Friedrichs's Lemma on Density in the Graph Norm

Math 213b (Spring 2012) Yum-Tong Siu 1
Friedrichs’s <strong>Lemma</strong> on Density in the Graph Norm
We derived the Bochner-Kodaira formula
□φ = − ( tr∇ ¯∇ ) φ − Rφ + Ricφ + Ωφ
for smooth V -valued (0, q)-form φ on a compact Kähler manifold M with V
being a holomorphic vector bundle over M with a smooth Hermitian metric
along its fibers. Here ∇ and ¯∇ are respectively the covariant differentiation
in the (1, 0)-direction and the (0, 1)-direction, R is the action of the curvature
operator of M, Ric is the Ricci curvature action, and Ω is the action of the
curvature operator of V .
In order to use the Bochner-Kodaira formula to get the vanishing theorem
of Kodaira for the case of V being a positive line bundle, we have to invert
the operator □ as a self-adjoint operator in the Hilbert space defined by L 2 .
However, our Bochner-Kodaira formula was derived under the assumption
that φ is smooth so that integration by parts can be justified. To pass from
a smooth φ to an L 2 φ, we need to use the smoothing process of local convolution
by approximating identity so that the density from approximation
holds for the graph norm of φ ↦→ (¯∂φ, ¯∂∗ φ ) .
By using a partition of unity to write φ = ∑ j ρ jφ, we can reduce the
problem of smoothing with density in the graph norm to the case where the
support of φ is contained in one coordinate chart. Then the problem is reduced
to the case of a first-order differential operator with smooth coefficients
on R m . The problem was solved in
K. O. Friedrichs, The identity of weak and strong extensions of differential
operators. Trans. Amer. Math. Soc. 55 (1944), 132-151.
The result is known as the Friedrichs lemma.
<strong>Lemma</strong> (Friedrichs). Let the first-order differential operator be L = a(x) ∂ + ∂x
b(x) on R m with a(x) and b(x) being smooth functions. Let χ(x) be a nonnegative
function supported on the unit open ball of R m . Let χ ε (x) = 1 χ( x ).
ε m ε m
Suppose u is an L 2 function on R m with compact support such that Lu in
the sense of distribution is again an L 2 function. Then both χ ε ∗ u → u and
L(χ ε ∗ u) → Lu in L 2 norm as ε → 0.
Proof. First we observe that, for any L 2 function f(x) on R m with compact
support, we always have f ∗ χ ε → f in the L 2 norm as ε → 0. The reason is

Math 213b (Spring 2012) Yum-Tong Siu 2
as follows. From
∫
(f ∗ χ φ ) (x) =
f(x − y)χ ε (y)dy
it follows that
∫
(f ∗ χ φ − f) (x) =
(f(x − y) − f(x)) χ ε (y)dy.
By the triangle inequality and the nonnegativity of χ ε ,
∫
∥f ∗ χ φ − f∥ L 2 = ∥f(x − y) − f(x)∥ L 2 (x) χ ε(y)dy,
where ∥·∥ L 2 (x) means the L2 of the function of the variable x. Since
we have f ∗ χ ε → f as ε → 0.
∥f(x − y) − f(x)∥ L 2 (x)
→ 0 as y → 0,
Since Lu taken in the sense of distributions is assumed to be L 2 , it follows
that χ ε ∗Lu → Lu in L 2 norm as ε → 0. In order to show that L(χ ε ∗u) → Lu
in L 2 norm as ε → 0, it suffices to show that χ ε ∗ Lu − L(χ ε ∗ u) approaches
0 in L 2 norm as ε → 0. This is clearly true when u belongs to the dense
subset of smooth functions. So it suffices to show that χ ε ∗ Lu − L(χ ε ∗ u)
is bounded in L 2 norm independently of ε when u belongs to a set bounded
in L 2 norm. The zero-order part b(x) of L clearly has bounded contribution.
So we can assume without loss of generality that b(x) = 0. Then
(
χ ε ∗ Lu − L(χ ε ∗ u) = χ ε ∗ a ∂u )
− a ∂
∂x ∂x (χ ε ∗ u)
= χ ε ∗ ∂
(
∂x (au) − χ ε ∗ u ∂a ) ( ) ∂
− a
∂x ∂x χ ε ∗ u
( )
(
∂
=
∂x χ ε ∗ (au) − χ ε ∗ u ∂a ) ( ) ∂
− a
∂x ∂x χ ε ∗ u .
Clearly the second term on the right-hand side is bounded. So we can drop
it. We have
(( ) ( ))
∂
∂
∂x χ ε ∗ (au) − a
∂x χ ε ∗ u (x)

Math 213b (Spring 2012) Yum-Tong Siu 3
∫ ( )
( )
∂ ∂
=
∂y χ ε (y)a(x − y)u(x − y) − a(x)
∂y χ ε (y)u(x − y)dy
∫ ( ∂
=
ε)
∂y χ (y) (a(x − y) − a(x)) u(x − y)dy.
|y|