PARABOLIC BOUNDARY INTEGRAL OPERATORS SYMBOLIC REPRESENTATION AND BASIC PROPERTIES M. COSTABEL AND J. SARANEN In this paper we develop a theory of parabolic pseudodifferential operators in anisotropic spaces. We construct a symbolic calculus for a class of symbols globally defined on Rn+1 × Rn+1 , and then develop a periodisation procedure for the calculus of symbols on the cylinder Tn × R. We show G˚ arding’s inequality for suitable operators and precise estimates for the essential norm in anisotropic Sobolev spaces. These new mapping properties are needed in localization arguments for the analysis of numerical approximation methods.
1. Introduction The motivation for writing the present paper comes from the analysis of some boundary element methods for the heat equation. As in the case of elliptic boundary value problems, the analysis of pseudodifferential operators can be used for this purpose. But unlike for elliptic operators, the pseudodifferential operators arising from the boundary reduction of parabolic operators have not found a complete treatment in the literature. Thus we could not find G˚ arding’s inequality nor the sharp norm estimates in anisotropic Sobolev spaces needed for localization techniques. There exist parts of a general theory of parabolic pseudodifferential operators. In [17], [18], Piriou develops a theory of parabolic boundary value problems and parabolic pseudodifferential operators by using expansions in quasi-homogeneous functions to represent the symbols of parabolic pseudodifferential operators. Piriou’s work, whose main results were prepared in [11], [12], covers standard results of the calculus of pseudodifferential operators except for those mentioned above. In the present paper, we develop a calculus of pseudodifferential operators with anisotropic symbols which contains the above-mentioned boundary integral operators for parabolic boundary value problems. The class of operators considered here is also covered by the calculus of general pseudodifferential operators of Beals [3]. Beals proves L2 and Sobolev space continuity based on the Calder´on-Vaillancourt-Cotlar-Knapp-Stein theorems. In the present paper, we prefer to follow the more elementary approach of H¨ormander [7], [8]. Like Beals’, this presentation starts with a symbolic calculus for a class of globally defined symbols, thus avoiding the notion of properly supported operators. It is more elementary in that it obtains L2 continuity from simple properties of weakly singular integral operators. The present approach also provides natural tools to carry out a periodisation in order to cover the case of cylindrical domains. Received by the editors August 28, 2000.
After completing the symbolic calculus, we investigate the corresponding operators in the framework of anisotropic Sobolev spaces. New mapping properties related to localization are shown in Corollaries 3.13, 4.6 and 5.4. These results are very important (necessary in some cases) when discussing approximation methods for the solution of boundary integral equations. The essence of the localization technique is that it allows to extend the results obtained for convolutional operators to the case of more general pseudodifferential operators, in particular to the case where the spatial domain is not a two-dimensional disc. We need the localization result of Corollary 5.4 in the forthcoming paper [6] where we extend the spline collocation results from circular cylinders, see [5], to the case of cylinders defined by a smooth closed curve. In addition to the sharp norm estimates we prove G˚ arding’s inequality. For the basic parabolic boundary integral operators (single layer and the hypersingular operator), we thus recover the result of [15], [2], [4], [9]. There are some earlier studies which are close to our work, in particular the paper [19] of Rabinovich is worth mentioning, see also [20]. 2. Anisotropic symbols In this section we introduce a class of anisotropic symbols and present some basic properties for this class. We follow closely the presentation by H¨ormander [8] §18 of the isotropic case. We fix some notations as follows. Let m ∈ R be given such that m ≥ 1. With this m we associate an anisotropic distance ρ in the space Rn+1 defined by 1
(2.1)
ρ(ζ) = ρm (ζ) = |ξ| + |η| m ;
ζ = (ξ, η) ,
ξ ∈ Rn , η ∈ R.
β For the definition of the symbol class Sm , we use the following notations: 1 1 2 2 n+1 1 2 For z = (x, t) ∈ R and ν = (ν , ν ) ∈ Nn0 × N0 , ∂zν = ∂xν ∂tν , similarly ∂ζµ = ∂ξµ ∂ηµ for µ = (µ1 , µ2 ) ∈ Nn0 × N0 . In addition to the conventional notation |ν| = |ν 1 | + ν 2 where |ν 1 | = ν11 + · · · + νn1 for ν 1 = (ν11 , . . . , νn1 ), we define |µ|m = |µ1 | + mµ2 .
Definition 2.1. Let β ∈ R and m ∈ R, m ≥ 1. We say that a function a ∈ C ∞ (Rn+1 ×Rn+1 ) z ζ n+1 n+1 β 1 2 1 2 is a symbol of the class Sm if for all ν = (ν , ν ) ∈ N0 , µ = (µ , µ ) ∈ N0 there exists a constant Cν,µ such that β−|µ|m (2.2) |∂zν ∂ζµ a(z, ζ)| ≤ Cν,µ 1 + ρ(ζ) for all z, ζ ∈ Rn+1 . β Given a symbol a(z, ζ) ∈ Sm , we define the pseudodifferential operator a(z, D) via the expression Z −(n+1) (2.3) a(z, D) u(z) = (2π) eihz , ζi a(z, ζ)ˆ u(ζ) dζ.
By S(Rn+1 ) we denote the Schwartz space of rapidly decreasing functions and by S 0 (Rn+1 ) its dual space. Here u ∈ S(Rn+1 ) and uˆ ∈ S(Rn+1 ) is its Fourier transform defined by Z uˆ(ζ) = e−ihz , ζi u(z) dz. The bilinear form hz , ζi is the usual scalar product in Rn+1 . It is clear that if a(z, ζ) = β ζj (j = 1, . . . , n + 1), then a(z, D) = Dj = −i ∂z∂ j . We shall write a(z, D) ∈ Op Sm for β operators defined in (2.3) for symbols a ∈ Sm . 2
In [3], the analysis of pseudodifferential operators with anisotropic symbols is based on the notion of “weight vectors”. For m = 2, see [3, Example 2.11]. It is not hard to see that our symbol classes are contained in those considered in [3], if we define the weight vector (ϕ1 , . . . , ϕn+1 , Φ1 , . . . , Φn+1 ) by ϕj (z, ζ) = 1 (j = 1, . . . , n + 1) and Φj (z, ζ) = 1 + ρ(ζ) (j = 1, . . . , n), Φn+1 (z, ζ) = (1 + ρ(ζ))m . The decisive feature of the anisotropic distance ρ which proves that the axioms (2.1)–(2.7) of weight vectors in [3] are satisfied is that ρ satisfies, for any m ≥ 1, the triangle inequality ρ(ζ + ζ 0 ) ≤ ρ(ζ) + ρ(ζ 0 ) (ζ, ζ 0 ∈ Rn+1 ) .
(2.4)
Note that for m > 1, the anisotropic balls defined by ρ(ζ) < R are not convex. This triangle inequality will be important also for our approach. β With fixed parameters β and m, the symbol class Sm is a topological space such β that the topology is defined by means of the seminorms qν,µ , ν, µ ∈ Nn+1 , 0 |µ|m −β β (2.5) qν,µ (a) = sup |∂zν ∂ζµ a(z, ζ) 1 + ρ(ζ) |. z,ζ
β β By definition a sequence of symbols aj ∈ Sm converges to a ∈ Sm if β qν,µ (aj − a) → 0 ,
(2.6)
j→∞
for all ν, µ ∈ Nn+1 , and {aj } is bounded if 0 β sup qν,µ (aj ) ≤ Cν,µ
(2.7)
j
Nn+1 . 0
for all ν, µ ∈ Having given a symbol a(z, ζ) we consider a family of symbols aε (z, ζ) , ε ≥ 0 which are defined by (2.8)
ζε := (εξ, εm η).
aε (z, ζ) = a(z, ζε ) ,
The proofs of the following two lemmas can be adapted from the proofs of Propositions 18.1.2 and 18.1.3 in [8]. For the proofs one may use the characteristic property of ρ(ζ), ρ(ζε ) = ερ(ζ), ε > 0 , and in addition for Lemma 2.2 the property: Given χ ∈ C0∞ (Rn+1 ) and ε > 0, there exists ζ δε > 0 such that |ζ| ≥ δε ⇒ χ(ζε ) = 0.
(2.9)
0 Lemma 2.1. Assume a ∈ Sm and 0 ≤ ε ≤ 1. Then the family aε , 0 ≤ ε ≤ 1 is bounded in 0 β Sm , and aε → a0 in Sm for all β > 0. More precisely, there holds
(2.10)
0 sup qν,µ (aε ) ≤ Cν,µ ,
0≤ε≤1
(2.11)
β qν,µ (aε − a0 ) ≤ Cν,µ εmin{1,β} β
if β > 0.
β0 Lemma 2.2. Assume that aj ∈ Smj , βj → −∞. Then there exists a symbol a ∈ Sm such that k−1 X βk (2.12) a− aj ∈ S m , k ∈ N. j=0
3
−∞ The symbol a(z, ζ) is uniquely defined S∞ modulo an additive term in Sm . Furthermore a(z, ζ) can be chosen such that supp a ⊂ j=0 supp aj .
The situation described in Lemma 2.2 will be abbreviated by ∞ X a ∼ aj . j=0
Remark 2.1. It is not required that the orders are monotonically decreasing. But by taking partial sums, we can always achieve this, and so we are allowed to assume that the numbers βj satisfy β0 > β1 > · · · . In this case we say that a0 is a principal symbol of a. 0 ) Lemma 2.3. Assume that there are given real valued symbols a1 , . . . , ak ∈ Sm (Rn+1 × Rn+1 z ζ ∞ k and suppose F ∈ C (R ). Define (2.13) F (a1 , . . . , ak )(z, ζ) = F a1 (z, ζ), . . . , ak (z, ζ) , z, ζ ∈ Rn+1 . 0 Then we have F (a1 , . . . , ak ) ∈ Sm .
Proof. Clearly F (a1 , . . . , ak ) ∈ C ∞ (Rn+1 × Rn+1 ) and it remains to verify that (2.2) holds z ζ 0 with β = 0. Now, since the functions al ∈ Sm are bounded and ∂ak F is continuous (infinitely smooth) we get |∂zj F (a1 , . . . , ak )(z, ξ, η)| ≤ C −1 , j = 1, . . . , n |∂ξj F (a1 , . . . , ak )(z, ξ, η)| ≤ C 1 + ρ(ζ) (2.14) −m . |∂η F (a1 , . . . , ak )(z, ξ, η)| ≤ C 1 + ρ(ζ) By (2.14), estimate (2.2) is valid with |ν| = 1, |µ| = 1, and the general case follows by induction. 3. Basic calculus of anisotropic pseudodifferential operators In this section, we prove the basic properties of pseudodifferential operators on β R as defined in (2.3) with anisotropic symbols from the classes Sm . The results up to the boundedness properties in anisotropic Sobolev spaces are consequences of Beals’ general calculus of pseudodifferential operators [3]. As in the previous section we follow for this part the more elementary presentation by H¨ormander [8]. For the part concerning sharp norm estimates and G˚ arding’s inequality, we apply the approach of [21], Chapter II. The first observation shows in particular that a(z, D) maps S(Rn+1 ) into itself. Its proof is identical to the case of isotropic symbols given by H¨ormander [8]. n+1
β Lemma 3.1. For a ∈ Sm and u ∈ S(Rn+1 ), (2.3) defines a function a(z, D)u ∈ S(Rn+1 ). The mapping (a, u) 7→ a(z, D)u : Sβm × S(Rn+1 ) → S(Rn+1 ) is continuous. Define the commutators between a(z, D) and Dj or zj by
[a(z, D), Dj ]u = a(z, D)Dj u − Dj a(z, D)u [a(z, D), zj ]u = a(z, D)(zj u) − zj a(z, D)u. Then we have for j = 1, . . . , n + 1 [a(z, D), Dj ] = i(∂zj a)(z, D) [a(z, D), zj ] = −i(∂ζj a)(z, D). 4
Remark 3.1. In some applications (in particular for parabolic operators), there appear symbols which do not have the smoothness required in Lemma 3.1. Typically, for a parabolic boundary integral operator the symbol is not smooth with respect to the variable ζ at ζ = 0 ; for an example see Example 5.1. To cover also situations of this kind, suppose that a(z, ζ) ∈ C ∞ (Rn+1 , E 0 (Rn+1 )) , z ζ where E 0 (Rn+1 ) denotes the space of compactly supported distributions in Rn+1 . Then for u ∈ S(Rn+1 ) there is defined a(z, D)u ∈ C ∞ (Rn+1 ) by a natural extension of (2.3), and the mapping u 7→ a(z, D)u is continuous from S(Rn+1 ) to C ∞ (Rn+1 ) . The next step is to show the basic results about adjoints and products of pseudodifferential operators and about their continuity in anisotropic Sobolev spaces on Rn+1 . The first result corresponds to [8, Th.18.1.7]. We only sketch its proof. β Theorem 3.2. Let m ≥ 1 and β ∈ R. Let a ∈ Sm and b(z, ζ) = eihDz , Dζ i a(z, ζ). Then
(i )
β b ∈ Sm
(ii )
a(z, D) = b(z, D)∗ ∞ X X ∂zµ Dζµ k a(z, ζ) (ihDz , Dζ i) b(z, ζ) ∼ ∼ a(z, ζ). k! µ! µ k=0
(iii )
Proof. The definition of b means that (3.1)
ˆ ˆ ˆb(ˆ ˆ = eihˆz , ζi b a(ˆ z , ζ), z , ζ)
ˆ are the dual variables of (z, ζ). In order where ˆb is the Fourier transform in S 0 of b and (ˆ z , ζ) to show (ii), it is sufficient to consider a ∈ S(R2(n+1) ). In this case, one can represent the Schwartz kernel K(x, y) ∈ S(R2(n+1) ) of a(z, D) in the form Z −(n+1) (3.2) K(z, y) = (2π) eihz−y , ζi a(z, ζ) dζ . For the Fourier transform of a, this representation implies Z n+1 ˆ = e−ihˆz , zi K(z, z + ζ)(2π) ˆ a ˆ(ˆ z , ζ) dz. The fact that the kernel of the adjoint operator is K ∗ (z, y) = K(y, z) then leads to (3.1), and (ii) is shown. β For (iii) (which implies (i)) and a general a ∈ Sm , one uses a Littlewood-Paley decomposition of a(z, ζ) and b(z, ζ) using anisotropic annular domains defined by 2l−1 < ρ(ζ) < 2l ,
,l ∈ N.
For the estimation of the individual terms in the decomposition, one uses H¨ormander’s stationary phase estimate which we quote for the sake of completeness below in Lemma 3.3. This estimate invokes the euclidean distance d(z, ζ) of a point (z, ζ) to an anisotropic ball defined by ρ(ζ 0 ) < 2l0 (l0 ≥ 0), where ζ is such that 2l0 +1 < ρ(ζ) ≤ 2l0 +2 . One needs here the following elementary properties of ρ (3.3)
∀ζ, ζ 0 ∈ Rn+1 : ρ(ζ) ≤ |ζ| + 1
and ρ(ζ + ζ 0 ) ≤ ρ(ζ) + ρ(ζ 0 ), 5
which allow to estimate for ζ, ζ 0 as above: |ζ − ζ 0 | > 51 (ρ(ζ) + 1), hence 1 (3.4) d(z, ζ) ≥ ρ(ζ) + 1 . 5 With these additional observations, H¨ormander’s proof goes through for our case. Lemma 3.3 ([7], Lemma 7.6.4 and Theorem 7.6.5). Let A be a complex symmetric N × N matrix defining a quadratic form x 7→ A(x) := hAx , xi on RN and a convolution operator Z −A(D) −N e u(x) := (2π) eihx , ξi−A(ξ) uˆ(ξ) dξ for u ∈ S(RN ). We assume that ∀x ∈ RN : Re A(x) ≥ 0 and 1
|A(x)| 2 ≤ 1. kAk := sup |x| x6=0 Let s be an integer, s > N2 . For every integer k ≥ s, there exists a constant Ck such that for any u ∈ S(RN ), x ∈ RN −A(D)
|e
u(x) −
k−1 X
X
(−A(D))j u(x)/j!| ≤ Ck · max{1, d(x)}−k kAkk ·
j=0
sup|Dα u| ,
|α|≤s+2k
where d(x) is the distance between x and supp u. Note that Ck does not depend on u, x, and A. The proof for the asymptotic expansion of the symbol of the product of two pseudodifferential operators uses a similar procedure as the one for the adjoint operator above. Here we reproduce some of the details of the proof which will be needed in the following section for a corresponding result in the periodic case. β
Theorem 3.4. Let aj ∈ Smj (j = 1, 2). Then with (3.5)
b(z, ζ) = eihDy , Dθ i a1 (z, θ)a2 (y, ζ) y=z,θ=ζ
β1 +β2 one has b ∈ Sm ,
(3.6)
b(z, D) = a1 (z, D)a2 (z, D),
and the asymptotic expansion ∞ X 1 (ihDy , Dθ i)k a1 (x, θ)a2 (y, ζ) θ=ζ,y=x k! k=0 X 1 µ ∼ ∂ζ a1 (z, ζ)Dzµ a2 (z, ζ). µ! µ
b(z, ζ) ∼ (3.7)
Proof. Define B(z, ζ, y, θ) = eihDy , Dθ i a1 (z, θ)a2 (y, ζ). We consider a(y, θ) = a1 (z, θ)a2 (y, ζ) as a symbol in (y, θ) where z and ζ appear as parameters. This a satisfies β −|µ|m β |∂yν ∂θµ a(y, θ)| ≤ Cµ,ν 1 + ρ(θ) 1 · 1 + ρ(θ) 2 . 6
We can now use Theorem 3.2 and its proof and obtain estimates k−1 X j ∂zκ ∂ζλ ∂yν ∂θµ B(z, ζ, y, θ) − ihDy , Dθ i a(y, θ) (3.8) j=0 β −|µ|m −k β −|λ|m ≤ Ckκλµν 1 + ρ(θ) 1 1 + ρ(ζ) 2 . If we define according to (3.5): b(z, ζ) = B(z, ζ, z, ζ), β1 +β2 we obtain therefore b ∈ Sm . From the proof of Theorem 3.2 follows also that the constants in (3.8) depend only on a finite number of seminorms of a1 and a2 , so that the mapping (a1 , a2 ) 7→ b is continuous β1 β2 β1 +β2 from Sm × Sm to Sm . In order to show (3.6), we approximate aj by aεj , where we define with a cut-off function χ ∈ C0∞ (Rn+1 ), χ = 1 in neighborhood of 0:
aεj (z, ζ) = χ(zε )χ(ζε )aj (z, ζ). β +1
We know from Theorem 2.1 that χ(ζε )aj (z, ζ) → aj (z, ζ) in Smj as ε → 0. If u ∈ S(Rn+1 ) we have therefore that aεj (z, D)u → aj (z, D)u in S(Rn+1 ), hence aε1 (z, D)aε2 (z, D)u → a1 (z, D)a2 (z, D)u in S(Rn+1 ). β1 +β2 Let bε be defined from aεj as b from aj in (3.5). Then (bε ) is bounded in Sm and converges 0 2(n+1) to b in S (R ). Therefore the proof will be finished if we show that (3.6) holds for bε :
bε (z, D) = aε1 (z, D)aε2 (z, D). To show this, we drop the index ε and assume just that aj ∈ S(R2(n+1) ) (j = 1, 2). Let b0 (z, D) = a1 (z, D)a2 (z, D). Then for u ∈ S(Rn+1 ) Z 0 −(n+1) (3.9) b (z, D)u(z) = (2π) eihz , ζi a1 (z, ζ) a2 (z, D)u b(ζ) dζ. From the definition −(n+1)
a2 (y, D)u(y) = (2π)
Z
eihy , θi a2 (y, θ)ˆ u(θ) dθ
we obtain its Fourier transform Z b a2 (z, D)u (ζ) = e−ihζ , yi a2 (y, D)u(y) dy Z −(n+1) = (2π) e−ihζ , yi+ihy , θi a2 (y, θ)ˆ u(θ) dθ dy, hence from (3.4) 0
−2(n+1)
Z
Z
a1 (z, ζ) e−ihζ , yi+ihy , θi a2 (y, θ)ˆ u(θ) dθ dy Z Z −2(n+1) ihz , θi = (2π) e eihz−y , ζ−θi a1 (z, ζ)a2 (y, θ) dydζ uˆ(θ) dθ.
b (z, D)u(z) = (2π)
ihz , ζi
e
This shows that the symbol of b0 is (we exchange ζ and θ) Z 0 −(n+1) (3.10) b (z, ζ) = (2π) e−ihz−y , ζ−θi a1 (z, θ)a2 (y, ζ) dy dθ . 7
Writing (3.10) as b0 (z, ζ) = B 0 (z, ζ, z, ζ) with Z 0 0 0 0 0 −(n+1) B (z, ζ, z , ζ ) = (2π) e−ihz −y , ζ −θi a1 (z, θ)a2 (y, ζ) dy dθ , we have with θ − ζ 0 = zˆ0 : 0
0
0
−(n+1)
Z
−ihz 0 , ζ 0 −θi
B (z, ζ, z , ζ ) = (2π)
−(n+1)
e Z
ihz 0 , zˆ0 i
= (2π)
Z
e
Z
0 eihy , ζ −θi a1 (z, θ)a2 (y, ζ) dy dθ
0 0 e−ihy , zˆ i a1 (z, ζ 0 + zˆ0 )a2 (y, ζ) dy dˆ z.
Taking Fourier transforms with respect to z 0 , this means Z Z 0 −ihˆ z0 , z0 i 0 0 0 0 e B (z, ζ, z , ζ ) dz = e−ihy , zˆ i a1 (z, ζ 0 + zˆ0 )a2 (y, ζ) dy . Hence Z
0
e−ihˆz
, z 0 i−ihζˆ0 , ζ 0 i
Z
0
B 0 (z, ζ, z 0 , ζ 0 ) dz 0 dζ 0 = ˆ0
0
e−ihy , zˆ i−ihζ , ζ i a1 (z, ζ 0 + zˆ0 )a2 (y, ζ) dy dζ 0 Z 0 ˆ0 ihˆ z 0 , ζˆ0 i =e e−ihˆz , yi−ihζ , θi a1 (z, θ)a2 (y, ζ) dy dθ.
=
0
This means that the Fourier transform of B 0 (z, ζ, z 0 , ζ 0 ) with respect to (z 0 , ζ 0 ) is eihˆz the Fourier transform of a1 (z, θ)a2 (y, ζ) with respect to (y, θ). In other words
, ζˆ0 i
times
B 0 (z, ζ, y, θ) = eihDy , Dθ i a1 (z, θ)a2 (y, ζ) = B(z, ζ, y, θ). Hence b0 (z, ζ) = B 0 (z, ζ, z, ζ) = B(z, ζ, z, ζ) = b(z, ζ) where b(z, ζ) is defined by (3.5). As an application of Lemma 2.3, we obtain the following result concerning the existence of a parametrix (pseudoinverse) for elliptic symbols (operators). We state the result in the form of a theorem: β Theorem 3.5. Suppose that there is given a symbol a ∈ Sm such that with some positive numbers ca and c00 there holds
(3.11)
|a(z, ζ)| ≥ ca ρ(ζ)β if ρ(ζ) ≥ c00 .
−β Then there exists b ∈ Sm such that
(3.12)
−∞ a(z, ζ)b(z, ζ) − 1 ∈ Sm ,
(3.13)
−1 , a(z, D)b(z, D) − I ∈ Op Sm
(3.14)
−1 b(z, D)a(z, D) − I ∈ Op Sm .
Proof. Observe that the condition (3.11) is equivalent to the condition |a(z, ζ)| ≥ ca ρ(ζ)β if |ζ| ≥ c0 . with some c0 depending on c00 . Assume first that β = 0. Consider the complex function ω1 for ω = ω1 + iω2 , ω 6= 0 and choose an infinitely differentiable function H(ω) = H(ω1 , ω2 ) such that H(ω1 , ω2 ) = ω1 for |ω| ≥ ca . Write H = F + iG where F (ω1 , ω2 ) and G(ω1 , ω2 ) are 8
0 real valued functions, F, G ∈ C ∞ (R2 ). Since Re a, Im a ∈ Sm we have, by Lemma 2.3, the 0 symbols F (Re a, Im a), G(Re a, Im a) ∈ Sm . But H(a) := F (Re a, Im a) + iG(Re a, Im a) ∈ 0 Sm . If |ζ| ≥ c0 then F (ω1 , ω2 ) = Re ω1 and G(ω1 , ω2 ) = Im ω1 . Since β = 0 we have by (3.11) 1 |a(z, ζ)| ≥ ca for all |ζ| ≥ c0 , which yields H(a)(z, ζ) = H(a(z, ζ)) = a(z,ζ) for all |ζ| ≥ c0 . Thus we have proved with b = H(a)
a(z, ζ)b(z, ζ) − 1 = 0 if |ζ| ≥ c0
(3.15)
0 which implies (3.12) for β = 0. Assume now the general case β ∈ R. Define a0 ∈ Sm by −β/2m a0 (z, ζ) = a(z, ζ) 1 + |ξ|2m + |η|2 . 0 Let b0 ∈ Sm be a corresponding symbol such that (3.12) (or even (3.15)) holds for the pair 0 0 −β a , b . Then defining b ∈ Sm by −β/2m b(z, ζ) = b0 (z, ζ) 1 + |ξ|2m + |η|2 ,
we see that (3.12) is satisfied for any given β ∈ R. Properties (3.13), (3.14) follow by Theorem 3.4. Remark 3.2. The results (3.13) and (3.14) can be improved. By the assumptions of Theo−β rem 3.5 one can show that it is in fact possible to choose a symbol b ∈ Sm such that (3.12) is valid and that (3.13), (3.14) are strengthened to (3.16)
−∞ a(z, D)b(z, D) − I ∈ Op Sm ;
(3.17)
−∞ b(z, D)a(z, D) − I ∈ Op Sm .
Remark 3.3. In fact one can show that the existence of a pseudo-inverse characterizes elliptic β −β symbols. More precisely, if a ∈ Sm is given such that there exists b ∈ Sm with −1 a(z, ζ)b(z, ζ) − 1 ∈ Sm
(3.18)
then a is elliptic, i.e. it satisfies the condition (3.11). 0 Next we will prove that for any symbol a ∈ Sm the operator a(z, D) defines a 2 n+1 continuous mapping in L (R ). We recall Schur’s Lemma R Lemma 3.6. Assume that (A u)(z) = K(z, zˆ)u(ˆ z ) dˆ z is an integral operator in Rn+1 such that K(z, zˆ) is continuous and Z Z (3.19) sup |K(z, zˆ)| dˆ z ≤ CA , sup |K(z, zˆ)| dz ≤ CA0 . z
zˆ
p Then kAuk ≤ CA CA0 kuk, u ∈ L2 (Rn+1 ). To apply Lemma 3.6, we first give a sufficient condition on β which guarantees that β an operator a(z, D) ∈ Op Sm has a continuous kernel. β Lemma 3.7. Assume that a(z, ζ) ∈ Sm and β < −n − m. Then the kernel of a(z, D) is continuous and bounded.
Proof. The kernel K(z, zˆ) of a(z, D) is given by (see (3.2)) Z −(n+1) K(z, zˆ) = (2π) eihz−ˆz , ζi a(z, ζ) dζ, Rn+1
9
and we obtain −(n+1)
|K(z, zˆ)| ≤ (2π) Z ≤C
Z |a(z, ζ)| dζ Rn+1 1
(1 + |ξ| + |η| m )β dξ dη ,
Rn ξ ×Rη
It is elementary to check that the last integral is finite if β < −n − m. Hence K(z, zˆ) is bounded, and continuous since the integrand is continuous. In order to show that for certain operators the conditions of Schur’s Lemma are satisfied, we investigate the behavior of the kernel outside of the diagonal. More precisely, we estimate the values of kernel with respect to the distance |z − zˆ|. The following result shows that for operators of any order α ∈ R, the kernel is continuous outside the diagonal and decays with any polynomial rate of |z − zˆ| for large distances. α Lemma 3.8. Assume A = a(z, D) ∈ Op Sm (Rn+1 ) , α ∈ R. Then the kernel K(z, zˆ) is continuous for z 6= zˆ and we have for any q > α + n + m
|K(z, zˆ)| ≤ Cp |z − zˆ|−q
(3.20)
(z 6= zˆ).
Proof. Consider the commutator [A, zj ] = Azj − zj A, where z = (x, t) ,
zj = xj ,
1 ≤ j ≤ n,
zn+1 = t.
For q ∈ N, the kernel of [A, zj ]q is related to the kernel of A by the formula K(z, zˆ)(zj − zˆj )q = K[A,zj ]q (z, zˆ).
(3.21)
On the other hand, the commutator has the symbol representation α−q [A, zj ]q = (−i)q (∂ζqj a(z, D)) ∈ Sm (Rn+1 ).
Choose now q large enough such that the kernel of [A, zj ]q is continuous and bounded, i.e., q > α + n + m, by Lemma 3.7. By this condition it follows from (3.21) |K(z, zˆ)| ≤ Cq |zj − zˆj |−q ,
(j = 1, . . . , n ; z 6= zˆ).
From these estimates in coordinate directions one can deduce |K(z, zˆ)| ≤ Cq0 |z − zˆ|−q
(z 6= zˆ)
which proves the lemma. We can now combine Lemma 3.7 and Lemma 3.8 to get β Lemma 3.9. Assume that a(z, ζ) ∈ Sm and β < −n − m. Then A(z, D) defines a bounded 2 n+1 operator in L (R ).
Proof. Take q ∈ N such that q > β + n + m. Now we obtain Z Z Z |K(z, zˆ)| dˆ z≤ |K(z, zˆ)| dˆ z+ Rn+1
|ˆ z −z|≥1
|ˆ z −z|≤1
where by the boundedness of the kernel Z |K(z, zˆ)| dˆ z≤C |ˆ z −z|≤1
10
|K(z, zˆ)| dˆ z
and requiring also q ≤ m + 2 Z Z |K(z, zˆ)| dˆ z ≤ Cq |ˆ z −z|≥1
|z − zˆ|−q dz ≤ Cq0 .
|ˆ z −z|≥1
Hence the first condition in (3.19) is satisfied. The second condition in (3.19) follows similarly and Schur’s Lemma concludes the proof. β In the next lemma we show that any symbol a ∈ Sm of a negative order defines a 2 n+1 bounded operator in L (R ). β Lemma 3.10. Assume a(z, ζ) ∈ Sm and β < 0. Then a(z, D) is a bounded operator in 2 n+1 L (R ). β0 Proof. By Lemma 3.9 we know that b(z, D) is bounded in L2 (Rn+1 ) if b(z, ζ) ∈ Sm with β0 /2 n+1 β0 := −n − m − 1. Consider an operator a(z, D) such that a(z, ζ) ∈ Sm (R ). For functions u ∈ S(Rn+1 ) we get
ka(z, D) uk2 = (a(z, D) u , a(z, D) u) = (b(z, D) u , u) β0 with b(z, D) := a(z, D)∗ a(z, D) ∈ Op Sm . By the L2 continuity of b(z, D)
ka(z, D) uk2 ≤ kb(z, D) uk kuk ≤ C kuk2 β /2
which shows that a(z, D) : L2 (Rn+1 ) → L2 (Rn+1 ) is bounded for any a ∈ Sm0 . Now we can β /2 β0 repeat the argument starting from the class Sm0 (Rn+1 ) instead of Sm , and find that any β0 /4 2 n+1 symbol a ∈ S defines a continuous operator in L (R ).Obviously we can repeat the β /2k process infinitely and find that a(z, D) ∈ Op Sm0 (Rn+1 ) defines a L2 (Rn+1 )-operator for all k ∈ N. Taking k large enough such that β0 /2k ≥ β we get the assertion of the lemma. 0 Lemma 3.11. Let a(z, ζ) ∈ Sm . Then a(z, D) defines a continuous mapping in L2 (Rn+1 ). −1 Moreover, putting M (a) = supz,ζ |a(z, ζ)|, we find for any M > M (a) an R(z, ζ) ∈ Sm such that
(3.22)
ka(z, D) uk ≤ M kuk + kR(z, D) uk ,
u ∈ L2 (Rn+1 ).
Proof. Writing M = M (a) + δ we have M 2 − |a(z, ζ)|2 = (M (a) + δ)2 − |a(z, ζ)|2 ≥ δ 2 > 0. Hence there exists a C ∞ -smooth function F : R → R such that F (t) = by Lemma 2.3 there holds p 0 . C(z, ζ) := M 2 − |a(z, ζ)|2 ∈ Sm
√
By Theorems 3.2 and 3.4 we also have C(z, D)∗ C(z, D) = M 2 − a(z, D)∗ a(z, D) + r(z, D) −1 where r(z, D) ∈ Op Sm . This yields
kC(z, D) uk2 + ka(z, D) uk2 = M 2 kuk2 + (r(z, D) u, u). In particular ka(z, D) uk2 ≤ M 2 kuk2 + kr(z, D) uk kuk 1 ≤ (M kuk + k 2M r(z, D) uk)2 .
11
t for t ≥ δ 2 , and
According to Lemma 3.10, r(z, D) is bounded in L2 (Rn+1 ). Hence a(z, D) is bounded, too. 1 Setting R(z, ζ) = 2M r(z, ζ), we obtain the assertion of the lemma. From Lemma 3.11 we can derive quite easily a very useful result which shows that β any pseudodifferential operator a(z, D) ∈ Sm defines a bounded mapping in a whole scale of s n+1 the anisotropic Sobolev spaces Hm (R ). These spaces are defined by the norms Z 12 −(n+1)/2 2 m 2 s/m 2 (3.23) kuks,m = (2π) (1 + |ξ| ) + |η| |ˆ u(ζ)| dζ ; Rn+1
more precisely, consists of those distributions u ∈ S 0 (Rn+1 ) for which kuks = kuks,m is finite. To simplify our notations, we introduce the symbols β/2m β (3.24) Λβm (ζ) := (1 + |ξ|2 )m + |η|2 ∈ Sm , β ∈ R. s Hm (Rn+1 )
Obviously the corresponding pseudodifferential operator (Bessel potential operator) Λβm (D) defines an isometric isomorphism (3.25)
s s−β Λβm (D) : Hm (Rn+1 ) → Hm (Rn+1 ) ,
s ∈ R.
Now the continuity part of the mapping in (3.25) remains valid for any operator a(z, D) ∈ β Op Sm . Moreover, as in Lemma 3.11 we can control the highest derivatives by a quantity β coming from the symbol a(z, ζ). For a(z, ζ) ∈ Sm we define M (a) = sup|Λ−β m (ζ) a(z, ζ)|.
(3.26)
z,ζ
0 n+1 Observe that M (a) is finite since Λ−β ). m (ζ) a(z, ζ) ∈ Sm (R β Theorem 3.12. Assume a(z, ζ) ∈ Sm , β ∈ R. Then a(z, D) defines a bounded operator s s−β a(z, D) : Hm (Rn+1 ) → Hm (Rn+1 )
(3.27)
s for any s ∈ R. Moreover, we have with any fixed M > M (a) for all s ∈ R and u ∈ Hm (Rn+1 ),
ka(z, D) uks−β ≤ M kuks + CM kuks−1 .
(3.28)
Proof. Consider the operator −s 0 n+1 Λs−β ). m (D) a(z, D) Λm (D) ∈ Op Sm (R
This operator has a principal symbol Λ−β m (ζ)a(z, ζ) and hence we find by Lemma 3.11 a −1 symbol R ∈ Sm such that (3.29)
−s kΛs−β m (D) a(z, D) Λm (D) vk0 ≤ M kvk0 + kR(z, D) vk0
0 s n+1 for any v ∈ Hm (Rn+1 ) = L2 (Rn+1 ). Putting u = Λ−s ) and using the m (D)v ∈ Hm (R isometric mapping property (3.25), we see in a first step that
ka(z, D) uks−β ≤ Ckuks , −1 hence (3.27). Applying this result to R(z, D) ∈ Op Sm , we get
kR(z, D) vk0 ≤ CM kvk−1 = CM kuks−1 . Thus (3.29) implies (3.28). As a special application of Theorem 3.12, we give a result which is useful when applying localization techniques for some approximation methods for boundary integral equations. 12
β Corollary 3.13. Assume a(z, ζ) ∈ Sm . Then there exists a constant C0 such that for any ∞ n+1 r > 0 and any function θr ∈ C (R ) satisfying θr (z) = 0 for |z| ≥ r and 0 ≤ θr (z) ≤ 1, there holds
(3.30)
kθr (z − z0 )(a(z, D) − a(z0 , D)) uks−β ≤ C0 r kuks + Ckuks−1
s for all u ∈ Hm (Rn+1 ). The constant C0 depends on a(z, ζ) and is independent of r, z0 and s. β Proof. Since a ∈ Sm , we have
|a(z, ζ) − a(z0 , ζ)| ≤ |(∇z a)(ˆ z , ζ) · (z − z0 )| ≤ c00 (1 + ρm (ζ))β |z − z0 |. Write az0 for the symbol a(z0 , ζ) , ζ ∈ Rn+1 and θr,z0 for θr (z − z0 ) , z ∈ Rn+1 . There holds β , θr,z0 · (a − az0 ) ∈ Sm
and for the quantity M [·] of this symbol we get (ζ) θ (z − z )|a(z, ζ) − a(z , ζ)| M [θr,z0 (a − az0 )] = sup Λ−β r 0 0 m z,ζ −β (3.31) ≤ sup Λm (ζ) C00 (1 + ρm (ζ))β |z − z0 | ≤ C000 r . |z−z0 |≤r ,ζ∈Rn+1
Now the rest follows from (3.31) and Theorem 3.12. Finally we consider strongly elliptic operators and show the validity of G˚ arding’s β inequality also in the anisotropic case. An operator a(z, D) ∈ Op Sm (Rn+1 ) is strongly elliptic if there exists positive numbers C0 and r such that (3.32)
Re a(z, ζ) ≥ C0 Λβm (ζ)
1
if ρm (ζ) = |ξ| + |η| m ≥ r .
β Theorem 3.14 (G˚ arding’s inequality). Assume a(z, D) ∈ Op Sm (Rn+1 ) where the symbol a(z, ζ) satisfies (3.32). Then for any fixed 0 < C < C0 there holds with some C1 > 0,
(3.33)
Re (a(z, D) u, u) ≥ Ckuk2β/2 − C1 kuk2(β−1)/2 ,
β/2 u ∈ Hm (Rn+1 ).
0 Proof. Considering first the case β = 0, we have a(z, ζ) ∈ Sm (Rn+1 ) such that
(3.34)
Re a(z, ζ) ≥ C0
if ρm (ζ) ≥ r.
n+1 Since the set {ζ : ρm (ζ) ≤ r} is compact and Λ−1 we have m (ζ) a positive function on Rζ
min Λ−1 m (ζ) =: λ0 > 0.
|ζ|≤r
Now, instead of a(z, ζ) we consider the modified symbol 0 n+1 ) a ˜(z, ζ) = κ Λ−1 m (ζ) + a(z, ζ) ∈ Sm (R
where we choose κ > 0 large enough such that for a ˜(z, ζ) the condition (3.34) holds for all ζ ∈ Rn+1 ; (3.35)
Re a ˜(z, ζ) ≥ C0
for all z, ζ ∈ Rn+1 .
Here it is enough to take κ ≥ λ−1 0 (supz,ζ |a(z, ζ)| + C0 ). Notice that the constant C0 in (3.35) is same as in (3.34). Choosing 0 < C < C 0 < C0 there holds Re a ˜(z, ζ) − C 0 ≥ C0 − C 0 > 0 , 13
z, ζ ∈ Rn+1 .
and we can, as in Lemma 3.11, take a square root p 0 ˜(z, ζ) − C 0 ∈ Sm (Rn+1 ). b(z, ζ) = Re a For the corresponding operator this implies (3.36)
b(z, D)∗ b(z, D) + CI = (Re a ˜)(z, D) + R1 = (Re a)(z, D) + R2
−1 −1 where R1 , R2 ∈ Op Sm (Rn+1 ). Since (Re a)(z, D) = Re a(z, D) + R3 with R3 ∈ Sm (Rn+1 ) where
Re a(z, D) := 12 (a(z, D) + a(z, D)∗ )
(3.37)
we deduce from (3.36),(3.37) for suitable C10 and C1 , (3.38)
Re (a(z, D) u, u) = (Re a(z, D) u, u) = kb(z, D) uk2 + Ckuk2 + (R4 u, u) ≥ C 0 kuk20 − C10 kuk−1 kuk0 ≥ Ckuk20 − C1 kuk2−1 .
By (3.38) the theorem is proved for β = 0. β 0 Assume now the general case a ∈ Sm , β ∈ R. The symbol a0 ∈ Sm (Rn+1 ), defined −β/2 −β/2 by a0 (z, ζ) = Λm (ζ) a(z, ζ)Λm (ζ), satisfies Re a0 (z, ζ) ≥ C0
(3.39)
0 with the constant C0 of (3.32). From the first part of the proof we get for all v ∈ Hm (Rn+1 )
Re (Λ−β/2 (D) a(z, D) Λ−β/2 (D) v, v) ≥ C kvk20 − C1 kvk2−1 , m m or Re (a(z, D)Λ−β/2 (D) v, Λ−β/2 (D)v) ≥ Ckvk20 − C1 kvk2−1 . m m −β/2
Putting u = Λm
(D)v we obtain the assertion of the theorem.
4. Pseudodifferential operators on the cylinder Tn × R We consider functions u(x, t) , z = (x, t) ∈ Rn+1 which are 1-periodic in x1 , . . . , xn : u(x + k, t) = u(x, t)
∀k ∈ Zn ; (x, t) ∈ Rn+1 .
If u is continuous, it can be considered as a continuous function on the cylinder Tn × R, where Tn = (R/Z)n is the n-torus. As representative for Tn we can take Qn = [0, 1]n , and we can consider u also as a continuous function on Qn × R with periodic boundary conditions on ∂Qn × R. If u is polynomially bounded, it has two Fourier transforms 1. uˆ(ζ) , ζ ∈ Rn+1 , the usual Fourier transform as defined before for u ∈ S 0 (Rn+1 ). If u ∈ S(Rn+1 ) (which is not the case for our periodic function) we have the definition for ζ = (ξ, η) ∈ Rn+1 : Z Z uˆ(ξ, η) = e−i(hξ , xi+tη) u(x, t) dt dx. Rn x
Rt
2. uˆ(ζ) for ζ = (k, η) ∈ Zn × R , defined as Fourier coefficients in the space variables: Z Z uˆ(k, η) = e−i(2πhk , xi+tη) u(x, t) dt dx. Qn x
Rt
14
We write this also as Z uˆ(ζ) =
e−ihζ , zi u(z) dz.
Tn ×R
The inverse transform is given by Z Z 1 1 X i(2πhk , xi+tη) eihζ , zi uˆ(ζ) dζ. u(z) = e uˆ(k, η) dη =: 2π k∈Zn Rη 2π Znk ×Rη Note that we use here side by side the two scalar products hζ , zi = hξ , xi + tη and hζ , zi = 2πhk , xi + tη for ζ = (ξ, η) ∈ Rn+1 , ζ = (k, η) ∈ Tn × R, and z = (x, t) ∈ Rn+1 . The dual group of Tn is (2πZ)n which we parametrize by k ∈ Zn , for convenience. We can now define symbols on Tn × R. We choose those symbols a(z, ζ) on Rn+1 that are 1-periodic in z1 , . . . , zn and define: β β Sm,per = a(z, ζ) ∈ Sm | a(x + k, t; ξ, η) = a(x, t; ξ, η) ∀k ∈ Zn . β β So these are C ∞ functions on (Tn × R) × Rn+1 . Since Sm,per ⊂ Sm , we can associate with β a ∈ Sm,per two pseudodifferential operators. We need the scaled symbol
(4.1)
a2π (x, t; ξ, η) = a(x, t; 2πξ, η). We define
1. a(z, D) as previously defined, acting on functions on Rn+1 , thus for z ∈ Rn+1 : Z −(n+1) a(z, D) u(z) = (2π) eihz , ζi a(z, ζ) uˆ(ζ) dζ. Rn+1
2. The pseudodifferential operator a(z, D) on Tn × R defined by Z 1 a(z, D) u(z) = eihζ , zi a2π (z, ζ) uˆ(ζ) dζ. 2π Zn ×R Recall that hζ , zi = 2πhk , xi + ηt for ζ = (k, η) ∈ Zn × R, z = (x, t) ∈ Tn × R. It is clear that a(z, D) u is 1-periodic in z1 , . . . , zn . It is important to note that the definition of a(z, D) uses only the values of a2π (z, ζ) for ζ = (k, η) ∈ Zn × R. Thus the operator a(z, D) determines the symbol a(z, ζ) only on (Tn × R)z × ((2πZ)n × R)ζ . This is in contrast to a(z, D) which determines a(z, ζ) on all of R2(n+1) . We can obtain quite simple proofs of the basic results about these pseudodifferential operators periodic in the space variables, by showing that the mapping a(z, D) 7→ a(z, D) is in fact a ∗-homomorphism between the two algebras of pseudodifferential operators on Rn+1 and Tn × R that is compatible with the symbolic calculus. β Theorem 4.1. Let a, b ∈ Sm,per such that
a(z, D)∗ = b(z, D) ,
(adjoint w.r.t. L2 duality on Rn+1 ).
Then a(z, D)∗ = b(z, D) ,
(adjoint w.r.t. L2 duality on Tn × R). 15
Proof. It is sufficient to consider a ∈ S((Tn ×R)×Rn+1 ) = C ∞ (Tn ; S(Rn+2 )). Then a(z, D) u has an integral representation with a kernel K(z, z 0 ) (z, z 0 ∈ Tn × R), and there holds : Z a(z, D) u(z) = K(z, z 0 ) u(z 0 ) dz 0 Tn ×R Z 1 = eihζ , zi a2π (z, ζ) uˆ(ζ) dζ 2π Zn ×R Z Z 1 0 = eihζ , zi a2π (z, ζ)e−ihζ , z i u(z 0 ) dz 0 dζ 2π (Zn ×R)ζ (Tn ×R)z0 ! Z Z 1 0 = eihz−z , ζi a2π (z, ζ) dζ u(z 0 ) dz 0 . 2π n n (Z ×R)ζ (T ×R)z0 By comparison, we obtain for z = (x, t) ∈ Tn × R (identified with Qn × R ), Z 1 0 0 K(z, z ) = eihz−z , ζi a2π (z, ζ) dζ 2π Zn ×R Z 1 X 0 0 = ei(2πhk , x−x i+η(t−t )) a(x, t; 2πk, η) dη. 2π Zn Rη k
This is similar to equation (3.2) in the non-periodic case. Now we introduce the special ˆ (ˆ z , ζ) z ∈ Zn × R ; ζˆ ∈ Tn × R) which corresponds to a finite Fourier Fourier transform ac 2π (ˆ transform on Tn for z1 , . . . , zn and to a discrete Fourier transform with respect to ζ1 , . . . , ζn : Z Z ˆ ˆ (4.2) ac z , ζ) := e−i(hˆz , zi+hζ , ζi) a2π (z, ζ) dζ dz. 2π (ˆ (Tn ×R)z
(Zn ×R)ζ
This can be written in terms of the kernel K as follows : Z Z ˆ ˆ e−ihˆz , zi eihz−(z+ζ) , ζi a2π (z, ζ) dζ dz ac z , ζ) = 2π (ˆ (Tn ×R)z
(4.3)
Z
(Zn ×R)ζ
ˆ dz . e−ihˆz , zi · 2πK(z, z + ζ)
= (Tn ×R)
z
Now we use the fact that the adjoint operator a2π (z, D)∗ has kernel K ∗ (z, z 0 ) = K(z 0 , z) and obtain for its symbol, which we call c2π , Z ˆ = ˆ z) dz. cc z , ζ) e−ihˆz , zi · 2πK(z + ζ, 2π (ˆ (Tn ×R)z
Z
ˆ ˆ dy e−ihˆz , y−ζi · 2πK(y, y − ζ)
= (Tn ×R)y
(4.4)
ˆ ihˆ z , ζi
Z
=e
ˆ dy eihˆz , yi · 2πK(y, y − ζ)
(Tn ×R)y ˆ ˆ = eihˆz , ζi ac z , −ζ) 2π (−ˆ ˆ ˆ , = eihˆz , ζi ac z , ζ) 2π (ˆ
(ˆ z ∈ Zn × R ; ζˆ ∈ Tn × R) . 16
Here a is the conjugate symbol a(z, ζ) = a(z, ζ). Now we compare (4.4) with the corresponding formula (3.1) from Theorem 3.2 : ˆ ˆ , ˆb(ˆ ˆ = eihˆz , ζi ˆ(ˆ a z , ζ) z , ζ)
(4.5)
(ˆ z , ζˆ ∈ Rn+1 ).
In order to complete the proof of the theorem, we show that (4.4) and (4.5) imply that ˆ = bc ˆ ∀ zˆ ∈ Zn × R , ζˆ ∈ Tn × R : cc z , ζ) z , ζ). 2π (ˆ 2π (ˆ
(4.6)
This will then give a(z, D)∗ = b(z, D). For the proof of (4.6), we need the following relations between the Fourier transform and Fourier series of a periodic function and between the discrete and continuous Fourier transform, both given by versions of the Poisson summation formula : For a function u ∈ S(Tn × R), we have for (ξ, η) ∈ Qn × R : uˆ(ξ, η) = (2π)n
(4.7)
X
δ(ξ − 2πk) uˆ(k, η).
k∈Zn
For a function v ∈ S(Rn+1 ), define its discrete Fourier transform v˜ for z = (x, t) as Z
−ihz , ζi
v˜(z) =
e
v(ζ) dζ =
Zn ×R
XZ Zn k
e−i(2πhx , ki+tη) vˆ(k, η) dη,
Rη
and the continuous Fourier transform as Z
e−ihz , ζi v(ζ) dζ.
vˆ(z) = Rn+1
Then we have (4.8)
v˜(x, t) =
X
vˆ(2π(x + p), t).
p∈Zn
For the passage from a to ac 2π , we have both types of Fourier transforms, and we obtain therefore for zˆ = (ξ, η): (2π)n (4.9)
X
ˆ ˆ) = δ(ξ − 2πk)ac 2π (k, η; ξ, η
X
ac z ; 2π(ξˆ + p), ηˆ) 2π (ˆ
p∈Zn
k∈Zn
= (2π)−n
X p∈Zn
17
a ˆ(ˆ z ; ξˆ + p, ηˆ).
Using this for cc 2π in (4.4) with (4.5), we obtain (4.10) (2π)n
X
ˆ ˆ) = (2π)n δ(ξ − 2πk)cc 2π (k, η; ξ, η
X
ˆ ˆ ˆ) δ(ξ − 2πk)ei(2πhk , ξi+η ηˆ) · ac 2π (k, η; ξ, η
k∈Zn
k∈Zn
ˆ
= (2π)n ei(hξ , ξi+η ηˆ)
X
ˆ ˆ) δ(ξ − 2πk)ac 2π (k, η; ξ, η
k∈Zn X ˆ −n i(hξ , ξi+η ηˆ)
= (2π)
e
ˆ(ˆ a z ; ξˆ + p, ηˆ)
p∈Zn ˆ
X
= (2π)−n ei(hξ , ξi+η ηˆ)
ˆ e−i(hξ , ξ+pi+ηηˆ)ˆb(ˆ z ; ξˆ + p, ηˆ)
p∈Zn
= (2π)−n
X
e−ihξ , piˆb(ˆ z ; ξˆ + p, ηˆ)
p∈Zn
= (2π)−n
X
ˆb(ˆ z ; ξˆ + p, ηˆ)
p∈Zn
= (2π)n
X
ˆ ˆ). δ(ξ − 2πk)bc 2π (k, η; ξ, η
k∈Zn
Here the second-to-last equality holds because the support of ˆb(ξ, η; ξˆ + p, ηˆ) is contained in (2πZ)nξ × Rη × Tn × R, and there e−ihξ , pi = 1 for p ∈ Zn . Now (4.10) implies (4.6), and the theorem is proved. β β−1 Corollary 4.2. For a ∈ Sm,per there is r ∈ Sm,per such that
a(z, D)∗ = a(z, D) + r(z, D). β−1 Proof. By Theorem 3.2 there is r ∈ Sm,per such that
a(z, D)∗ = a(z, D) + r(z, D) . Hence by Theorem 4.1 we have with b(z, ζ) = a(z, D) + r(z, D) : a(z, D)∗ = b(z, ζ) = a(z, D) + r(z, D) . β
j β1 +β2 Theorem 4.3. Let aj ∈ Sm,per , (j = 1, 2) and b ∈ Sm,per .
If b(z, D) = a1 (z, D) · a2 (z, D) , then b(z, D) = a1 (z, D) · a2 (z, D) . Proof. According to Theorem 3.4, one has (4.11)
b(z, ζ) = eihDy , Dθ i a1 (z, θ) a2 (y, ζ)|y=z,θ=ζ .
As in the previous proof, we will show that for c(z, ζ), defined by c(z, D) = a1 (z, D) · a2 (z, D), an analogous formula holds; and we will deduce from this that c2π (z, ζ) = b2π (z, ζ) for all z ∈ Tn × R, ζ ∈ Zn × R, which implies the result of the theorem. The computations are analogous to those in the proof of Theorem 3.4: We can always assume 18
that aj ∈ S(Tn × R × Rn+1 ) , (j = 1, 2) and u ∈ S(Tn × R). With w(z) = a2 (z, D)u(z), we have Z 1 eihz , ζi a1,2π (z, ζ)b w(ζ) dζ c(z, D) = 2π (Zn ×R)ζ
1 = (2π)2
Z
ihz , ζi
e
Z
−ihζ , yi
e
a1,2π (z, ζ)
(Zn ×R)ζ
Z
eihθ , yi a2,2π (y, θ)ˆ u(θ) dθ dy dζ
(Zn ×R)θ
(Tn ×R)y
=
1 (2π)2
Z
eihθ , zi
Z
Z
ˆ (θ) dθ . eihz−y , ζ−θi a1,2π (z, ζ)a2,2π (y, θ) dζ dy u
(Tn ×R)y (Zn ×R)ζ
(Zn ×R)θ
This shows that the symbol of c(z, D) is for z ∈ Tn × R , ζ ∈ Zn × R determined by Z Z 1 eihz−y , θ−ζi a1,2π (z, θ)a2,2π (y, ζ) dθ dy. c2π (z, ζ) = 2π (Tn ×R)y (Zn ×R)θ Therefore we have c2π (z, ζ) = B(z, ζ, z, ζ), if we define Z Z 1 0 0 0 0 (4.12) B(z, ζ, z , ζ ) = eihz −y , ζ −θi a1,2π (z, θ)a2,2π (y, ζ) dθ dy. 2π (Tn ×R)y (Zn ×R)θ In the proof of the Theorem 3.4 we saw that b(z, ζ) = B(z, ζ, z; ζ) with ZZ 0 0 0 0 −(n+1) B(z, ζ, z , ζ ) = (2π) eihz −y , ζ −θi a1 (z, θ)a2 (y, ζ) dy dθ. R2(n+1)
Hence b2π (z, ζ) = B2π (z, ζ, z, ζ) with
(4.13)
0
−(n+1)
ZZ
0
0
B2π (z, ζ, z , ζ ) = (2π) eihz −y , ζ −θi a1 (z, θ)a2,2π (y, ζ) dθ dy R2(n+1) Z 1 0 0 eihz −y , ζ −θi a1,2π (z, θ)a2,2π (y, ζ) dθ dy. = 2π R2(n+1) 0
By using the Poisson summation formulas (4.7) and (4.8), we can see that for ζ 0 ∈ Zn × R, (4.12) and (4.13) define in fact the same function, so that for ζ ∈ Zn × R we find b2π (z, ζ) = c2π (z, ζ). β
j Corollary 4.4. Let aj ∈ Sm,per (j = 1, 2). Then
(i )
β1 +β2 −1 a1 (z, D)a2 (z, D) = (a1 · a2 )(z, D) + r(z, D) with r ∈ Sm,per
(ii )
β1 +β2 −1 [a1 (z, D), a2 (z, D)] ∈ Op Sm,per .
Theorems 4.1 and 4.3 provide the basic ingredients for the calculus of pseudodifferential operators on the cylinder Tn ×R. It is now very easy to obtain the complete calculus as in the non-periodic case in Section 3, including G˚ arding’s inequality and anisotropic Sobolev norm estimates. We define the anisotropic Sobolev spaces on the cylinder Tn × R in the same way n+1 as on R : 21 X Z s/m −1/2 (4.14) kuks = kuks,m = (2π) (1 + |2πk|2 )m + |η|2 |ˆ u(k, η)|2 dη ; k∈Zn
Rη
19
Then the Bessel potential operators s s−β Λβm (D) : Hm (T n × R) → Hm (Tn × R)
are isometric isomorphisms, with Λβm (ζ) defined in (3.25). It is clear that the proofs and results of Lemma 3.6 (“Schur’s lemma”), Lemma 3.7 β for β < −n − m”), and Lemma 3.8 (“singularity (“continuous kernel for a(z, D) if a ∈ Sm,per of the kernel on the diagonal”) carry over immediately to the periodic case. Therefore also 0 ”) are immediately available. the Lemmas 3.9–3.11 (“L2 continuity of a(z, D) for a ∈ Sm,per Moreover, we have the continuity in Sobolev spaces as in Theorem 3.12. Let the number M (a) be defined as in (3.26). β Theorem 4.5. Assume a(z, ζ) ∈ Sm,per , β ∈ R. Then a(z, D) defines a bounded operator
(4.15)
s s−β a(z, D) : Hm (Tn × R) → Hm (Tn × R)
for any s ∈ R. Moreover, we have with any fixed M > M (a) for all s ∈ R and u ∈ s Hm (Tn × R), (4.16)
ka(z, D) uks−β ≤ M kuks + CM kuks−1 . In the same manner, we also get the localization result of Corollary 3.13 :
β Corollary 4.6. Assume a(z, ζ) ∈ Sm,per . Then there exists a constant C0 such that for any r with 0 < r ≤ 1 and any function θr ∈ C ∞ (Tn × R) satisfying θr (z) = 0 for |z − z0 | ≥ r and 0 ≤ θr (z) ≤ 1, there holds
(4.17)
kθr (z)(a(z, D) − a(z0 , D)) uks−β ≤ C0 r kuks + Ckuks−1
s for all u ∈ Hm (Tn × R). The constant C0 depends on a(z, ζ) and is independent of r, z0 and s.
5. Parabolic symbols and operators Here we introduce a class of anisotropic symbols which define parabolic operators or operators of Volterra type. By the (rough) definition an operator A acting on functions in Rnx × Rt is of Volterra type if the following holds for each t: If u vanishes in the domain τ < t, then Au also has this property. A sufficient condition for the Volterra type can be given by requiring analyticity property of the symbol (see [17]). More precisely there holds β Theorem 5.1. Assume that a(z, ζ) ∈ Sm (Rn+1 × Rn+1 ) and that the mapping η 7→ a(z, ξ, η) z ζ has an analytic continuation into the domain η − iσ , σ > 0 such that this continuation is continuous for σ ≥ 0 and satisfies
(5.1)
1
|a(z, ξ, η − iσ)| ≤ C(1 + |ξ| + |η − iσ| m )β ,
σ ≥ 0.
Then the operator a(z, D) is of Volterra type. ) for the class of symbols satisfying the For shortness we write Vmβ (Rn+1 × Rn+1 z ζ assumption of Theorem 5.1. In connection with parabolic boundary value problems one needs, in addition to boundary conditions, an initial condition. The initial condition is needed also for boundary integral solution of those problems. Typically one considers the vanishing initial condition and this condition is also imposed here for our class pseudodifferential operators of Volterra 20
s s ˜m ˜m type. Introduce the anisotropic Sobolev space H (Rn+1 ) = H (Rnx × Rt ) which takes the vanishing initial condition at t = 0 into account, s s ˜m H (Rn+1 ) := u ∈ Hm (Rn+1 ) : supp u ⊂ Rnx × [0, ∞] .
For a finite time interval we put with Rn+1 := Rnx × (0, T ) , T > 0 T n o s s n+1 ˜m ˜ n H (Rn+1 ) := u = U | : U ∈ H (R ) . Rx ×(−∞,T ) m T s ˜m The norm in H (Rn+1 T ) is given by
kuks;T = inf kU ks : u = U |Rnx ×(−∞,T ) . Theorem 5.2. Assume a(z, ζ) ∈ Vmβ (Rn+1 × Rn+1 ). Then a(z, D) defines for all s ∈ R z ζ bounded operators s s−β ˜m ˜m (i ) a(z, D) : H (Rn+1 ) → H (Rn+1 ) , (ii ) (iii )
s ˜m ˜ s−β n+1 a(z, D) : H (Rn+1 T ) → Hm (RT ) ,
ka(z, D)kL(H˜ ms (Rn+1 ),H˜ ms−β (Rn+1 )) ≤ ka(z, D)kL(H˜ ms (Rn+1 ),H˜ ms−β (Rn+1 )) . T
T
Proof. The assertion (i) follows directly by the boundedness of the mapping s s−β a(z, D) : Hm (Rn+1 ) → Hm (Rn+1 ) s ˜m ˜ s n+1 ) such that and the Volterra property. For (ii) let u ∈ H (Rn+1 T ) and take U ∈ Hm (R u = U |Rnx ×(−∞,T ) . We can define
a(z, D)u := a(z, D)U |Rnx ×(−∞,T ) since by the Volterra property the term on the right is independent of U . s−β ˜m Moreover, a(z, D)u ∈ H (Rn+1 T ) and we obtain ka(z, D)uks−β;T = inf {kF ks−β : F |Rnx ×(−∞,T ) = a(z, D)u} ≤ inf {ka(z, D)U ks−β : U |Rnx ×(−∞,T ) = u} ≤ inf {ka(z, D)kL(H˜ ms (Rn+1 ),H˜ ms−β (Rn+1 )) kU ks : U |Rnx ×(−∞,T ) = u} = ka(z, D)kL(H˜ ms (Rn+1 ),H˜ ms−β (Rn+1 )) kuks;T . This implies (ii) as well as (iii) . Using the results of Section 4, we easily obtain the corresponding results for the anisotropic Sobolev spaces on the finite cylinder QT = Tn × (0, T ). As above, we define s s ˜m ˜m the spaces H (Tn × R) as the space of functions which vanish for negative t and H (QT ) as space of restrictions on QT . β Theorem 5.3. Assume a(z, ζ) ∈ Vmβ (Rn+1 × Rn+1 ) ∩ Sm,per . Then a(z, D) defines for all z ζ s ∈ R bounded operators s s−β ˜m ˜m (i ) a(z, D) : H (Tn × R) → H (Tn × R) ,
(ii ) (iii )
s s−β ˜m ˜m a(z, D) : H (QT ) → H (QT ) ,
ka(z, D)kL(H˜ ms (QT ),H˜ ms−β (QT )) ≤ ka(z, D)kL(H˜ ms (Tn ×R),H˜ ms−β (Tn ×R)) . 21
Remark 5.1. If we have a symbol which is not smooth in the ζ variable, but whose singularities are confined to a compact set (the origin in many examples, see below), then the preceding results remain valid modulo C ∞ functions, see Remark 3.1. These C ∞ functions will, in general, not be integrable at infinity, so the global norm estimates in our anisotropic Sobolev spaces will not remain true for such symbols. But the mapping property (ii) and the estimate (iii) of the preceding theorem remain valid. For the case of a finite cylinder QT , we now give a localization result of a slightly different flavor than Corollaries 3.13 and 4.6 which is needed in some applications (see [6]). We localize only in space direction and assume that the pseudodifferential operator does not depend on time. Because of the appearance of two different norms on the right hand side of the estimate and the definition of the norms on QT as infimums we need an extension operator (5.2)
s s ˜m ˜m E:H (QT ) → H (Tn × R) ;
Eu|QT = u .
Such an operator which is continuous for all s ∈ [−N, N ] can easily be constructed by the reflection method. β Corollary 5.4. Let a(z, ζ) ∈ Vmβ (Rn+1 × Rn+1 ) ∩ Sm,per . Assume that a(x, t; n, η) is indez ζ pendent of t ∈ R. Then there exists a constant C0 such that for any r with 0 < r ≤ 1 and any function θr ∈ C ∞ (Tn ) satisfying for some x ∈ Tn : θr (x) = 0 for |x − x0 | ≥ r and 0 ≤ θr (x) ≤ 1, there holds
(5.3)
kθr (x)(a(x, t; D) − a(x0 , t; D)) ukH˜ ms−β (QT ) ≤ C0 r kukH˜ ms (QT ) + CkukH˜ ms−1 (QT )
s ˜m for all u ∈ H (QT ). The constant C0 depends on a(z, ζ) and is independent of r and x0 .
Proof. Let U = Eu with the extension operator E from (5.2). Then we have that a(z; D)U is an extension of a(z; D)u, and therefore kθr (x)(a(x, t; D) − a(x0 , t; D)) ukH˜ ms−β (QT ) ≤ kθr (x)(a(x, t; D) − a(x0 , t; D)) U kH˜ ms−β (Tn ×R) ≤ C0 r kEukH˜ ms (Tn ×R) + CkEukH˜ ms−1 (Tn ×R) ≤ C1 C0 r kukH˜ ms (QT ) + CkukH˜ ms−1 (QT ) . s s−1 ˜m ˜m Here C1 is an estimate for both operator norms of E in H and in H . Replacing C1 C0 by C0 , we obtain (5.3).
Example 5.1. Let E(·, ·) be the fundamental solution of the heat equation in the plane and consider the single layer and the hypersingular heat operators VΓ and HΓ , Z tZ (5.4)
uΓ (y, τ )E(x − y, t − τ ) dsy dτ , Z tZ ∂ ∂ HΓ uΓ (x, t) = − uΓ (y, τ ) E(x − y, t − τ ) dsy dτ ∂nx 0 Γ ∂ny VΓ uΓ (x, t) =
0
(5.5)
Γ
22
on the smooth closed curve Γ. Having the parametric representation θ 7→ x(θ) of Γ we put u(θ, t) = uΓ (x(θ), t) and define the operator Z tZ 1 (5.6) V u(θ, t) = u(φ, τ )E(x(θ) − x(φ), t − τ ))|x0 (φ)| dφ dτ , 0 0 Z tZ 1 ∂ ∂ u(φ, τ ) (5.7) Hu(θ, t) = − E(x(θ) − x(φ), t − τ ))|x0 (φ)| dφ dτ . ∂nx(θ) 0 0 ∂nx(φ) The operators V and H are integral operators of Volterra type and they can be considered essentially (cf. Remark 5.1) as parabolic pseudodifferential operators. The principal symbols of V and H are − 21 21 1 h 2πξ i2 1 h 2πξ i2 + iη + iη , a (θ, ξ, η) = . (5.8) aV (θ, ξ, η) = H 2 |x0 (θ)| 2 |x0 (θ)| These formulas can be derived by using the method of [18]. Here we get them in a different way as follows (some details are omitted). Applying the Laplace transform Z ∞ (Lu)(s) = e−st u(t) dt 0
and using [1], 29.3.120 we obtain Z 1 √ 1 (Lu)(φ, s)K0 (|x(θ) − x(φ)| s) |x0 (φ)| dφ . (LV u)(θ, s) = 2π 0 Since u(θ, t) = 0 for t < 0 we obtain the Fourier transform by putting s = i η , Z 1 p 1 c (5.9) (V u)(θ, η) = (Lu)(φ, η)K0 (|x(θ) − x(φ)| iη) |x0 (φ)| dφ . 2π 0 Next we write the kernel as p K0 (|x(θ) − x(φ)| iη) = K0 (2z sin π(θ − φ)) where
p |x(θ) − x(φ)| z = κ iη , κ = κ(θ, φ) = . 2 sin π(θ − φ) Applying [16], 4.32 we further get X p p p K0 (|x(θ) − x(φ)| iη) = I|n| (κ(θ, φ) iη) K|n| (κ(θ, φ) iη)ein2π(θ−φ) n∈Z
which yields the representation Z Z 1 X 1 (5.10) V u(θ, t) = a(θ, φ, n, η)ˆ u(φ, η)ein2π(θ−φ)+iηt dφ dη 2π n∈Z 0 Rη with p p 1 I|n| (κ(θ, φ) iη) K|n| (κ(θ, φ) iη)|x0 (φ)| . 2π Comparing the representation (5.10) with the definition of a pseudodifferential operator on T1 × R (with z = (θ, t) ∈ T1 × R) : Z 1 X b(z, D) u(z) = ei(2πθn+tη) b2π (θ, t; n, η) uˆ(n, η) dη 2π n∈Z Rη
(5.11)
a(θ, φ, n, η) =
23
R1 with uˆ(n, η) = 0 e−2πinφ uˆ(φ, η) dφ , we obtain as in the proofs of Theorem 3.4 and Theorem 4.3 the relation b2π (θ, t; n, η) = e2πiDφ Dξ a(θ, φ, ξ, η) φ=θ,ξ=n . The corresponding asymptotic expansion shows that the principal symbol is obtained from (5.11) putting θ = φ and separating the leading term: p p 1 a(θ, n, η) := a(θ, θ, n, η) = I|n| (κ(θ) iη) K|n| (κ(θ) iη) |x0 (θ)| 2π 0
(θ)| with κ(θ) = |x2π . Applying the formulae 9.7.7, 9.7.8 and 9.3.9 in [1] we find the asymptotic behavior for n 6= 0 , p p 1 I|n| (κ(θ) iη) K|n| (κ(θ) iη) ∼ p 2 2 n + κ(θ)2 iη
which leads to the principal symbol aV given above in (5.8). The operators VΓ and HΓ are related to each other by the formula VΓ HΓ = 14 I − DΓ2 where the DΓ is the double layer heat operator, which has a negative order; see [4], [10] and [18]. Hence the formula for aH also follows. We can now represent the operators V and H as Z 1 X (5.12) aV (θ, n, η)ˆ u(n, η)ein2πθ+iηt dη + BV u(θ, t) , V u(θ, t) = 2π n∈Z Rη Z 1 X (5.13) Hu(θ, t) = aH (θ, n, η)ˆ u(n, η)ein2πθ+iηt dη + BH u(θ, t) , 2π n∈Z Rη where BV and BH are operators of Volterra type. For a finite time interval they define bounded operators ˜ 2s (QT ) → H ˜ 2s+2 (QT ) , BH : H ˜ 2s (QT ) → H ˜ 2s (QT ) . (5.14) BV : H
(5.15)
Next we discuss G˚ arding’s inequality. Introduce the symbol β/2 aβ (θ, ξ, η) = α(θ) |ξ|2 + i η
where α(θ) is a smooth 1-periodic function and α(θ) 6= 0 , θ ∈ R . Consider an operator of the form (5.16)
L = aβ (z, D) + B
where B is of Volterra type and, for a finite time interval, defines a bounded operator ˜ 2s−β+1 (QT ) . ˜ 2s (QT ) → H (5.17) B :H The class of operators defined by (5.16), (5.17) covers V and H. Notice also that the symbol aβ is not smooth but satisfies the strong ellipticity condition (cf. (3.32)), (5.18)
Re a(θ, ξ, η) ≥ C0 (|ξ|2 + |η|)β/2
if |ξ| + |η|1/2 ≥ r
if (and only if) |β| < 2 . For aβ the smoothness property is violated at ξ = 0, η = 0, and in order to apply the previous theoretical results, we slightly modify the representation (5.16). Choose a smooth positive function χ on R such that χ(ξ) = |ξ| if |ξ| ≥ 1 , 24
and introduce a modified symbol a ˜β , (5.19)
a ˜β (θ, ξ, η) := aβ (θ, χ(ξ), η) .
One can easily verify that this modification preserves the strong ellipticity; a ˜β is strongly β β elliptic if |β| < 2 . Consider now the corresponding symbol a ˜β ∈ S2,per ∩ V2 (R2z × R2ζ ) and write ˜. (5.20) L=a ˜β (z, D) + B Lemma 5.5. Let 0 < T < ∞. Then we have the following bounded operators of the Volterra type: ˜ 2s (QT ) → H ˜ 2s−β (QT ) , a ˜β (z, D) : H (i ) (ii ) (iii )
˜:H ˜ 2s (QT ) → H ˜ 2s−β+1 (QT ) , B ˜ 2s (QT ) → H ˜ 2s−β (QT ) . L:H
Proof. The assertion (i) is included in Theorem 5.3 and it is enough to verify (ii). From (5.16), (5.20) we obtain ˜ = B + aβ (z, D) − a ˜β (z, D) =: B + E . B Since B has the required mapping property, it suffices to consider E for which we further write E = R + C , where Z 1 Ru(θ, t) = aβ (θ, 0, η) − a ˜β (θ, 0, η) uˆ(0, η)eitη dη , 2π |η|≤1 Z 1 aβ (θ, 0, η) − a ˜β (θ, 0, η) uˆ(0, η)eitη dη . Cu(θ, t) = 2π |η|≥1 The operator R is an integral operator with a smooth kernel Z Z 1 Ru(θ, t) = K(θ, t, τ ) u(φ, τ ) dφdτ , Rτ 0 Z 1 eiη(t−τ ) aβ (θ, 0, η) − a K(θ, t, τ ) = ˜β (θ, 0, η) dη ; 2π |η|≤1 observe that aβ (θ, 0, η) is integrable at η = 0 by the condition |β| < 2 . Thus R defines a ˜ 2s (QT ) → H ˜ 2r (QT ) for any s , r ∈ R . Moreover there holds bounded operator H |aβ (θ, 0, η) − a ˜β (θ, 0, η)| ≤ c |η|(β−1)/2 if |η| ≥ 1 ˜ 2s (T × R) → H ˜ 2s−β+1 (T × R) , and in which implies that C defines a bounded operator H ˜ 2s (QT ) → H ˜ 2s−β+1 (QT ) . Hence (ii) follows. particular H After these remarks we can prove G˚ arding’s inequality for the operator L when the time interval is finite. Theorem 5.6. Suppose that the operator L is given by (5.16), (5.17) such that |β| < 2 . Let T be a given positive number. Then with some C0 > 0 , C1 ≥ 0 there holds for all ˜ 2β/2 (QT ) u∈H (5.21)
Re (Lu, u)T ≥ C0 kuk2H˜ β/2 (Q
T)
2
25
− C1 kuk2H˜ (β−1)/2 (Q ) . 2
T
Proof. Let v be a smooth function in T1 × (0, ∞) such that supp v ⊂ T1 × [0, ∞). Then we have with constants C00 > 0, C10 , C100 ≥ 0 ˜ v)T Re (Lv, v)T = Re (˜ aβ (z, D)v, v) + Re (Bv, (5.22)
≥ C00 kvk2H˜ β/2 (Q 2
≥ ≥
∞)
C00 kvk2H˜ β/2 (Q ) T 2 C0 kvk2H˜ β/2 (Q ) T
− C10 kvk2H˜ (β−1)/2 (Q
∞)
2
− −
2
C100 kvkH˜ (β−1)/2 (QT )
kvkH˜ β/2 (QT )
C1 kvk2H˜ (β−1)/2 (Q ) T
.
2
2
˜ ˜ −β/2 kvkH˜ β/2 (QT ) − kBvk (QT ) H 2
2
2
˜ 2r (QT ) with r < 1 (see e.g. [13], [14]; observe that By the density of the functions v in H ˜ 2r (QT )), it follows that (5.21) is the smoothness with respect to the time variable is r/2 in H β/2 ˜ 2 (QT ) , β < 2 . valid for all u ∈ H In case of the single layer and the hypersingular heat operators we obtain an improvement with respect to (5.21) as follows Corollary 5.7. Let T be a given positive number. Then for the operators V and H defined in (5.6) and (5.7) there holds with some C0 > 0 (5.23)
Re (V u, u)T ≥ C0 kuk2H˜ −1/2 (Q
T)
2
(5.24)
Re (Hu, u)T ≥ C0 kuk2H˜ 1/2 (Q
T)
2
if if
−1/2
˜2 u∈H
(QT ) ,
˜ 21/2 (QT ) . u∈H
Proof. The operators V and H are known to be positive , i.e. we have ˜ 2−1/2 (QT ) , u 6= 0 , Re (V u, u)T > 0 if u ∈ H Re (Hu, u)T > 0
˜ 21/2 (QT ) , u 6= 0 . if u ∈ H
By using these and compactness of the embedding ˜ 2r (QT ) ⊂ H ˜ 2s (QT ) (r > s , 0 < T < ∞) H one can conclude that (5.22) implies (5.23) and (5.24); for details compare [4] and [10]. References 1. M. Abramowitz and I. A. Stegun (eds.), Handbook of mathematical functions with formulas, graphs and mathematical tables, U.S. Government Printing Office, Washington DC, 1971. 2. D. N. Arnold and P. J. Noon, Coercivity of the single layer heat potential, J. Comput. Math. 7 (1983), 100–104. 3. R. Beals, A general calculus of pseudodifferential operators, Duke Math. J. 42 (1975), 1–42. 4. M. Costabel, Boundary integral operators for the heat equation, Integral Equations Operator Theory 13 (1990), no. 4, 498–552. 5. M. Costabel and J. Saranen, Spline collocation for convolutional parabolic boundary integral equations, Numer. Math. 84 (2000), 417–449. 6. , The spline collocation for parabolic boundary integral equations on smooth curves, to appear. 7. L. H¨ ormander, The analysis of linear partial differential operators I, Springer-Verlag, Berlin, 1983. 8. , The analysis of linear partial differential operators III, Springer-Verlag, Berlin, 1985. 9. G. C. Hsiao and J. Saranen, Coercivity of the single layer heat operator, Technical report 89–2, Department of Mathematical Sciences, University of Delaware, 1989. 10. , Boundary integral solution of the two-dimensional heat equation, Math. Methods Appl. Sci. 16 (1993), 87–114. 11. C. Hunt and A. Piriou, Op´erateurs pseudo-differentiels anisotropes d’ordre variable, C.R. Acad. Sc. Paris, s´erie A 268 (1969), 28–31. 26
12. 13. 14. 15. 16. 17. 18. 19. 20. 21.
, Majorations L2 et inegalit´e sous-elliptique pour les op´erateurs pseudo-differentiels anisotropes d’ordre variable, C.R. Acad. Sc. Paris, s´erie A 268 (1969), 214–217. J. L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications I, SpringerVerlag, Berlin, Heidelberg, New York, 1972. , Non-homogeneous boundary value problems and applications II, Springer-Verlag, Berlin, Heidelberg, New York, 1972. J. P. Noon, The single layer heat potential and Galerkin boundary element methods for the heat equation, Ph.D. thesis, University of Maryland, 1988. F. Oberhettinger, Fourier expansions, Academic Press, New York, London, 1973. A. Piriou, Une classe d’op´erateurs pseudo-diff´erentiels du type de Volterra, Ann. Inst. Fourier Grenoble 20 (1970), 77–94. , Probl`emes aux limites g´en´eraux pour des op´erateurs diff´erentiels paraboliques dans un domaine born´e, Ann. Inst. Fourier Grenoble 21 (1971), 59–78. V. S. Rabinovich, Quasi-elliptic pseudodifferential operators and the Cauchy problem for parabolic equations, Soviet Math. Dokl. 12 (1971), no. 6, 1791–1796. S. Rempel and B.-W. Schulze, Index theory of elliptic boundary value problems, Akademie-Verlag, Berlin, 1982. M. E. Taylor, Pseudodifferential operators, Princeton Univ. Press, Princeton – New Jersey, 1981.
M.Costabel IRMAR ´ de Rennes 1 Universite Campus de Beaulieu 35042 Rennes Cedex, France E-mail address:
[email protected] J. Saranen University of Oulu Department of Mathematical Sciences P.O. Box 3000 90014 Oulu, Finland E-mail address:
[email protected]
1991 Mathematics Subject Classification. 45P05, 35K05, 47G30.
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