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 and also absolutely integrable. Specifically, for 0 < p < ∞, the Bergman space A^p(D) is the space of all holomorphic functions $f$ in D for which the p-norm is finite:

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

The quantity $\Vert f{\Vert }_{A^p(D)}$ is called the norm of the function f; it is a true norm if $p\ge 1$, thus A^p(D) is the subspace of holomorphic functions of the space L^p(D). The Bergman spaces are Banach spaces for $0<p<\mathrm{\infty }$, which is a consequence of the following estimate that is valid on compact subsets K of D:$$\underset{z\in K}{\sup }|f(z)|\le C_K\Vert f{\Vert }_{L^p(D)}.$$Convergence of a sequence of holomorphic functions in L^p(D) thus implies 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:

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

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

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

that is, A² is isometrically isomorphic to the weighted ℓ^p(1/(n + 1)) space. In particular, not only are the polynomials dense in A², but every function $f\in A^2$ can be uniformly approximated by radial dilations of functions $g$ holomorphic on a disk $D_R(0)$, where $R>1$ and the radial dilation of a function is defined by $g_r(z):=g(rz)$ for $0<r<1$.

Similarly, if D = $\mathbb{ℂ}$₊, the right (or the upper) complex half-plane, then:

$\Vert F{\Vert }_{A^2({\mathbb{ℂ}}_{+})}^2:=\frac{1}{\pi }\underset{{\mathbb{ℂ}}_{+}}{\int }|F(z){|}^{2}dz=\underset{0}{\overset{\mathrm{\infty }}{\int }}|f(t){|}^2\frac{dt}{t},$

where $F(z)=\underset{0}{\overset{\mathrm{\infty }}{\int }}f(t)e^{-tz}dt$, that is, A²($\mathbb{ℂ}$₊) 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.,

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

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

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

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

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

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

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

Here Γ denotes the Gamma function.

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

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

It is possible to generalise $A^2$ to the (weighted) Bergman space of vector-valued functions, defined by$$A_{\alpha }^2(\mathbb{𝔻};{\mathscr{H}}):=\{f:\mathbb{𝔻}\to {\mathscr{H}}|f\text{ analytic and }\Vert f{\Vert }_{2,\alpha }<+\mathrm{\infty }\},$$and the norm on this space is given as$$\Vert f{\Vert }_{2,\alpha }={(\underset{\mathbb{𝔻}}{\int }\Vert f(z){\Vert }_{{\mathscr{H}}}^{2}d{\mu }_{\alpha }(z))}^{\frac{1}{2}}.$$The measure ${\mu }_{\alpha }$ is the same as the previous measure on the weighted Bergman space over the unit disk, ${\mathscr{H}}$ is a Hilbert space. In this case, the space is a Banach space for $0<p\le \mathrm{\infty }$ and a (reproducing kernel) Hilbert space when $p=2$.

Reproducing kernels

The reproducing kernel $k_z^{A^2}$ of A² at point $z\in \mathbb{𝔻}$ is given by:

$k_z^{A^2}(\zeta )=\frac{1}{(1-\bar{z}\zeta {)}^2}(\zeta \in \mathbb{𝔻}),$

and similarly, for $A^2({\mathbb{ℂ}}_{+})$ we have:

$k_z^{A^2({\mathbb{ℂ}}_{+})}(\zeta )=\frac{1}{(\bar{z}+\zeta {)}^2}(\zeta \in {\mathbb{ℂ}}_{+}),$

In general, if $\phi$ maps a domain $\mathrm{\Omega }$ conformally onto a domain $D$, then:

$k_z^{A^2(\mathrm{\Omega })}(\zeta )=k_{\phi (z)}^{{{𝒜}}^2(D)}(\phi (\zeta ))\overline{{\phi }^{\prime }(z)}{\phi }^{\prime }(\zeta )(z,\zeta \in \mathrm{\Omega }).$

In weighted case we have:

$k_z^{A_{\alpha }^2}(\zeta )=\frac{\alpha +1}{(1-\bar{z}\zeta {)}^{\alpha +2}}(z,\zeta \in \mathbb{𝔻}),$

and:

$k_z^{A_{\alpha }^2({\mathbb{ℂ}}_{+})}(\zeta )=\frac{2^{\alpha }(\alpha +1)}{(\bar{z}+\zeta {)}^{\alpha +2}}(z,\zeta \in {\mathbb{ℂ}}_{+}).$

In any reproducing kernel Bergman space, functions obey a certain property. It is called the reproducing property. This is expressed as a formula as follows: For any function $f\in A^2$ (respectively other Bergman spaces that are RKHS), it is true that$$f(z)=⟨f,k_z^{A^2}{⟩}_2=\underset{\mathbb{𝔻}}{\int }\frac{f(\zeta )}{(1-z\bar{\zeta }{)}^2}dA.$$

Further reading

• Bergman, Stefan (1970), The kernel function and conformal mapping, Mathematical Surveys, 5 (2nd ed.), American Mathematical Society 
• Hedenmalm, H.; Korenblum, B.; Zhu, K. (2000), Theory of Bergman Spaces, Springer, ISBN 978-0-387-98791-0, https://www.springer.com/mathematics/analysis/book/978-0-387-98791-0 
• Hazewinkel, Michiel, ed. (2001), "Bergman spaces", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=B/b120130 .

See also

• Bergman kernel
• Banach space
• Hilbert space
• Reproducing kernel Hilbert space
• Hardy space
• Dirichlet space