Let
G
(
P
;
Q
)
G(P;Q)
be the discrete Green’s function over a discrete h-convex region
Ω
\Omega
of the plane; i.e.,
a
(
P
)
G
x
x
¯
(
P
;
Q
)
+
c
(
P
)
G
y
y
¯
(
P
;
Q
)
=
−
δ
(
P
;
Q
)
/
h
2
a(P){G_{x\bar x}}(P;Q) + c(P){G_{y\bar y}}(P;Q) = - \delta (P;Q)/{h^2}
for
P
∈
Ω
h
,
G
(
P
;
Q
)
=
0
P \in {\Omega _h},G(P;Q) = 0
for
P
∈
∂
Ω
h
P \in \partial {\Omega _h}
. Assume that
a
(
P
)
a(P)
and
c
(
P
)
c(P)
are Hölder continuous over
Ω
\Omega
and positive. We show that
|
D
(
m
)
G
(
P
;
Q
)
|
≦
A
m
/
ρ
P
Q
m
|{D^{(m)}}G(P;Q)| \leqq {A_m}/\rho _{P\;Q}^m
and
|
D
~
(
m
)
G
(
P
;
Q
)
|
≦
B
m
d
(
Q
)
/
ρ
P
Q
m
+
1
|{\tilde D^{(m)}}G(P;Q)| \leqq {B_m}d(Q)/\rho _{P\;Q}^{m + 1}
, where
D
(
m
)
{D^{(m)}}
is an mth order difference quotient with respect to the components of P or Q, and
D
~
(
m
)
{\tilde D^{(m)}}
denotes an mth order difference quotient only with respect to the components of P.