12 2. FUNDAMENTAL SOLUTION AND COVARIANCE FUNCTION
and that the integrals
|w|≤2
dw
|w|−d+b,
|w|≤2
dw |e ±
w|−d+b,
converge for each b 0.
We next study
|w| 2
dw |D2kd−b(w, e)|. Set
Φ(λ, µ) = kd−b (w µ)e) , λ, µ [0, 1].
Then
D2kd−b(w,
e) =
1
0

1
0

∂2Φ
∂λ∂µ
(λ, µ).
Elementary computations lead to
∂2Φ
∂λ∂µ
(λ, µ) C kd−b+2 (w µ)e) .
Therefore, by Fubini’s theorem,
|w| 2
dw
|D2kd−b(w,
e)| C
1
0

1
0

|w| 2
dw |w
µ)e|−d+b−2.
The integral
|w| 2
dw |w
µ)e|−d+b−2
converges for any b 2. Consequently,
(d) is proved.
The proof of (e) is analogous to that of (a). It suffices to apply the identity
(2.11) four times, and to use the change of variables z = cw.
Lemma 2.6 is the basis for the main technical estimates of this paper, whose
statements and proofs are deferred to Chapter 6.
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