AbstractLet $$u_{k}$$
u
k
be a solution of the Helmholtz equation with the wave number k, $$\varDelta u_{k}+k^{2} u_{k}=0$$
Δ
u
k
+
k
2
u
k
=
0
, on (a small ball in) either $${\mathbb {R}}^{n}$$
R
n
, $${\mathbb {S}}^{n}$$
S
n
, or $${\mathbb {H}}^{n}$$
H
n
. For a fixed point p, we define $$M_{u_{k}}(r)=\max _{d(x,p)\le r}|u_{k}(x)|.$$
M
u
k
(
r
)
=
max
d
(
x
,
p
)
≤
r
|
u
k
(
x
)
|
.
The following three ball inequality $$M_{u_{k}}(2r)\le C(k,r,\alpha )M_{u_{k}}(r)^{\alpha }M_{u_{k}}(4r)^{1-\alpha }$$
M
u
k
(
2
r
)
≤
C
(
k
,
r
,
α
)
M
u
k
(
r
)
α
M
u
k
(
4
r
)
1
-
α
is well known, it holds for some $$\alpha \in (0,1)$$
α
∈
(
0
,
1
)
and $$C(k,r,\alpha )>0$$
C
(
k
,
r
,
α
)
>
0
independent of $$u_{k}$$
u
k
. We show that the constant $$C(k,r,\alpha )$$
C
(
k
,
r
,
α
)
grows exponentially in k (when r is fixed and small). We also compare our result with the increased stability for solutions of the Cauchy problem for the Helmholtz equation on Riemannian manifolds.