- DIFFERENTIATING A CONVOLUTION WITH THE PARAMETRIX 247
(I-3) Estimating^82 ~Nj -^82 PNj. We shall show that
fJx't,,uxx ax1axz
(23.102)
a^2 .. PN - a^2 .. PN (x,z,t-s)sC ( 1+ d^2 (x ' z)) (t-s)-2e-4Ct-sJ, n d^2 (x,z)
axi x axJ x axi z axJ z t - swhere the function on the LHS of (23.102) has support in Minj(g) x (0, oo).
This implies the absolute convergence of the integral1
t1 ( f)
2
. pN. - a2pN).. (x,z,t-s)G(z,y,s)dμ(z)ds.
O M ax~ox1 ax~ax{We now show that the absolute value of each of the seven lines on the
RHS of (23.101) is bounded by the RHS of (23.102):(I-3a) Since r/ (d (x, z)) = 0 ford (x, z) < injdg), the absolute value of the
n d^2 (x,z)last three lines is bounded by C (t - s)-2 e-4(t-s).
(I-3b) The same is true for the fourth line since^82 <P;~x,;) and^82 </Jk(x,z)
OXx Xx OX10X~
are bounded.
In the first three lines of the RHS of (23.101) we need to control the t factor.(I-3c) Since \ a;d:~x,;) Xx Xx - a;d:~';) Xz Xz \ s Cd^2 (x, z) by (23.99), the absolute
d2( ) n d^2 (x,z)value of the third line is bounded by C t:!._~z (t - s)-2 e -^4 (t-s).
(I-3d) For the first two lines, when k 2: 1 in the summation, the factor
uk controls the t factor, so that these terms are bounded in absolute value
n _ d^2 (x,z)by C (t - s)-2 e 4(t-s).
(I-3e) For k = 0 in the first two lines, we now show that the combination
of the ~~ °tx^0 terms are bounded by Cd^2. Let (3 (x) ~ </>o (x, x), which is
identically equal to 1. Then(23.103) 0 =-a a(3. (x) = -. a¢o I (x,z) + -. a¢o I (x,z),
xJ ax1 x=z ax{ z=x
which implies(23.104) a¢? (x, z) + a¢? (x, z) =^0 (d (x, z))
ax1 ax{since the functions on the LHS of (23.104) are smooth. From this and (23.98)
we see that the k = 0 terms of the first two lines are bounded in absolute
d2( ) n d^2 (x,z)
value by C t'!!..: (t - s)-2 e-^4 <t-s). This completes the proof of (23.102).
(II) Next we estimate the second term in (23.96). Since PN (x, ·, t - s)
has compact support in B (x, inj (g)) C U, we may integrate by parts to