Bergman kernel explained

In the mathematical study of several complex variables, the Bergman kernel, named after Stefan Bergman, is the reproducing kernel for the Hilbert space (RKHS) of all square integrable holomorphic functions on a domain D in Cⁿ.

In detail, let L²(D) be the Hilbert space of square integrable functions on D, and let L^2,h(D) denote the subspace consisting of holomorphic functions in L²(D): that is,

L^2,h(D)=L^2(D)\capH(D)

where H(D) is the space of holomorphic functions in D. Then L^2,h(D) is a Hilbert space: it is a closed linear subspace of L²(D), and therefore complete in its own right. This follows from the fundamental estimate, that for a holomorphic square-integrable function ƒ in Dfor every compact subset K of D. Thus convergence of a sequence of holomorphic functions in L²(D) implies also compact convergence, and so the limit function is also holomorphic.

Another consequence of is that, for each z ∈ D, the evaluation

\operatorname{ev}_z:f\mapstof(z)

is a continuous linear functional on L^2,h(D). By the Riesz representation theorem, this functional can be represented as the inner product with an element of L^2,h(D), which is to say that

\operatorname{ev}_zf=\int_Df(\zeta)\overline{η_{z(\zeta)}d\mu(\zeta).}

The Bergman kernel K is defined by

K(z,\zeta)=\overline{η_z(\zeta)}.

The kernel K(z,ζ) is holomorphic in z and antiholomorphic in ζ, and satisfies

f(z)=\int_DK(z,\zeta)f(\zeta)d\mu(\zeta).

One key observation about this picture is that L^2,h(D) may be identified with the space of

L²

holomorphic (n,0)-forms on D, via multiplication by

dz^1\wedge … \wedgedzⁿ

. Since the

L²

inner product on this space is manifestly invariant under biholomorphisms of D, the Bergman kernel and the associated Bergman metric are therefore automatically invariant under the automorphism group of the domain.

The Bergman kernel for the unit disc D is the function $K(z,\zeta) = \frac\frac.$

Bergman kernel explained

See also

References