- Source: Bergman space
In complex analysis, functional analysis and operator theory, a Bergman space, named after Stefan Bergman, is a function space of holomorphic functions in a domain D of the complex plane that are sufficiently well-behaved at the boundary that they are absolutely integrable. Specifically, for 0 < p < ∞, the Bergman space Ap(D) is the space of all holomorphic functions
f
{\displaystyle f}
in D for which the p-norm is finite:
‖
f
‖
A
p
(
D
)
:=
(
∫
D
|
f
(
x
+
i
y
)
|
p
d
x
d
y
)
1
/
p
<
∞
.
{\displaystyle \|f\|_{A^{p}(D)}:=\left(\int _{D}|f(x+iy)|^{p}\,\mathrm {d} x\,\mathrm {d} y\right)^{1/p}<\infty .}
The quantity
‖
f
‖
A
p
(
D
)
{\displaystyle \|f\|_{A^{p}(D)}}
is called the norm of the function f; it is a true norm if
p
≥
1
{\displaystyle p\geq 1}
. Thus Ap(D) is the subspace of holomorphic functions that are in the space Lp(D). The Bergman spaces are Banach spaces, which is a consequence of the estimate, valid on compact subsets K of D:
Thus convergence of a sequence of holomorphic functions in Lp(D) implies also compact convergence, and so the limit function is also holomorphic.
If p = 2, then Ap(D) is a reproducing kernel Hilbert space, whose kernel is given by the Bergman kernel.
Special cases and generalisations
If the domain D is bounded, then the norm is often given by:
‖
f
‖
A
p
(
D
)
:=
(
∫
D
|
f
(
z
)
|
p
d
A
)
1
/
p
(
f
∈
A
p
(
D
)
)
,
{\displaystyle \|f\|_{A^{p}(D)}:=\left(\int _{D}|f(z)|^{p}\,dA\right)^{1/p}\;\;\;\;\;(f\in A^{p}(D)),}
where
A
{\displaystyle A}
is a normalised Lebesgue measure of the complex plane, i.e. dA = dz/Area(D). Alternatively dA = dz/π is used, regardless of the area of D.
The Bergman space is usually defined on the open unit disk
D
{\displaystyle \mathbb {D} }
of the complex plane, in which case
A
p
(
D
)
:=
A
p
{\displaystyle A^{p}(\mathbb {D} ):=A^{p}}
. In the Hilbert space case, given:
f
(
z
)
=
∑
n
=
0
∞
a
n
z
n
∈
A
2
{\displaystyle f(z)=\sum _{n=0}^{\infty }a_{n}z^{n}\in A^{2}}
, we have:
‖
f
‖
A
2
2
:=
1
π
∫
D
|
f
(
z
)
|
2
d
z
=
∑
n
=
0
∞
|
a
n
|
2
n
+
1
,
{\displaystyle \|f\|_{A^{2}}^{2}:={\frac {1}{\pi }}\int _{\mathbb {D} }|f(z)|^{2}\,dz=\sum _{n=0}^{\infty }{\frac {|a_{n}|^{2}}{n+1}},}
that is, A2 is isometrically isomorphic to the weighted ℓp(1/(n + 1)) space. In particular the polynomials are dense in A2. Similarly, if D =
C
{\displaystyle \mathbb {C} }
+, the right (or the upper) complex half-plane, then:
‖
F
‖
A
2
(
C
+
)
2
:=
1
π
∫
C
+
|
F
(
z
)
|
2
d
z
=
∫
0
∞
|
f
(
t
)
|
2
d
t
t
,
{\displaystyle \|F\|_{A^{2}(\mathbb {C} _{+})}^{2}:={\frac {1}{\pi }}\int _{\mathbb {C} _{+}}|F(z)|^{2}\,dz=\int _{0}^{\infty }|f(t)|^{2}{\frac {dt}{t}},}
where
F
(
z
)
=
∫
0
∞
f
(
t
)
e
−
t
z
d
t
{\displaystyle F(z)=\int _{0}^{\infty }f(t)e^{-tz}\,dt}
, that is, A2(
C
{\displaystyle \mathbb {C} }
+) is isometrically isomorphic to the weighted Lp1/t (0,∞) space (via the Laplace transform).
The weighted Bergman space Ap(D) is defined in an analogous way, i.e.,
‖
f
‖
A
w
p
(
D
)
:=
(
∫
D
|
f
(
x
+
i
y
)
|
2
w
(
x
+
i
y
)
d
x
d
y
)
1
/
p
,
{\displaystyle \|f\|_{A_{w}^{p}(D)}:=\left(\int _{D}|f(x+iy)|^{2}\,w(x+iy)\,dx\,dy\right)^{1/p},}
provided that w : D → [0, ∞) is chosen in such way, that
A
w
p
(
D
)
{\displaystyle A_{w}^{p}(D)}
is a Banach space (or a Hilbert space, if p = 2). In case where
D
=
D
{\displaystyle D=\mathbb {D} }
, by a weighted Bergman space
A
α
p
{\displaystyle A_{\alpha }^{p}}
we mean the space of all analytic functions f such that:
‖
f
‖
A
α
p
:=
(
(
α
+
1
)
∫
D
|
f
(
z
)
|
p
(
1
−
|
z
|
2
)
α
d
A
(
z
)
)
1
/
p
<
∞
,
{\displaystyle \|f\|_{A_{\alpha }^{p}}:=\left((\alpha +1)\int _{\mathbb {D} }|f(z)|^{p}\,(1-|z|^{2})^{\alpha }dA(z)\right)^{1/p}<\infty ,}
and similarly on the right half-plane (i.e.,
A
α
p
(
C
+
)
{\displaystyle A_{\alpha }^{p}(\mathbb {C} _{+})}
) we have:
‖
f
‖
A
α
p
(
C
+
)
:=
(
1
π
∫
C
+
|
f
(
x
+
i
y
)
|
p
x
α
d
x
d
y
)
1
/
p
,
{\displaystyle \|f\|_{A_{\alpha }^{p}(\mathbb {C} _{+})}:=\left({\frac {1}{\pi }}\int _{\mathbb {C} _{+}}|f(x+iy)|^{p}x^{\alpha }\,dx\,dy\right)^{1/p},}
and this space is isometrically isomorphic, via the Laplace transform, to the space
L
2
(
R
+
,
d
μ
α
)
{\displaystyle L^{2}(\mathbb {R} _{+},\,d\mu _{\alpha })}
, where:
d
μ
α
:=
Γ
(
α
+
1
)
2
α
t
α
+
1
d
t
{\displaystyle d\mu _{\alpha }:={\frac {\Gamma (\alpha +1)}{2^{\alpha }t^{\alpha +1}}}\,dt}
(here Γ denotes the Gamma function).
Further generalisations are sometimes considered, for example
A
ν
2
{\displaystyle A_{\nu }^{2}}
denotes a weighted Bergman space (often called a Zen space) with respect to a translation-invariant positive regular Borel measure
ν
{\displaystyle \nu }
on the closed right complex half-plane
C
+
¯
{\displaystyle {\overline {\mathbb {C} _{+}}}}
, that is:
A
ν
p
:=
{
f
:
C
+
⟶
C
analytic
:
‖
f
‖
A
ν
p
:=
(
sup
ε
>
0
∫
C
+
¯
|
f
(
z
+
ε
)
|
p
d
ν
(
z
)
)
1
/
p
<
∞
}
.
{\displaystyle A_{\nu }^{p}:=\left\{f:\mathbb {C} _{+}\longrightarrow \mathbb {C} {\text{ analytic}}\;:\;\|f\|_{A_{\nu }^{p}}:=\left(\sup _{\varepsilon >0}\int _{\overline {\mathbb {C} _{+}}}|f(z+\varepsilon )|^{p}\,d\nu (z)\right)^{1/p}<\infty \right\}.}
Reproducing kernels
The reproducing kernel
k
z
A
2
{\displaystyle k_{z}^{A^{2}}}
of A2 at point
z
∈
D
{\displaystyle z\in \mathbb {D} }
is given by:
k
z
A
2
(
ζ
)
=
1
(
1
−
z
¯
ζ
)
2
(
ζ
∈
D
)
,
{\displaystyle k_{z}^{A^{2}}(\zeta )={\frac {1}{(1-{\overline {z}}\zeta )^{2}}}\;\;\;\;\;(\zeta \in \mathbb {D} ),}
and similarly, for
A
2
(
C
+
)
{\displaystyle A^{2}(\mathbb {C} _{+})}
we have:
k
z
A
2
(
C
+
)
(
ζ
)
=
1
(
z
¯
+
ζ
)
2
(
ζ
∈
C
+
)
,
{\displaystyle k_{z}^{A^{2}(\mathbb {C} _{+})}(\zeta )={\frac {1}{({\overline {z}}+\zeta )^{2}}}\;\;\;\;\;(\zeta \in \mathbb {C} _{+}),}
In general, if
φ
{\displaystyle \varphi }
maps a domain
Ω
{\displaystyle \Omega }
conformally onto a domain
D
{\displaystyle D}
, then:
k
z
A
2
(
Ω
)
(
ζ
)
=
k
φ
(
z
)
A
2
(
D
)
(
φ
(
ζ
)
)
φ
′
(
z
)
¯
φ
′
(
ζ
)
(
z
,
ζ
∈
Ω
)
.
{\displaystyle k_{z}^{A^{2}(\Omega )}(\zeta )=k_{\varphi (z)}^{{\mathcal {A}}^{2}(D)}(\varphi (\zeta ))\,{\overline {\varphi '(z)}}\varphi '(\zeta )\;\;\;\;\;(z,\zeta \in \Omega ).}
In weighted case we have:
k
z
A
α
2
(
ζ
)
=
α
+
1
(
1
−
z
¯
ζ
)
α
+
2
(
z
,
ζ
∈
D
)
,
{\displaystyle k_{z}^{A_{\alpha }^{2}}(\zeta )={\frac {\alpha +1}{(1-{\overline {z}}\zeta )^{\alpha +2}}}\;\;\;\;\;(z,\zeta \in \mathbb {D} ),}
and:
k
z
A
α
2
(
C
+
)
(
ζ
)
=
2
α
(
α
+
1
)
(
z
¯
+
ζ
)
α
+
2
(
z
,
ζ
∈
C
+
)
.
{\displaystyle k_{z}^{A_{\alpha }^{2}(\mathbb {C} _{+})}(\zeta )={\frac {2^{\alpha }(\alpha +1)}{({\overline {z}}+\zeta )^{\alpha +2}}}\;\;\;\;\;(z,\zeta \in \mathbb {C} _{+}).}
References
Further reading
Bergman, Stefan (1970), The kernel function and conformal mapping, Mathematical Surveys, vol. 5 (2nd ed.), American Mathematical Society
Hedenmalm, H.; Korenblum, B.; Zhu, K. (2000), Theory of Bergman Spaces, Springer, ISBN 978-0-387-98791-0
Richter, Stefan (2001) [1994], "Bergman spaces", Encyclopedia of Mathematics, EMS Press.
See also
Bergman kernel
Banach space
Hilbert space
Reproducing kernel Hilbert space
Hardy space
Dirichlet space
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