 0). Moreover, let I, J ∈ IN be given. Then there existsan L, depending on I, J , s, s′, t′ and n, but independent of δ and θ, such that

∣∣∣r(α)γ,θ (β)(x, ξ)

∣∣∣ ≤ crαβγ 〈ξ〉t+|γ|δ+2Lδ+|β|δ 〈ξ′〉−|α|(4.14)

for all |γ| ≥ s+ s′ + t′ and all α, β with |α| ≤ I, |β| ≤ J .

Proof. The proof employs a technique of Kumano-go, [Kg, Chapter 2, Lemma 2.4].Due to the more subtle estimates (4.11) and (4.12), however, a more careful analysisis required.We first note that

r(α)γ,θ(β)(x, ξ) = Os−

∫ ∫e−iyη

∑α1+α2=αβ1+β2=β

cα1α2β1β2 p(α1+γ)(β1)

(x, ξ′ + θη)q(α2)(γ+β2)

(x+ y, ξ)dydη.

It is sufficient to consider the terms under the summation separately.Step 1. Fix γ, α1, α2, β1, β2 and let

Iθ(x, ξ) = Os−∫ ∫

e−iyηp(α1+γ)(β1)

(x, ξ′ + θη)q(α2)(γ+β2)

(x+ y, ξ)dydη.

Now let l0 =[

n2

]+ 1. The fact that

e−iyη =(1 + 〈ξ′〉2δ |y|2

)l0 (1 + 〈ξ′〉2δ (−∆η)

)l0e−iyη


together with integration by parts gives

Iθ(x, ξ) = Os−∫ ∫

e−iyηhθ(x, ξ, y, η)dydη ,

wherehθ(x, ξ, y, η) =

(1 + 〈ξ′〉2δ |y|2

)−l0 (1 + 〈ξ′〉2δ (−∆η)

)l0 ·

·[p(γ+α1)(β1)

(x, ξ′ + θη)q(α2)(γ+β2)

(x+ y, ξ)].(4.15)

Since 2l0 > n, integration over y is well-defined.Let

Ω1 =η : |η| ≤ 〈ξ′〉δ/2

, Ω2 =

η : 〈ξ′〉δ/2 < |η| < 〈ξ′〉/2

, Ω3 = η : |η| ≥ 〈ξ′〉/2 .

On Ω1 ∪ Ω2〈ξ′〉2

≤ 〈ξ′ + θη〉 ≤ 32〈ξ′〉(4.16)

and〈ξ′〉2

≤ 〈ξ′ + θη′〉 ≤ 32〈ξ′〉(4.17)

We will now analyze the integrals

Iθ,j = Os−∫

Ωj

∫

IRn

e−iyηhθ(x, ξ, y, η)dydη, j = 1, 2, 3.

Step 2. On Ω1 ∪ Ω2, (4.11), (4.12), (4.16), and (4.17) imply that

|hθ(x, ξ, y, η)| ≤(1 + 〈ξ′〉2δ |y|2

)−l0 ·

·

l0∑

j=0

(l0j

)〈ξ′〉2δj

θ2j∑

|α(j)|=2j

cpγ+α1+α(j),β1〈ξ′ + θη〉s+|β1|δ〈ξ′ + θη′〉s

′−|α1+γ|−2j

· cqα2,γ2+β2〈ξ〉t+|γ+β2|δ 〈ξ′〉t

′−|α2|

≤ cpq1αβγ

(1 + 〈ξ′〉2δ |y|2

)−l0 〈ξ〉t+|γ|δ+|β|δ 〈ξ′〉s+s′+t′−|γ|−|α|,(4.18)

with a suitable constant cpq1αβγ .

Step 3. Since l0 > n2 ,

∫

Ω1

∫ (1 + 〈ξ′〉2δ |y|2

)−l0dydη = 〈ξ′〉−δn

∫

Ω1

∫〈w〉−2l0 dwdη = cl0n

independent of δ and ξ′, cf. [Kg, Chapter 2, (2.31)].Therefore

|Iθ,1(x, ξ)| ≤ c 〈ξ〉t+|γ|δ+|β|δ 〈ξ′〉s+s′+t′−|γ|−|α|.

Since we had assumed that |γ| ≥ s+s′+t′, the first term satisfies the desired estimate.


Step 4. In order to estimate Iθ,2, we first note that∣∣∣∂µ

y (1 + 〈ξ′〉2δ |y|2)−l0∣∣∣ ≤ cµl0 〈ξ′〉δ|µ| (1 + 〈ξ′〉2δ |y|2)−l0 .(4.19)

Together with the estimate in (4.18) this gives

|(−∆y)l0hθ(x, ξ, y, η)| ≤ cl0(1 + 〈ξ′〉2δ |y|2)−l0 ··cpq2

αβγ 〈ξ〉t+|γ|δ+|β|δ+2l0δ 〈ξ′〉s+s′+t′−|γ|−|α|.

Integration by parts yields

|Iθ,2(x, ξ)| =∣∣∣∣∫

Ω2

|η|−2l0

∫e−iyη(−∆y)l0hθ(x, ξ, y, η)dydη

∣∣∣∣

≤ cpq2αβγcl0n 〈ξ〉t+|γ|δ+|β|δ+2l0δ 〈ξ′〉s+s′+t′−|γ|−|α|−2l0δ

.(4.20)

Here we have used the inequality∫

Ω2

|η|−2l0

∫(1 + 〈ξ′〉2δ |y|2)−l0dydη ≤ cl0n 〈ξ′〉−2l0δ

.

Note that (4.20) again has the required from.Step 5. On Ω3, 〈ξ′ + θη′〉 ≤ 〈ξ′ + θη〉 ≤ 3|η|. Hence (4.11), (4.12), and (4.19) imply

the preliminary estimate

|(−∆y)Lhθ(x, ξ, y, η)| ≤ cLl0(1 + 〈ξ′〉2δ |y|2)−l0 ·· cpq3

αβγ |η|s++s′++t′++|β|δ+2l0δ+|α| 〈ξ〉t+|γ|δ+2Lδ+|β|δ 〈ξ′〉−|α| .(4.21)

Here s+ = maxs, 0, analogously we define s′+, t′+; so far L ∈ IN is arbitrary.Given I and J , we now choose L so that

I + J + 2l0 + s+ + s′+ + t′+ ≤ 2L− n− 1

This is independent of γ, δ, and θ. It allows us to estimate the exponent of |η| in(4.21) by 2L− n− 1. Integrating by parts

|Iθ,3(x, ξ)| ≤∣∣∣∣∫

Ω3

|η|−2L

∫e−iyη(−∆y)L hθ(x, ξ, y, η)dydη

∣∣∣∣

≤ cpq4αβγL

∫

Ω3

|η|−n−1dη

∫(1 + 〈ξ′〉2δ |y|2)−l0dy 〈ξ〉t+|γ|δ+2Lδ+|β|δ 〈ξ′〉−|α| .

The integration over y and η gives a finite value, independent of ξ′; so we obtainprecisely the estimate required for (4.14). This concludes the proof. 2

Remark 4.6. Looking more closely at the estimates, it is easy to check that weneed derivatives of p up to order |α| + |γ| + n + 1 in ξ, and order |β| in x. For q weneed the derivatives with respect to ξ up to order |α|, and with respect to x up toorder |β|+ |γ|+ 2L.


5. Proof of the Extension Theorem

The idea in this section is the following. The hypoellipticity of the symbol a givesan embedding of W 1,a′

2 (IRn−1) in Hm′−1/22 (IRn−1). On this space, however, there exist

many linear continuous extension operators to Hm′2 (IRn). We choose a particularly

simple one. Since m′ > 1/2, the restriction operator γ0 is well-defined on Hm′2 (IRn).

Restricting the extension operator to W 1,a′2 (IRn−1), we immediately get a right inverse

to γ0. The difficult task in this section is to show that the extension operator in factmaps W 1,a′

2 (IRn−1) to W 1,a2 (IRn). This again requires a careful analysis. We proceed

in a series of lemmata. As before A, A′, Λ, and Λ′ are the operators with symbolsa(x, ξ), a′(x′, ξ′) = 〈ξ′〉−1/2 #a(x′, 0, ξ′, 0), 〈ξ〉 and 〈ξ′〉, respectively. B′ = Op b′ is aparametrix to A′ over IRn−1.

Lemma 5.1. For a fixed φ ∈ C∞0 (IR) with φ ≡ 1 near 0 define

E : S(IRn−1) → S(IRn)

by(E u)(x′, xn) = F−1

ξ′→x′ φ(〈ξ′〉xn)Fx′→ξ′ u.

Then E extends to a continuous operator

W 1,a′2 (IRn−1) → Hm′

2 (IRn);

moreover, γ0E = id on W 1,a′2 (IRn−1).

Proof. The operator E is suggested in [Kg, Chapter 6, Lemma 2.3]. The hypoellip-ticity of a implies that a′ ∈ S(m−1/2,m′−1/2, δ). If u ∈W 1,a′

2 (IRn−1), thenA′ u ∈ L2,hence u ∈ H

m′−1/22 (IRn−1). It is well-known that E : H

m′−1/22 (IRn−1) → Hm′

2 (IRn)is bounded and that γ0E = id on Hm′−1/2

2 (IRn−1), a forteriori on W 1,a′2 (IRn−1). 2

Lemma 5.2. Fix some k > κ. In order to prove Theorem 3.2 it is sufficient toshow that

Λ−k AEB′ Λ′k ∈ L(Hk2 (IRn−1), Hk

2 (IRn)).(5.1)

Proof. After Lemma 5.1 it remains to show that for u ∈ W 1,a′2 (IRn−1), we have

AE u ∈ L2(IRn); notice that, trivially, E u ∈ L2(IRn). In view of the fact thatW 1,a

2 (IRn) → Hm′2 (IRn), the continuity of E then is immediate from the closed graph

theorem.On the other hand, AE u ∈ L2(IRn) iff Λ−k AE u ∈ Hk

2 (IRn). Now

Λ−k AE u = Λ−k AEB′A′ u+ Λ−k AER′ u ,

where R′ = I − B′A′ ∈ S−∞. Therefore Λ−k AER′ u ∈ Hk2 (IRn); it is in fact much

better. Noting that A′ u ∈ L2(IRn−1) and Λ−k AEB′A′ = Λ−k AEB′ Λ′k Λ′−k A′ weobtain the assertion. 2


In the following we will consider the products Λ−k A and EB′ Λ′k separately. Wewill decompose each of them into a finite number of terms which are ”easy” to handleand remainders with sufficiently good mapping properties.For the analysis of EB′ Λ′k we rely on the theory of potential operators in Boutet deMonvel’s calculus.

Theorem 5.3. Let l = l(x′, ξ′, xn) be a C∞ function on IRn−1× IRn−1× IR, µ ∈ IR,and suppose that for all r, r′ ∈ IN0, α, β ∈ INn−1

0

‖xrnD

r′xnDα

ξ′ Dβx′ l(x

′, ξ′, xn) |L2(IRxn)‖ ≤ Cαβrr′ 〈ξ′〉µ−|α|−r+r′

.(5.2)

Then the operatorOpK l : Hs

2(IRn−1) → Hs−µ2 (IRn)

is bounded for every s ∈ IR. Here the potential operator OpK l associated with l isdefined by

(OpK l)f(x′, xn) = (2π)n−1

2

∫eix′ξ′ l(x′, ξ′, xn) f(ξ′)dξ′.

For every fixed s we only need a finite number of α, β, r, r′ in (5.2) in order toestimate ‖OpK l |L(Hs

2(IRn−1), Hs−µ2 (IRn))‖.

Proof. This is immediate e.g. from Theorem 2.5.1 and (2.3.26) in [Gr]. 2

Lemma 5.4. Let ψ ∈ S(IR) and l(x′, ξ′, xn) = ψ(〈ξ′〉xn). Then l satisfies theestimates in (5.2) for µ = −1/2.

Proof. This is an easy calculation, noting that ‖ψ(〈ξ′〉xn) |L2(IRxn)‖ ≤ C 〈ξ′〉−1/2.

2

Lemma 5.5. Let ψ ∈ S(IR), s ∈ Sµ1,δ(IR

n−1). Then

OpKψ(〈ξ′〉xn) Op s

=∑

|γ|≤N

OpK lγ + (N + 1) ·∑

|γ|=N+1

OpK

∫ 1

0

(1− θ)N

γ!rγ,θ(x′, ξ′, xn)dθ

withlγ(x′, ξ′, xn) = Dγ

ξ′ ψ(〈ξ′〉xn)∂γx′ s(x

′, ξ′)/γ!

and

rγ,θ(x′, ξ′, xn) = Os−∫ ∫

e−iy′η′Dγξ′ψ(〈ξ′ + θη′〉xn)∂γ

x′s(x′ + y′, ξ′)dy′ dη′.

Here, dη = (2π)1−ndη.


Proof. Writing out the integrals, OpK ψ(〈ξ′〉xn) Op s = OpK l with

l(x′, ξ′, xn) = Os−∫ ∫

e−iy′η′ ψ(〈ξ′ + η′〉xn) s(x′ + y′, ξ′) dy′dη′.

Taylor expansion with respect to η′ and integration by parts then gives the desiredformulas. 2

It is our next goal to estimate the symbols of the operators in Lemma 5.5. As apreparation we prove the following technical result.

Lemma 5.6. Let χ ∈ S(IR), p = p(ξ′) ∈ Sν1,0(IR

n−1), q ∈ Sν′1,δ(IR

n−1), 0 ≤ θ ≤ 1,and

kθ(x′, ξ′, xn) = Os−∫ ∫

e−iy′η′χ(〈ξ′ + θη′〉xn)p(ξ′ + θη′)q(x′ + y′, ξ′)dy′dη′ .

Then‖kθ(x′, ξ′, xn)|L2(IRxn)‖ ≤ C 〈ξ′〉ν+ν′−1/2

with a constant independent of θ.

Proof. We again use the technique of the proof of Lemma 4.5; now however, thedomain of the integration is IRn−1, and we need L2(IRxn)−estimates.Fixing l0 = [n−1

2 ] + 1 we have

kθ(x′, ξ′, xn) = Os−∫ ∫

e−iy′η′hθ(x′, ξ′, y′, η′, xn)dy′dη′

with

hθ(x′, ξ′, y′, η′, xn) =(1 + 〈ξ′〉2δ |y′|2

)−l0 ·

·(1 + 〈ξ′〉2δ (−∆η′)

)l0[χ(〈ξ′ + θη′〉xn)p(ξ′ + θη′)q(x′ + y′, ξ′)] .

Let

Ω1 =η′ : |η′| ≤ 〈ξ′〉δ/2

,

Ω2 =η′ : 〈ξ′〉δ/2 < |η′| < 〈ξ′〉/2

,

Ω3 = η′ : |η′| ≥ 〈ξ′〉/2 ,and consider the three integrals similarly as before. The identity

Dηjχ(〈ξ′ + θη′〉xn) = θ(ξ′ + θη′)j

〈ξ′ + θη′〉〈ξ′ + θη′〉−1 (tDtχ)(〈ξ′ + θη′〉xn)

then leads to the desired estimate, very much like before. 2


Corollary 5.7. We use the notation of Lemma 5.5. Then

‖Dαξ′D

βx′x

rnD

r′xnrγ,θ(x′, ξ′, xn)|L2(IRxn)‖ ≤ C 〈ξ′〉µ−(N+1)(1−δ)−|α|+|β|δ−r+r′−1/2

.

The constant is independent of θ.

Proof. We have

Dαξ′D

βx′x

rnD

r′xnrγ,θ(x′, ξ′, xn)

=∑

α1+α2=α

cα1α2 Os−∫ ∫

e−iy′η′xrnD

r′xnDα1+γ

ξ′ ·ψ(〈ξ′+ θη′〉xn)Dα2ξ′ D

β+γx′ s(x′+ y′, ξ′)dy′dη′.

We now have the following identities, j = 1, . . . , n− 1:

xrnD

r′xnψ(〈ξ′ + θη′〉xn) = 〈ξ′ + θη′〉r

′−r (trDr′t ψ)(〈ξ′ + θη′〉xn) ,

Dξjψ(〈ξ′ + θη′〉xn) =(ξ′ + θη′)j

〈ξ′ + θη′〉〈ξ′ + θη′〉−1 (tDtψ)(〈ξ′ + θη′〉xn) .(5.3)

The notation (trDr′t ψ)(〈ξ′ + θη′〉xn) indicates that we evaluate the function

t 7→ trDr′t ψ(t) at t = 〈ξ′ + θη′〉xn. Therefore xr

nDr′xnDα1+γ

ξ′ ψ(〈ξ′ + θη′〉xn) is alinear combination of terms of the form p(ξ′ + θη′)χ(〈ξ′ + θη′〉xn) withp ∈ S

−|α1|−|γ|−r+r′

1,0 (IRn−1) and χ ∈ S(IR). Note that both p and χ depend on thechoice of r and r′.Now the assertion follows from Lemma 5.6. 2

Lemma 5.8. The operator EB′Λ′k can be written in the form

EB′Λ′k =∑

|γ|≤N

1γ!OpKD

γξ′φ(〈ξ′〉xn)∂γ

x′b′(x′, ξ′) 〈ξ′〉k +RN ,(5.4)

where RN : Hk2 (IRn−1) −→ Hm

2 (IRn) is bounded, provided N is large enough. Here φis the fixed function of Lemma 5.1.

Proof. First apply Lemma 5.5 in order to obtain the representation (5.4). Lemma5.5 also gives the precise form of the remainder. Corollary 5.7 in connection withTheorem 5.3 implies the boundedness of RN provided N is large. 2

Lemma 5.9. We have the following estimates:

|Dαξ′D

βx′b

′(x′, ξ′)| ≤ C 〈ξ′〉1/2−|α|+δ|β| |a(x′, 0, ξ′, 0)|−1

for large |ξ′|.

Proof. The symbol b′ is a parametrix to a′(x′, ξ′) = 〈ξ′〉−1/2 #a(x′, 0, ξ′, 0) ∈S (m− 1/2 , m′ − 1/2, δ). By Remark 2.4(iii) we get the desired estimate. 2

Now we consider the composition of Λ−kA with EB′Λ′k. We start with a decompo-sition of the symbol of Λ−kA.


Lemma 5.10. Let L,M ∈ IN. Then we can write

〈ξ〉−k #a(x, ξ) =M∑

j=0

xjnaj(x′, 0, ξ) + xM+1

n aM+1(x, ξ) + rL(x, ξ),

where, for all multi-indices α, β, we have the estimates

|Dαξ D

βxaj(x′, 0, ξ)| ≤C|a(x′, 0, ξ)| 〈ξ〉−k+jδ−|α|+|β|δ

, j = 0, . . . ,M, |ξ| large,(5.5)

|Dαξ D

βxaj(x′, 0, ξ)| ≤C 〈ξ〉m−k+jδ−|α|+|β|δ

, j = 0, . . . ,M,(5.6)

|Dαξ D

βxaM+1(x, ξ)| ≤ C 〈ξ〉m−k+(M+1)δ−|α|+|β|δ

,(5.7)

|Dαξ D

βxrL(x, ξ)| ≤C 〈ξ〉m−k−L(1−δ)−|α|+|β|δ

.(5.8)

Proof.We use the asymptotic expansion formula for symbols

〈ξ〉−k #a(x, ξ) =∑

|ν|≤L

1ν!

(〈ξ〉−k)(ν)a(ν)(x, ξ) + rL(x, ξ).

In view of the fact that a ∈ S(m,m′, δ) we have∣∣∣[(〈ξ〉−k)(ν)a(ν)(x, ξ)]

(α)(β)

∣∣∣ ≤ C 〈ξ〉−k−|ν|(1−δ)−|α|+|β|δ |a(x, ξ)|,(5.9)

and (5.8) is the immediate standard estimate for the remainder. Now we use Taylor’sformula

f(xn) =M∑

j=0

xjn

j!f (j)(0) +

xM+1n

M !

∫ 1

0

(1− θ)Mf (M+1)(θxn)dθ

for the function∑|ν|≤L

1ν! (〈ξ〉−k)(ν)a(ν)(x, ξ). We obtain (5.5) from (5.9). Estimates

(5.6) and (5.7) also follow from (5.9), using that |a(x, ξ)| ≤ C 〈ξ〉m . 2

Remark 5.11. In order to show the boundedness of Λ−kAEB′Λ′k : Hk2 (IRn−1) →

Hk2 (IRn) we split the operator up into four parts. In the notation of Lemma 5.8 and

Lemma 5.10

Λ−kAEB′Λ′k =M∑

j=0

∑

|γ|≤N

Op(xjnaj(x′, 0, ξ))OpK(Dγ

ξ′φ(〈ξ′〉xn)∂γx′b

′(x′, ξ′) 〈ξ′〉k)/γ!

+∑

|γ|≤N

Op(xM+1n aM+1(x, ξ))OpK(Dγ


′(x′, ξ′) 〈ξ′〉k)/γ!(5.10)

+∑

|γ|≤N

Op rLOpK(Dγξ′φ(〈ξ′〉xn)∂γ

x′b′(x′, ξ′) 〈ξ′〉k)/γ!

+Λ−kARN .

Obviously the last summand does not require any attention, since RN : Hk2 (IRn−1) −→

Hm2 (IRn) is bounded.


In order to analyze the others we make the following observations

(i) Op (xjnc(x, ξ)) =

∑jl=0

(jl

)Op((−Dξn

)lc(x, ξ))Op (xj−ln ) for an arbitrary sym-

bol c.(ii) Dγ

ξ′φ(〈ξ′〉xn) is a linear combination of terms of the form p(ξ′)ψ(〈ξ′〉xn) where

p ∈ S−|γ|1,0 (IRn−1) and ψ ∈ S(IR). Noting that

Op(xn)OpK(ψ(〈ξ′〉xn)) = OpK(〈ξ′〉−1 (tψ)(〈ξ′〉xn))

(for the notation cf. the remark after (5.3)), the composition

Op (xj−ln )OpK(Dγ


′(x′, ξ′) 〈ξ′〉k)

is a linear combination of terms of the form

OpK(p(ξ′)ψ(〈ξ′〉xn)∂γx′b

′(x′, ξ′) 〈ξ′〉k)(5.11)

with p ∈ S−|γ|−j+l1,0 (IRn−1) and ψ ∈ S(IR).

(iii) In view of (ii), Lemma 5.9, and Theorem 5.3 we will have

OpK(Dγξ′φ(〈ξ′〉xn)∂γ

x′b′(x′, ξ′) 〈ξ′〉k) : Hk

2 (IRn−1) → Hm′2 (IRn)

bounded. Moreover Op rL : Hm′2 (IRn) → Hk

2 (IRn) will be bounded for largeL. Hence the third term in (5.10) induces a bounded operator from Hk

2 (IRn−1)to Hk

2 (IRn) and this is sufficient for Theorem 3.2 according to Lemma 5.2.

We now treat the second summand in (5.10).

Lemma 5.12. For every γ

Op(xM+1n aM+1(x, ξ))OpK(Dγ


′(x′, ξ′) 〈ξ′〉k) : Hk2 (IRn−1) −→ Hk

2 (IRn)

is bounded provided M is sufficiently large.

Proof.The considerations in Remark 5.11 show that we have to estimate the norm of an

expression of the form

Op (DlξnaM+1(x, ξ))OpK(p(ξ′)ψ(〈ξ′〉xn)∂γ

x′b′(x′, ξ′) 〈ξ′〉k)

where l = 0, . . . ,M + 1, p ∈ S−M−1−|γ|+l1,0 (IRn−1), and ψ ∈ S(IR). We now use the

estimates of Lemma 5.4 and Lemma 5.9 together with the fact that |a(x′, 0, ξ′, 0)|−1 ≤C 〈ξ′〉−m′

. We get

‖Dαξ′D

βx′x

rnD

r′xnp(ξ′)ψ(〈ξ′〉xn)∂γ

x′b′(x′, ξ′) 〈ξ′〉k|L2(IRxn)‖

≤ c 〈ξ′〉−|α|−r+r′−M−1−|γ|+l+|γ|δ+|β|δ+k−m′.


Using Theorem 5.3 this will induce a bounded operator from Hk2 (IRn−1) to

Hk+M−l−k+m′2 (IRn) = HM−l+m′

2 (IRn), noting that δ can be taken arbitrarily small,so that for the maximum |β| required for the boundedness estimate we still have|β|δ < 1/2.

The operator OpDlξnaM+1 maps HM−l+m′

2 (IRn) to Hk+M(1−δ)−δ−m+m′

2 (IRn) whichis a subspace of Hk

2 (IRn), provided M is large. 2

Now for the terms in the first summand in (5.10). We start with a simple observation.

Lemma 5.13. Let q ∈ ⋂δ>0

Sµ1,δ(IR

n−1), ψ ∈ S(IR). Then

OpK(ψ(〈ξ′〉xn)q(x′, ξ′)) = Op q(x′, ξ′)OpK(ψ(〈ξ′〉xn)) .

Notice that the mapping (x′, xn, ξ′, ξn) 7→ q(x′, ξ′) defines a symbol in S

µ++ε0,0 (IRn) for

every ε > 0, so that the operator Op q is well-defined on each Hs2(IRn).

Lemma 5.14. We use the notation of Lemma 5.10 and Lemma 5.11.

Op(xjnaj(x′, 0, ξ))OpK(Dγ


′(x′, ξ′) 〈ξ′〉k) : Hk2 (IRn−1) −→ Hk

2 (IRn)

is bounded for 0 ≤ j ≤M and all multi-indices γ, |γ| ≤ N . Without loss of generalitywe assume that M and N are fixed so large that the boundedness results of Lemma 5.8and Lemma 5.12 are valid.

Proof. As pointed out in Remark 5.11 it is sufficient to consider

Op(Dlξnaj(x′, 0, ξ))OpK(p(ξ′)ψ(〈ξ′〉xn)∂γ

x′b′(x′, ξ′) 〈ξ′〉k)

for l = 0, . . . , j, p ∈ S−|γ|−j+l1,0 (IRn−1), ψ ∈ S(IR). Using the boundedness of

OpK(〈ξ′〉1/2ψ(〈ξ′〉xn)) : Hk

2 (IRn−1) −→ Hk2 (IRn)

and Lemma 5.13, all we have to show is the boundedness of

Op(Dlξnaj(x′, 0, ξ))Op(p(ξ′)∂

γx′b

′(x′, ξ′) 〈ξ′〉k−1/2) : Hk2 (IRn) −→ Hk

2 (IRn) .

For fixed j, γ and l we denote the first symbol by q1(x′, ξ) and the second by q2(x′, ξ′).By Lemma 2.1 we get

(q1#q2)(x′, ξ) =∑

|α|≤N

1α!q(α)1 (x′, ξ)q2(α)(x′, ξ′) + rN (x′, ξ)(5.12)

where rN (x′, ξ) is given in Lemma 2.1 and depends, of course, on j, γ and l.Now we show that each term of the sum of the right hand side of (5.12) definesa bounded operator in Hk

2 (IRn). The boundedness of the operator defined by the


remainder term in (5.12) will be shown in Lemma 5.16, below. Consider an arbitraryterm under the summation in (5.12). By Lemma 5.10 we have

|Dαξ D

βx(∂α

ξ Dlξnaj(x′, 0, ξ)p(ξ′) 〈ξ′〉k−1/2

Dαx∂

γx′b

′(x′, ξ′))|≤ c|a(x′, 0, ξ)| 〈ξ〉−k+jδ−l−|α|+|β|δ |a(x′, 0, ξ′, 0)|−1

· 〈ξ′〉1/2−|γ|−j+l+(k−1/2)+|α|δ+|γ|δ−|α|

≤ c

∣∣∣∣a(x′, 0, ξ′, ξn)a(x′, 0, ξ′, 0)

∣∣∣∣( 〈ξ〉〈ξ′〉

)−k+ρ|α|−l+jδ

〈ξ〉−ρ|α|+δ|β|

· 〈ξ′〉−|γ|(1−δ)−|α|(1−δ)−j(1−δ)−|α|(1−ρ)

≤ c 〈ξ〉−ρ|α|+δ|β|( 〈ξ〉〈ξ′〉

)κ−k+ρ|α|+jδ−l

,

because of the assumption of Theorem 3.2.Let I and J again denote the numbers of derivatives required in Theorem 2.2 for s = k.Then we have for |α| ≤ I, |β| ≤ J

|Dαξ D

βx(∂α

ξ q1Dαx q2)| ≤ c 〈ξ〉−ρ|α|+δ|β|

( 〈ξ〉〈ξ′〉

)κ−k+ρI+δM−l

≤ c 〈ξ〉−ρ|α|+δ|β|

provided that we choose ρ = min(

κ−kI+M , 1

)and δ < ρ. This is possible, since we

fixed k > κ in Lemma 5.2, and since a(x, ξ) belongs to S(m,m′; δ) for all δ > 0by assumption. Then the last inequality implies that ∂α

ξ q1Dαx q2 ∈ S0

ρ,δ(I, J) andby Theorem 2.2 we get the boundedness in Hk

2 (IRn) of all summands of the sum inthe right hand side of (5.12). The proof is complete except for the estimates of theremainder terms which will be given in the next two lemmas. 2

To estimate the terms rN (x, ξ) we need yet another modification of Lemma 2.4 in[Kg, Chapter 2].

Lemma 5.15. Let 0 ≤ δ ≤ 12 and let q1, q2 be two symbols for which the following

estimates hold

|q(α)1(β)(x, ξ)| ≤ c1αβ 〈ξ〉s−|α|+|β|δ

|q(α)2(β)(x, ξ)| ≤ c2αβ 〈ξ〉|β|δ 〈ξ′〉t

′−|α|.

For |θ| ≤ 1 then define

rγ,θ(x, ξ) = Os−∫ ∫

e−iyηq(γ)1 (x, ξ + θη)q2(γ)(x+ y, ξ′)dy dη ;

where ξ′ = (ξ1, . . . , ξn−1, 0). Moreover, let I, J ∈ IN be given. Then

|r(α)γ,θ(β)(x, ξ)| ≤ crαβγ

for all α, β with |α| ≤ I, |β| ≤ J and all sufficently large γ ; in fact it is sufficient tohave |γ| ≥ 2(s+ + t′+) + J . As before we use the notation σ+ = maxσ, 0.


We omit the complete proof. It is very similar to that of Lemma 4.5; in fact it evenis slightly easier. Here we split IRn

η with respect to 〈ξ〉 instead of 〈ξ′〉. Therefore thethird term Iθ,3(x, ξ) permits such good estimates as in [Kg, Chapter 2, p.72].Indeed, we get

|r(α)γ,θ(β)(x, ξ)| ≤ crαβγ 〈ξ〉s++t′+−|γ|(1−δ)+|β|δ 〈ξ′〉−|α| .

Since δ ≤ 12 , the first exponent is negative for |γ| ≥ 2(s+ + t′+) + J .

Lemma 5.16. We use the notation of Lemma 5.14. For N sufficiently large theremainder terms rN (x′, ξ) in (5.12) define bounded operators in Hk

2 (IRn).

Proof. From Lemma 5.10 and Lemma 5.14 we know that the symbols q1 and q2introduced before fulfill the estimates in Lemma 5.15 with s = m − k + jδ − l andt′ = −|γ| − j + l−m′ + |γ|δ+ k ; where 0 ≤ j ≤M and M was fixed by Lemma 5.12.We let I and J denote the numbers of derivatives in x and ξ, respectively, requiredfor the continuity of symbols in S0

0,0(I, J) on Hk2 (IRn) by Theorem 2.2. Lemma 5.15

then yields the assertion. 2

This completes the proof of Lemma 5.14, hence that of Theorem 3.2.

References

[Be] R.Beals: Weighted distribution spaces and pseudodifferential operators. J. d’AnalyseMathematique 39 (1981), 131-187.

[Bea] B.Beauzamy: Espaces de Sobolev et de Besov d’ordre variable definis sur Lp.C. R. Acad. Sci. Paris 274 (1972), 1935-1938.

[Bo] J.-M.Bony and J.-Y.Chemin: Espaces fonctionnels associes au calcul de Weyl-Hormander.Bull. Soc. Math. France, 122 (1994), 119-145.

[CV1] A.P.Calderon and R.Vaillancourt: On the boundedness of pseudodifferential operators.J. Math. Soc. Japan 23 (1971), 374-378.

[CV2] A.P.Calderon and R.Vaillancourt: A class of bounded pseudodifferential operators.Proc. Nat. Acad. Sci. USA (1972), 1185-1187.

[Fe] C.Fefferman: Lp-bounds for pseudo-differential operators, Israel J. Math. 14 (1973), 413-417.

[Gr] G.Grubb: Functional Calculus for Boundary Value Problems. Birkhauser, Boston, Basel1986.

[JL] N.Jacob and H.-G. Leopold: Pseudo differential operators with variable order of differenti-ation generating Feller semigroups. Integral Equations Operator Theory 17 (1993), 544-553.

[Ka] G.A.Kalyabin: Spaces of traces for generalized Liouville anisotropic classes. (Russ.) Izv.Akad. Nauk SSSR Ser. Mat. 42 (1978), 305-314.

[KL] G.A.Kalyabin and P.I.Lizorkin: Spaces of Functions of Generalized Smoothness.Math. Nachr. 133 (1987), 7-32.

[KN] K.Kikuchi and A.Negoro: Pseudodifferential operators and Sobolev spaces of variableorder of differentiation. Rep.Fac. Liberal Arts, Shizuoka University, Sciences, 31 (1995),19-27.

[Kg] H.Kumano-go: Pseudo-Differential operators. Massachusetts Institute of Technology Press,Cambridge (Massachusetts)-London 1981.


[Le1] H.-G.Leopold: Pseudodifferentialoperatoren und Funktionenraume variabler Glattheit.Dissertation B. Jena: Friedrich-Schiller-Univ. 1987.

[Le2] H.-G.Leopold: On Besov Spaces of Variable Order of Differentiation. Z. Anal. Anw. 8,Heft 1 (1989), 69-82.

[Le3] H.-G.Leopold: On function spaces of variable order of differentiation. Forum Math.3 (1991), 1-21.

[LS1] H.-G.Leopold and E.Schrohe: Spectral invariance for algebras of pseudodifferential oper-ators on Besov spaces of variable order of differentiation. Math. Nachr. 156 (1992), 7-23.

[LS2] H.-G.Leopold and E.Schrohe: Spectral invariance for algebras of pseudodifferential oper-ators on Besov-Triebel-Lizorkin spaces, manuscripta mathematica 78 (1993), 99-110.

[Ne] A.Negoro: Stable-like process: construction of the transition density and the behavior ofsample paths near t=0 . Osaka J. Math. 31 (1994), 189-214.

[Pe] I.Pesenson: The trace problem and Hardy operator for non-isotropic function spaces onthe Heisenberg group. Comm. Part. Diff. Eq. 19 (1994), 655-676.

[ScS] E.Schrohe and B.-W.Schulze: Boundary Value Problems in Boutet de Monvel’s Algebrafor Manifolds with Conical Singularities I, in: Pseudodifferential Calculus and MathematicalPhysics, Advances in Partial Differential Equations 1, Akademie Verlag, Berlin 1994.

[Sc] B.-W.Schulze: Pseudodifferential Operators on Manifolds with Singularities, North-Holland, Amsterdam, 1991.

[Un] A.Unterberger: Sobolev spaces of variable order and problems of convexity for partialdifferential operators with constant coefficients. Asterisque 2 et 3 Soc. Math. France, 1973,325-341.

[UB] A.Unterberger and J.Bokobza: Les operateurs pseudodifferentiels d’ordre variable.C. R. Acad. Sci. Paris 261 (1965), 2271-2273.

[VE1] M.I.Visik and G.I.Eskin: Elliptic convolution equations in a bounded region and theirapplications. (Russ.) Uspekhi Mat. Nauk 22.1 (1967), 15-76.

[VE2] M.I.Visik and G.I.Eskin: Convolution equations of variable order. (Russ.) TrudyMoskov. Mat. Obsc. 16 (1967), 26-49.

[VK] L.V.Volevic and V.M.Kagan: Pseudodifferential hypoelliptic operators in the theory offunction spaces. (Russ.) Trudy Moskov. Mat. Obsc. 20 (1969), 241-275.

Mathematisches InstitutFakultat fur Mathematikund InformatikFriedrich-Schiller-Universitat JenaD-07740 JENAGermany

Max-Planck-Arbeitsgruppe“Partielle Differentialgleichungenund Komplexe Analysis”Universitat PotsdamD-14415 POTSDAMGermany

trace theorems for sobolev spaces of variable order of ......on lp sobolev spaces requires that ρ =...

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