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Kernel in unit disc

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In the [[mathematics|mathematical]] study of [[several complex variables]], the '''Bergman kernel''', named after [[Stefan Bergman]], is the [[reproducing kernel]] for the [[Hilbert space]] ([[Reproducing kernel Hilbert space|RKHS]]) of all [[square integrable]] [[holomorphic function]]s on a domain ''D'' in '''C'''''n''.

In detail, let [[Lp space|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,
:<math>L^{2,h}(D) = L^2(D)\cap H(D)</math>
where ''H''(''D'') is the space of holomorphic functions in ''D''. Then ''L''2,''h''(''D'') is a Hilbert space: it is a [[closed set|closed]] linear subspace of ''L''2(''D''), and therefore [[complete metric space|complete]] in its own right. This follows from the fundamental estimate, that for a holomorphic square-integrable function ''&fnof;'' in ''D''
{{NumBlk|:|<math>\sup_{z\in K} |f(z)| \le C_K\|f\|_{L^2(D)}</math>|{{EquationRef|1}}}}
for every [[compact set|compact]] subset ''K'' of ''D''. Thus convergence of a sequence of holomorphic functions in ''L''2(''D'') implies also [[compact convergence]], and so the limit function is also holomorphic.

Another consequence of ({{EquationRef|1}}) is that, for each ''z'' ∈ ''D'', the evaluation
:<math>\operatorname{ev}_z : f\mapsto f(z)</math>
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
:<math>\operatorname{ev}_z f = \int_D f(\zeta)\overline{\eta_z(\zeta)}\,d\mu(\zeta).</math>
The Bergman kernel ''K'' is defined by
:<math>K(z,\zeta) = \overline{\eta_z(\zeta)}.</math>
The kernel ''K''(''z'',ζ) is holomorphic in ''z'' and antiholomorphic in ζ, and satisfies
:<math>f(z) = \int_D K(z,\zeta)f(\zeta)\,d\mu(\zeta).</math>

One key observation about this picture is that ''L''2,''h''(''D'') may be identified with the space of <math>L^2</math> holomorphic (n,0)-forms on D, via multiplication by <math>dz^1\wedge \cdots \wedge dz^n</math>. Since the <math>L^2</math> 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
<math display="block">K(z,\zeta) = \frac{1}{\pi}\frac{1}{(1-z\bar\zeta)^2}.</math>

==See also==
* [[Bergman metric]]
* [[Bergman space]]
* [[Szegő kernel]]

==References==
* {{Citation | last1=Krantz | first1=Steven G. | authorlink=Steven Krantz|title=Function Theory of Several Complex Variables | publisher=[[American Mathematical Society]] | location=Providence, R.I. | isbn=978-0-8218-2724-6 | year=2002}}.
* {{springer|title=Bergman kernel function|id=B/b015560|first=E.M.|last=Chirka}}.

[[Category:Several complex variables]]

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imported>Tito Omburo: Kernel in unit disc