We study integral operators u→
M
K(x,y)u(y)d My, x∈ M,
where M is an m-dimensional differentiable manifold without boundary (m≥1) having a positive measured M locally equivalent to m-dimensional Lebesgue measure, and K(x,ã) is absolutely integrable. (In our applications M =∂for some⊂Rn,m=n−1, andd M is the surface-area measure d A in∂induced fromn-dimensional Lebesgue measure inRn.) We study only a special class of kernels K(x,y) (classC∗r(α),r =integer≥0, α >0, defined below)r-times differentiable forx =ywith certain growth-restrictions on the derivatives asx−y→0. Standard tricks – partition of unity and local change of variables – permit us to work in EuclideanRm, and we do so until Assumptions 7.1.4. We will see later that fundamental solutions of second order elliptic equations are kernels of this class, as are the kernels of many integral operators associated with elliptic boundary value problems.
The space of value of the kernels K(x,y) has little importance here – we will take the values to be real or complex matrices of given dimensions, and whenever products occur, there is the implicit assumption that the matrices are compatible. The transpose of a matrixBis writtenB.
7.1 Weakly-Singular Integral Operators with C∗r(α)Kernels 117 Definition 7.1.1. Let Abe an open set inRm; a functionK(x,y) defined for x=yinAis akernel of class Cr∗(α)in A(r=integer≥0, α >0) if it isCr forx=yand for anyδ >0 and|i| + |j| + |k| ≤r
∂xi∂yj(∂x+∂y)kK(x,y)=O
1+ |x−y|α−m−|i|−|j|−δ
uniformly forx=yin compact subsets ofA. Ifα >m+ |i| + |j|, we require also that∂ix∂yj(∂x+∂y)kK(x,y) extend continuously to{x=y}; ifm>1 and
|i| + |j|<r, this condition is superfluous, sinceA×Ais not disconnected by the diagonal. We say the kernelK hasexponentα, and we will sometimes use
“operator of order−α” to mean an integral operator with kernel of exponentα.
(See Theorems 7.1.2 and 7.1.3 for partial justification of this usage.)
Remark. Ifφ:Rm\{0} →Ris differentiable, (∂x+∂y)φ(x,y)=0. Thus if φisCrand homogeneous of degreeα−monRm\{0}, φ(x−y) is aC∗rkernel.
K(x,y) is of classC∗r(α) if and only if∂xK(x,y) and∂yK(x,y) are of class Cr∗−1(α−1) and (∂x+∂y)K(x,y) isC∗r−1(α). Ifα >r+m, a “kernel of class Cr∗(α)” simply meansa Crfunction. IfK(x,y) is aC∗r(α) kernel inA⊂Rm,h : A0 ⊂Rm→ A is aCr+1 diffeomorphism andψ: A0×A0→Ris of class Cr, then (x,y)→ψ(x,y)K(h(x),h(y)) is aC∗r(α) kernel in A0. This permits definition ofC∗r(α) kernels onCr+1manifolds (Assumptions 7.1.4 below).
Example. LetQ(x,y, σ) beCr on (Rm)3, α >0,j =integer≥0 and K(x,y)= |x−y|α−m(log|x−y|)jQ
x,y, y−x
|y−x|
forx=y;
then K is aCr∗(α) kernel inRm. We will see many kernels of this kind later.
Here we have used the Euclidean norm| ã |but we have the same conclusion with| ã |x,yin place of| ã |, where
|v|x,y=(vãG(x,y)v)1/2,
G(x,y) is aCr function whose values are symmetric positive matrices. For this case, in a neighborhood of a given x0∈Rm, let|v|0= |v|x0,x0 and write K(x,y) as a finite sum of kernels|x−y|α+0 i−m(log|x−y|0)kQi k(x,y,|yy−−xx|0) [wherei≥0,0≤k≤ j,Qi k(x,y, σ) isCr], plus aCr function ofx,y.
Theorem 7.1.2. Let K be a C∗r(α)kernel in Rm with compact support and values inCa×b (= the a×b complex matrices), and letK denote the corre-˜ sponding integral operator
K u(x)˜ =
RmK(x,y)u(y)d y, x∈Rm.
118 Chapter 7. Boundary Operators for Second-Order Elliptic Equation ThenK is a compact operator in any of the following pairs of spaces˜ (with j,k integers in0≤ j ≤k≤r; 1≤ p,q ≤ ∞; 0≤σ, τ ≤1),
(1) Wj,p(Rm,Cb)→Wk,q(Rm,Ca)if j−m/p+α >k−m/q (2) Cj,σ(Rm,Cb)→Wk,q(Rm,Ca)if j+σ +α >k−m/q (3) Wj,p(Rm,Cb)→Ck,τ(Rm,Ca)if j−m/p+α >k+τ,k<r
(4) Cj,σ(Rm,Cb)→Ck,τ(Rm,Ca)if j+σ+α >k+τ,k<r,k< j+α.
Recalling that j−m/p is the “net smoothness” of functions in Wj,p(Rm), an approximate summary is: for 0< α≤r,K is smoothing of order˜ α, or an operator of order−α.
Proof. The identities
∂
∂x
K(x,y)u(y)d y=
∂xK(x,y)u(y) (whenα >1,r ≥1)
=
(∂x+∂y)K(x,y)u(y) +
K(x,y)∂yu(y) (whenα >0,r≥1) allow us to reduce the case (α,r; j,k) to (α−1,r−1;j,k−1) whenα >1, r ≥1, or to (α,r−1;j,k−1) and (α,r;j−1,k−1) when α >0,r ≥1.
Using this and the Sobolev embeddings (in the cases α >r;r≥α >k;r≥ k≥α), we may reduce to one of the following cases:
(1) ˜K :Lp →Lq is compact ifα >0,r ≥0, α−m/p>−m/q,p≤q; (2) ˜K :Lp →Cτ is compact if 0< α≤1=r, α−m/p> τ;
(3) ˜K :Cσ →Cτ is compact if 0< α≤1=r, α+σ > τ. HereCσ =C0,σ orC0,σ+forσ <1, and forσ =1 may denoteC1orC0,1(=Lipschitz), whichever gives the stronger conclusion.
In cases (2) and (3), it suffices to prove continuity; compactness follows from the compactness of the support ofK and the compactness of the inclusion Cσ+ ⊂Cσ( >0).
(1) Continuity follows immediately from Young’s inequality (f ∗gLq≤ fLsgLpwhen 1≤ p,q,s≤ ∞and 1/q = 1/p+1/sfor the conjugate exponents). If θ:Rn→RisC∞with compact support,θ ≡1 near 0, andK (x,y)=(1−θ(x−y))K(x,y) for >0, then ˜K is certainly compact for any >0 and Young’s inequality shows K˜ −K˜ L(Lp,Lq)→0 as →0, so ˜K is compact.
7.1 Weakly-Singular Integral Operators with C∗r(α)Kernels 119 (2) Forλ < α−m(arbitrarily close) we have
|K(x,y)−K(x,y)| ≤Cmin{|x−y|λ+ |x−y|λ,
|x−x|(|x−y|λ−1+ |x−y|λ−1}
≤C|x−x|θ(|x−y|λ−θ+ |x−y|λ−θ) for 0≤θ ≤1 (constantsC,C).
Takeθ=τ and (2) follows from H¨older’s inequality.
(3) SayK(x,y)=0 if|x|>Ror|y|>R. For|x| ≤R,|x+h| ≤R K u(x˜ +h)−K u(x)˜ =
|y|<3R(K(x+h,y)−K(x,y))(u(y)−u(x))d y +
|y|<3R(K(x+h,y+h)−K(x,y))d y u(x) Ifτ ≤min{1, α+σ −δ}for someδ >0, this implies
|K u(x˜ +h)−K u˜ (x))≤const.|h|τ uCσ. In caseα+σ >1 this gives continuity intoC0,1(=Lipschitz); but it is easy to see ˜K(C1)⊂C1so in fact ˜K(Cσ)⊂C1.
Theorem 7.1.3. Let K,L be kernels with compact support of class C∗r(α), Cr∗(β)respectively in Rm; we suppose their values can be multiplied. Then K ◦L defined by
K ◦L(x,y)=
Rm K(x,z)L(z,y)d z
is a kernel of class C∗r(α+β)inRmwith compact support. Ifα+β >r+m, K ◦L is a Cr function onRm×Rm.
Proof. The caser =0 is well-known (see, for example, W. Pogorzelski,Inte- gral Equations and Their Applications, vol. I (Pergamon Press, 1966)).
Supposer ≥1, α >1; we have (∂x+∂y)K◦L(x,y)=
(∂x+∂y)K(x,z)L(z,y) +
K(x,z)(∂z+∂y)L(z,y)
∂xK◦L(x,y)=
∂xK(x,z)L(z,y)
∂yK◦L(x,y)=
∂zK(x,z)L(z,y)+
K(x,z)(∂z+∂y)L(z,y)
120 Chapter 7. Boundary Operators for Second-Order Elliptic Equation so the case (r, α, β) reduces to (r−1, α−1, β), whenr≥1, α >1. Similarly (r, α, β) reduces to (r−1, α, β−1) whenr ≥1, β >1. Thus we reduce even- tually to eitherr =0 – which is known – or tor≥1 with 0< α, β≤1, which we now examine.
Letθ:Rn→RbeC∞, θ(x)=0 for|x| ≤1, θ(x)=1 for|x| ≥2, and for any >0 define
K (x,y)=θ
y−x
K(x,y), K˜ (x,y)=
1−θ
y−x
K(x,y), and similarly defineL ,L˜ . We estimate the derivatives ofK◦Lnear a given (x0,y0),x0=y0. For any >0,
K ◦L =K ◦L +K ◦L˜ +K˜ ◦L +K˜ ◦L˜
and we choose =15(x0−y0). For (x,y) in a neighborhood of (x0,y0) we have 4 <|x−y|<6 so ˜K ◦L (x,y)=0; and since K ,L areCr func- tions, we seeK ◦LisCrnear (x0,y0). Now for|i| + |j| + |k| ≤r, ∂xi∂yj(∂x+
∂y)kK˜ ◦L (x,y) is a sum of terms const. 1−θ
x−z θ
y−z
(∂x+∂z)k+iK(x,z)
×∂zi∂yj(∂y+∂z)kL(y,z) with k+k=k, i+i=i. The terms containing some derivative of θ(y−z) vanish, since this gives disjoint supports. Since 0< α, β≤m and
|y−z|> ,|x−z|<2 on the support of the integrand, we may estimate the integral by
const.
|x−z|<2
|x−z|α−m−δ β−m−i−j−δd z=O( α+β−m−i−j−2δ)
=O(|x−y|α+β−m−i−j−2δ), for anyδ >0. Similarly forK ◦L˜ . In the case K ◦L , we have a sum of terms
const.
θ
x−z θ
z−y
∂ix(∂x+∂z)kK(x,z)∂yj(∂z+∂y)kL(z,y) (k+k=k), in addition to integrals where one or more of the derivatives∂xi∂yj
falls onθ(x−z) or θ(z−y). But in the latter,|x−z|or|z−y|is comparable to in the support of the integrand, and the estimates are like the previous case.
In the remaining integrals we have|x−z| ≥ ,|y−z| ≥ and we write the
7.1 Weakly-Singular Integral Operators with C∗r(α)Kernels 121 integrals as I+I I =
<|x−z|<8 <|y−z|
ã ã ã +
|x−z|≥8 ã ã ã. Then
|I| ≤
|x−z|<8
const. α+β−2m−i−j−2δ =O( α+β−m−i−j−2δ) and
|I I| ≤
|x−z|≥8
const.|x−z|α+β−2m−i−j−2δ=O( α+β−m−i−j−2δ), and is comparable to|x−y|, so the proof is complete.
Now we transfer our results to a differentiable manifoldM.
Assumptions 7.1.4. M is a compactm-dimensionalCr+1 manifold without boundary, provided with a positive measure d M which is Cr-equivalent to m-dimensional Lebesgue measure; that is, for any system of coordinatesxin M there is a positiveCr densityx→w(x) sod M =w(x)d x.
In our applications, M =∂ where is a bounded Cr+1 region in Rn, m=n−1, and d M=d A= surface area measure. Thus if ∂ is given locally as {x|xn =ψ(x1, . . . ,xn−1)}, then ψ is Cr+1 and d A=
1+n−1 1 (∂∂ψx
j)2d x1ã ã ãd xn−1.
We may define the spacesWj,p(M,Cq),Cj,σ(M,Cq) in the obvious ways (0≤ j ≤r), and a functionK(x,y) defined forx=yinMis a kernel of class Cr∗(α) on M provided it isCr for x=y and is aCr∗(α) kernel when written in any local coordinate system. Thus if h: (open set)⊂Rm→ M is aCr+1 embedding, (ξ, η)→K(h(ξ),h(η)) is aC∗r(α) kernel on the domain ofh.
Choose an open cover{Vj}Nj=1for M such that, for each j, there is aCr+1 diffeomorphismhj:Uj ⊂Rm→Vjand aCrwj:Uj →R+withd Mh j(ξ)= wj(ξ)dξ(ξ∈Uj). Choose a correspondingCr+1partition of unity{θj}Nj=1, θj≥ 0, suppθj ⊂Vj,N
1 θj =1 onM. Then
M
K(x,y)u(y)d My =
n
j=1
Vj
K(x,y)θj(y)u(y)d My
=
N
i,j=1
Uj
θi(x)K(x,hj(η))θj(hj(η))wj(η)u(hj(η))dη.
Now for x∈Vj,(x, η)→θi(x)K(x,hj(η))θj(hj(η))wj(η) isCrj and forx= hj(ξ)∈Vj,(ξ, η)→θi(hj(ξ))K(hj(ξ),hj(η))θj(hj(η))wj(η) is aC∗r(α) ker- nel onUj(vanishing ifηnears∂Uj). ThusK(x,y)=N
i,j=1Ki j(x,y) where Ki j(x,y)=θi(x)θj(y)K(x,y) is aCr∗(α) kernel supported in Vi×Vj. Using this representation, we may easily transfer our results above toMand conclude, in particular, that whenK,Lare kernels of classCr∗(α),C∗r(β) respectively on
122 Chapter 7. Boundary Operators for Second-Order Elliptic Equation M, thenK◦Lis kernel of classC∗r(α+β) onM, where
K◦L(x,y)=
M
K(x,z)L(z,y)d Mz.
IfK(x,y) is aC∗r(α) kernel onM andφ:M →CisCr+1, then the com- mutator [ ˜K, φ]=K˜φ−φK˜ is an integral operator with kernelCr∗(α+1)
[ ˜K, φ]u(x)=K˜(φu)(x)−φ(x) ˜K u(x)=
M
K(x,y)(φ(y)−φ(x))u(y)d My. IfK(x,y) is aC∗r(α) kernel onM,r ≥1, andV is aCrtangent vector field onM withL =V ã ∇Mthe corresponding first-order differential operator, the commutator [ ˜K,L] is an integral operator with kernel of classC∗r−1(α)
[ ˜K,L]u(x)= −
M
{K(x,y)divMV(y)+(Lx+Ly)K(x,y)}u(y)d My. If also α >1 then ˜K L and LK˜ are integral operators with kernel of class Cr∗−1(α−1).