Bergman space

${\displaystyle \Vert f{\Vert }_{A^p(D)}:={\left({\int }_D|f(x+iy){|}^p\mathrm{d}x\mathrm{d}y\right)}^{1/p}<\mathrm{\infty }.}$

The quantity ${\displaystyle \Vert f{\Vert }_{A^p(D)}}$ is called the norm of the function f; it is a true norm if ${\displaystyle p\ge 1}$. Thus A^p(D) is the subspace of holomorphic functions that are in the space L^p(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 L^p(D) implies also compact convergence, and so the limit function is also holomorphic.

If p = 2, then A^p(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:

${\displaystyle \Vert f{\Vert }_{A^p(D)}:={\left({\int }_D|f(z){|}^{p}dA\right)}^{1/p}(f\in A^p(D)),}$

where ${\displaystyle A}$ 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 ${\displaystyle \mathbb{D}}$ of the complex plane, in which case ${\displaystyle A^p(\mathbb{D}):=A^p}$. In the Hilbert space case, given:${\displaystyle f(z)=\sum _{n=0}^{\mathrm{\infty }}{a}_n{z}^n\in A^2}$, we have:

${\displaystyle \Vert f{\Vert }_{A^2}^2:=\frac{1}{\pi }{\int }_{\mathbb{D}}|f(z){|}^{2}dz=\sum _{n=0}^{\mathrm{\infty }}\frac{|a_n{|}^2}{n+1},}$

that is, A² is isometrically isomorphic to the weighted ℓ^p(1/(n + 1)) space. In particular the polynomials are dense in A². Similarly, if D = ${\displaystyle \mathbb{C}}$₊, the right (or the upper) complex half-plane, then:

${\displaystyle \Vert F{\Vert }_{A^2({\mathbb{C}}_{+})}^2:=\frac{1}{\pi }{\int }_{{\mathbb{C}}_{+}}|F(z){|}^{2}dz={\int }_0^{\mathrm{\infty }}|f(t){|}^2\frac{dt}{t},}$

where ${\displaystyle F(z)={\int }_0^{\mathrm{\infty }}f(t)e^{-tz}dt}$, that is, A²(${\displaystyle \mathbb{C}}$₊) is isometrically isomorphic to the weighted L^p_1/t (0,∞) space (via the Laplace transform).

The weighted Bergman space A^p(D) is defined in an analogous way, i.e.,

${\displaystyle \Vert f{\Vert }_{A_w^p(D)}:={\left({\int }_D|f(x+iy){|}^{2}w(x+iy)dxdy\right)}^{1/p},}$

provided that w : D → [0, ∞) is chosen in such way, that ${\displaystyle A_w^p(D)}$ is a Banach space (or a Hilbert space, if p = 2). In case where ${\displaystyle D=\mathbb{D}}$, by a weighted Bergman space ${\displaystyle A_{\alpha }^p}$ we mean the space of all analytic functions f such that:

${\displaystyle \Vert f{\Vert }_{A_{\alpha }^p}:={\left((\alpha +1){\int }_{\mathbb{D}}|f(z){|}^p(1-|z{|}^2{)}^{\alpha }dA(z)\right)}^{1/p}<\mathrm{\infty },}$

and similarly on the right half-plane (i.e., ${\displaystyle A_{\alpha }^p({\mathbb{C}}_{+})}$) we have:

${\displaystyle \Vert f{\Vert }_{A_{\alpha }^p({\mathbb{C}}_{+})}:={\left(\frac{1}{\pi }{\int }_{{\mathbb{C}}_{+}}|f(x+iy){|}^p{x}^{\alpha }dxdy\right)}^{1/p},}$

and this space is isometrically isomorphic, via the Laplace transform, to the space ${\displaystyle L^2({\mathbb{R}}_{+},d{\mu }_{\alpha })}$, where:

${\displaystyle d{\mu }_{\alpha }:=\frac{\mathrm{\Gamma }(\alpha +1)}{2^{\alpha }{t}^{\alpha +1}}dt}$

(here Γ denotes the Gamma function).

Further generalisations are sometimes considered, for example ${\displaystyle A_{\nu }^2}$ denotes a weighted Bergman space (often called a Zen space) with respect to a translation-invariant positive regular Borel measure ${\displaystyle \nu }$ on the closed right complex half-plane ${\displaystyle \overline{{\mathbb{C}}_{+}}}$, that is:

${\displaystyle A_{\nu }^p:=\left\{f:{\mathbb{C}}_{+}⟶\mathbb{C}\text{ analytic}:\Vert f{\Vert }_{A_{\nu }^p}:={\left(\underset{\epsilon >0}{sup}{\int }_{\overline{{\mathbb{C}}_{+}}}|f(z+\epsilon ){|}^{p}d\nu (z)\right)}^{1/p}<\mathrm{\infty }\right\}.}$