Nonlinear Equations - UFRJ

58 [CH. 5: REPRODUCING KERNEL SPACES
Exercise 5.1. Show that V is continuous in Bergman space A(M).
Hint: verify first that for u harmonic and r small enough,
∫
1
u(z) dz = u(p).
Vol B(p, r)
B(p,r)
5.2 Metric structure on root space
Because of Definition 5.2(2), K(·, y) ≠ 0. Thus, y ↦→ K(·, y) induces
a map from M to P(F). The differential form ω is defined as the
pull-back of the Fubini-Study form ω f of P(F) by y ↦→ K(·, y).
Recall from (4.5) that The Fubini-Study differential 1-1 form in
F \ {0} is defined by
√ −1
ω f =
2 ∂ ¯∂ log ‖f‖ 2
and is equivariant by scaling. Its pull-back is
√ −1
ω x =
2 ∂ ¯∂ log K(x, x).
When the form ω is non-degenerate for all x ∈ M, it induces a
Hermitian structure on M. This happens if and only if the fewspace
is a non-degenerate fewspace.
Remark 5.6. If F is the Bergman space, the kernel obtained above is
known as the Bergman Kernel and the metric induced by ω as the
Bergman metric.
Remark 5.7. If φ i (x) denotes an orthonormal basis of F (finite or
infinite), then the kernel can be written as
K(x, y) = ∑ φ i (x)φ i (y).
Remark 5.8. The form ω induces an element of the cohomology ring
H ∗ (M), namely the operator that takes a 2k-chain C to ∫ ω∧· · ·∧ω.
C
If F is a fewspace and x ∈ M, we denote by F x the space K(·, x) ⊥
of all f ∈ F vanishing at x.