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6.2 Elliptic Pseudodifferential Operators on Ω ⊂ IR n 337

Before we proceed to prove these assertions, we remark that N j (x, y)
can be computed explicitly by using the fundamental solution E 0 (x, z; y)in
(6.2.32); namely by the recursion procedure

N 0 (x, y) := E 0 (x, y; y)= L 0 (x, y) ,

N (x, y) = E 0 (x, z; y)T z (y)N −1 (z, y)dz (6.2.45)

Ω
for =1, 2,... ,j +1 .

We remark that the kernels in (6.2.45) are all of weakly singular type as
in (6.2.43) (with j ≥−1). The composite integrals of this kind can also be
analyzed according to estimates obtained by Sobolev [285] and Mikhlin [212]
(see Mikhlin and Pr¨oßdorf [215, p. 214]).
So, the Levi function L j (x, y) of order j has the form

j

L j (x, y)= N (x, y) . (6.2.46)
=0
With the Levi function available, we may seek a solution of the partial dif-
ferential equation (6.2.30) in the form

u(x)= L j (x, y)Φ(y)dy . (6.2.47)
Ω
By applying the distributions in (6.2.42) to u(x), we obtain the equation

f(x)= Φ(x) − T(y)N j+1 Φ(y)dy . (6.2.48)
Ω
This is a Fredholm integral equation of the second kind for the unknown
density Φ(x). The integral operator T in (6.2.48) has the kernel
∼j+1
T j+1 (x, y)= T(y)N j+1 (x, y)

belonging to C j+1−n (Ω × Ω)for j +1 − n ≥ 0. If the kernel T j+1 (x, y)is
properly supported, then the integral operator defines a continuous mapping
C (Ω) → C (Ω) and for (6.2.48), the classical Fredholm theory is available.
∞
∞
We now return to the proof of the above made assertions for N j (x, y)and
T j (x, y).
Lemma 6.2.5. Let N j (x, y) be the Schwartz kernel of a pseudodifferential
−j−2κ
operator N ∈L ,j ∈ IN 0 .Then
∼j
T(y)N j (x, y)=: T j (x, y) (6.2.49)