The weighted Kohn Laplacian
◻
φ
\Box _\varphi
is a natural second order elliptic operator associated to a weight
φ
:
C
n
→
R
\varphi :\mathbb {C}^n\rightarrow \mathbb {R}
and acting on
(
0
,
1
)
(0,1)
-forms, which plays a key role in several questions of complex analysis.
We consider here the case of model monomial weights in
C
2
\mathbb {C}^2
, i.e.,
\[
φ
(
z
,
w
)
:=
∑
(
α
,
β
)
∈
Γ
|
z
α
w
β
|
2
,
\varphi (z,w):=\sum _{(\alpha ,\beta )\in \Gamma }|z^\alpha w^\beta |^2,
\]
where
Γ
⊆
N
2
\Gamma \subseteq \mathbb {N}^2
is finite. Our goal is to prove coercivity estimates of the form
\[
(
∗
)
◻
φ
≥
μ
2
,
(*)\hspace {10pc} \Box _\varphi \geq \mu ^2,\hspace {10pc}
\]
where
μ
:
C
n
→
R
\mu :\mathbb {C}^n\rightarrow \mathbb {R}
acts by pointwise multiplication on
(
0
,
1
)
(0,1)
-forms, and the inequality is in the sense of self-adjoint operators. We proved in 2015 how to derive from
(
∗
)
(*)
new pointwise bounds for the weighted Bergman kernel associated to
φ
\varphi
. Here we introduce a technique to establish
(
∗
)
(*)
with
\[
μ
(
z
,
w
)
=
c
(
1
+
|
z
|
a
+
|
w
|
b
)
(
a
,
b
≥
0
)
,
\mu (z,w)=c(1+|z|^a+|w|^b) \qquad (a,b\geq 0),
\]
where
a
,
b
≥
0
a,b\geq 0
depend on (and are easily computable from)
Γ
\Gamma
. As a corollary we also prove that, for a wide class of model monomial weights, the spectrum of
◻
φ
\Box _\varphi
is discrete if and only if the weight is not decoupled, i.e.,
Γ
\Gamma
contains at least a point
(
α
,
β
)
(\alpha ,\beta )
with
α
≠
0
≠
β
\alpha \neq 0\neq \beta
.
Our methods comprise a new holomorphic uncertainty principle and linear optimization arguments.