Bergman and Szegő Kernels

Under The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), complex $L^2(\mu )$ with the pairing $⟨f,g⟩=\int f\bar{g}d\mu$ is a Hilbert space, with the pairing linear in its first variable and conjugate symmetric (The complex $L^2$ pairing on equivalence classes, $L^2$ with the integral pairing is a Hilbert space).

Under The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), every closed linear subspace of a Hilbert space has an orthogonal projection, determined by the unique orthogonal decomposition (Orthogonal decomposition by a closed subspace, The Hilbert orthogonal projection onto a closed subspace).

Under The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), every bounded linear functional on a Hilbert space has a unique Riesz representer, with $f(x)=⟨x,y⟩$ in the first-variable-linear convention (Riesz representation for Hilbert spaces).

The Axiom of Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$ is the principle defined by The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$). It is the only choice assumption below; the polar-measure and product-Lebesgue suppliers used here state it explicitly.

If $g$ is holomorphic on $D(b,R)$ and $0<s<R$, then $g(b)=(2\pi {)}^{-1}{\int }_0^{2\pi }g(b+s{e}^{i\theta })d\theta$ (A holomorphic function equals its average on every circle inside a larger concentric holomorphy disc).

The chart surface measure of $S^1$ agrees with its polar surface measure; under the angular chart $\omega =e^{i\theta }$ its density is $d\theta$, so $\sigma (S^1)=2\pi$ (Surface integration on compact C1 hypersurfaces, Agreement with the existing polar sphere measure).

For every nonnegative Borel $h$ on ${\mathbb{R}}^2$, polar coordinates give $\int hd{\lambda }_2={\int }_0^{\mathrm{\infty }}{\int }_{S^1}h(s\omega )sd\sigma (\omega )ds$ (The polar surface set function on the unit sphere, Polar coordinates decompose Lebesgue measure into r^{n-1} dr d sigma).

Lebesgue measure is translation invariant on measurable sets (Lebesgue outer measure, Lebesgue measurability and Lebesgue measure are unchanged by translation).

Every nonnegative measurable function is the increasing limit of nonnegative simple functions, and increasing limits pass through the Lebesgue integral; thus the setwise translation invariance in [F4] extends from indicators and simple functions to nonnegative Borel integrals (Every nonnegative measurable function admits an explicit increasing sequence of simple approximations, Monotone convergence for the integral).

Borel sets in a finite Euclidean product are product-measurable; Tonelli permits iterated integration of nonnegative product-measurable functions, and the finite product of planar Lebesgue measures agrees with Euclidean Lebesgue measure under ${\mathbb{R}}^{2m}\cong ({\mathbb{R}}^2{)}^m$ (The Borel product of R^m and R^n is the Borel sigma-algebra of R^{m+n}, Tonelli's theorem for nonnegative measurable functions on a sigma-finite product, On Borel subsets of R^{m+n}, the product lambda_m times lambda_n agrees with lambda_{m+n}).

A holomorphic function is continuous, hence $\mathrm{\mid }f{\mathrm{\mid }}^2$ is Borel (A holomorphic function of several variables is continuous and separately holomorphic); the one-variable case is also supplied by Complex differentiability at a point implies continuity there.

Restricting the complex linear derivative in the definition of holomorphy to a coordinate line shows that every coordinate slice of a holomorphic function is holomorphic (Holomorphic functions on an open subset of ${\mathbb{C}}^m$).

A polydisc is the product of its coordinate discs, and ${\mathbb{C}}^m$ is identified with ${\mathbb{R}}^{2m}$ with the corresponding Lebesgue convention (Balls, polydiscs and the distinguished boundary in ${\mathbb{C}}^m$, Complex $m$-space and its real coordinate dictionary).

Given: An integer $m\ge 1$, an open set $U\subseteq {\mathbb{C}}^m$, real-valued functions $u,v\in C^2(U)$, a point $a\in U$, a neighbourhood $V\subseteq U$ of $a$ on which $u\ge v$, and $u(a)=v(a)$.

The identification of ${\mathbb{C}}^m$ with ${\mathbb{R}}^{2m}$ transports open sets and real coordinate regularity, and $C^2$ means all ordered real coordinate derivatives through order two exist and are continuous (Complex $m$-space and its real coordinate dictionary, $C^k$ maps and multi-index derivative notation in Euclidean space).

For a real $C^2$ function on a real open set, its real Hessian quadratic form is nonpositive at an interior local maximum (The Hessian is negative semidefinite at an interior local maximum).

The Wirtinger operators are ${\mathrm{\partial }}_{z_k}=\frac{1}{2}({\mathrm{\partial }}_{x_k}-i{\mathrm{\partial }}_{y_k})$ and ${\mathrm{\partial }}_{{\bar{z}}_k}=\frac{1}{2}({\mathrm{\partial }}_{x_k}+i{\mathrm{\partial }}_{y_k})$ (Wirtinger operators in ${\mathbb{C}}^m$).

If a map has continuous coordinate partial derivatives on a neighbourhood, it is totally differentiable there, and the ordinary chain rule for total derivatives applies (If all partial derivatives exist on a neighbourhood and are continuous at a point, then the map is totally differentiable there with Jacobian derivative, The chain rule for total derivatives: $D(g\circ f)(a)=Dg(f(a))\circ Df(a)$). Applied to an affine complex line and to the first coordinate derivatives of a real $C^2$ function, it gives the line-composition derivative formulas used below.

The real coordinate mixed partial derivatives of a $C^2$ function commute (Clairaut--Schwarz theorem for continuous second partial derivatives).

The Levi form is ${\mathcal{L}}_u(a;X)={\sum }_{j,k=1}^m{u}_{z_j{\bar{z}}_k}(a)X_j\overline{X_k}$, with the one-based indices of The Levi form and strict plurisubharmonicity.

Given: An integer $m\ge 1$, the Axiom of Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$, and a multi-index $\alpha =({\alpha }_0,\dots ,{\alpha }_{m-1})\in {\mathbb{N}}^m$. Write $\alpha !:={\prod }_{j<m}{\alpha }_j!$, $\mathrm{\mid }\alpha \mathrm{\mid }:={\sum }_{j<m}{\alpha }_j$, and $z^{\alpha }:={\prod }_{j<m}{z}_j^{{\alpha }_j}$ using the conventions of $C^k$ maps and multi-index derivative notation in Euclidean space and Integer powers in the complex field.

For $d\ge 1$, $\beta \in {\mathbb{N}}^d$ and $t\in \mathbb{N}$, write $I_d(\beta ,t):={\int }_{{\mathbb{B}}^d}\mathrm{\mid }{z}^{\beta }{\mathrm{\mid }}^2(1-\mathrm{\mid }z{\mathrm{\mid }}^2{)}^{t}d{\lambda }_{2d}(z)$.

The only choice principle assumed is ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). It is required by the polar-coordinate, chart/polar surface, sigma-finiteness and product-Lebesgue suppliers below; no full Axiom of Choice or arbitrary-index selection is used.

The complex-real identification and Euclidean norm are those of Complex $m$-space and its real coordinate dictionary, and the open unit ball ${\mathbb{B}}^m$ and open unit polydisc ${\mathbb{D}}^m$ are those of Balls, polydiscs and the distinguished boundary in ${\mathbb{C}}^m$.

For every nonnegative Borel function on ${\mathbb{R}}^n$, polar coordinates give $\int fd{\lambda }_n={\int }_0^{\mathrm{\infty }}{\int }_{S^{n-1}}f(r\omega )r^{n-1}d\sigma (\omega )dr$ (The polar surface set function on the unit sphere, Polar coordinates decompose Lebesgue measure into r^{n-1} dr d sigma).

On each angular chart $X(\theta )=(\cos \theta ,\sin \theta )$ of $S^1$, the derivative has norm $1$, so the surface density is $1$; summing a partition of unity over one full turn gives chart surface measure ${\int }_0^{2\pi }1d\theta =2\pi$. The chart surface measure equals the polar measure $\sigma$ (Surface integration on compact C1 hypersurfaces, Agreement with the existing polar sphere measure).

For $p,q>0$, the Euler Beta integral is $B(p,q)={\int }_0^1{t}^{p-1}(1-t{)}^{q-1}dt$ and converges (Euler's real Beta integral, Euler's Beta integral converges exactly for two positive parameters); change of variables is valid on the improper interval after compact truncation (Change of variable in an improper integral).

The Beta-Gamma identity is $B(p,q)=\mathrm{\Gamma }(p)\mathrm{\Gamma }(q)\mathrm{/}\mathrm{\Gamma }(p+q)$, and $\mathrm{\Gamma }(n+1)=n!$ for every $n\in \mathbb{N}$ (The real Beta--Gamma identity, $\mathrm{\Gamma }(n+1)=n!$ for every natural number $n$, The factorial $n!$ and the falling factorial $n^{\underset{‾}{k}}$, defined by recursion in $\mathbb{N}$).

Nonnegative product-measurable functions satisfy Tonelli's iterated-integral identity on sigma-finite measure spaces; under ${\mathrm{A}\mathrm{C}}_{\omega }$, product Lebesgue measure agrees with Euclidean Lebesgue measure on Borel sets, and equality of these measures gives equality of nonnegative integrals by simple approximation and monotone convergence (Tonelli's theorem for nonnegative measurable functions on a sigma-finite product, The Borel product of R^m and R^n is the Borel sigma-algebra of R^{m+n}, On Borel subsets of R^{m+n}, the product lambda_m times lambda_n agrees with lambda_{m+n}, Lebesgue measure is sigma-finite, and every metrically bounded subset of ${\mathbb{R}}^n$ has finite outer measure, Every nonnegative measurable function admits an explicit increasing sequence of simple approximations, Monotone convergence for the integral).

In real coordinates the displayed integrands are nonnegative polynomials times indicators of open balls or polydiscs. Finite sums and products of coordinate polynomials are continuous by the $C^k$ algebra theorem; the preimage criterion makes continuous maps Borel measurable, and products of measurable functions remain measurable ($C^k$ Euclidean maps and diffeomorphisms, $C^k$ Euclidean maps are closed under componentwise algebra and composition, The Borel sigma-algebra of a topological space, A measurable function between measurable spaces, A continuous map has Borel preimages of Borel sets, Arithmetic and lattice operations preserve measurability whenever they are defined, Balls, polydiscs and the distinguished boundary in ${\mathbb{C}}^m$).

A nonnegative real has a unique nonnegative square root (Existence and uniqueness of $n$-th roots: a unique $a^{1\mathrm{/}n}\ge 0$ with $(a^{1\mathrm{/}n}{)}^n=a$).

Induction on the positive integer dimension is valid (The principle of mathematical induction).

Multi-indices, their factorials and lengths, and complex nonnegative integer powers have the conventions used in the statement ($C^k$ maps and multi-index derivative notation in Euclidean space, The factorial $n!$ and the falling factorial $n^{\underset{‾}{k}}$, defined by recursion in $\mathbb{N}$, Integer powers in the complex field).

Given: $m\ge 1$, the Axiom of Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$, an open set $\mathrm{\Omega }\subseteq {\mathbb{C}}^m$, a nonempty compact set $K\subseteq \mathrm{\Omega }$, and a holomorphic function $f\in L^2(\mathrm{\Omega })$.

The only choice principle is ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). It is used through the local mean-value lemma and the complex $L^2$ Hilbert-space supplier; no full Axiom of Choice or sequence-selection principle is used.

Read ${\mathbb{C}}^m$ as ${\mathbb{R}}^{2m}$ with its Euclidean metric and norm (Complex $m$-space and its real coordinate dictionary).

Open and closed polydiscs and balls have the coordinatewise definitions of Balls, polydiscs and the distinguished boundary in ${\mathbb{C}}^m$.

For a holomorphic $g$ on an open set containing a closed polydisc ${\bar{\mathrm{\Delta }}}_{\mathbf{r}}(a)$, the local mean-value lemma gives $\mathrm{\mid }g(a)\mathrm{\mid }\le ({\pi }^m{\prod }_{j<m}{r}_j^2{)}^{-1\mathrm{/}2}\mathrm{\parallel }g{\mathrm{\parallel }}_{L^2({\mathrm{\Delta }}_{\mathbf{r}}(a))}$ (The mean-value $L^2$ bound for holomorphic functions on a polydisc).

If $g$ is holomorphic on ${\mathrm{\Delta }}_{\mathit{\rho }}(a)$, $r_j<{\rho }_j$, and $M={\sup }_{{\mathrm{\Gamma }}_{\mathbf{r}}(a)}\mathrm{\mid }g\mathrm{\mid }$, then $\mathrm{\mid }{\mathrm{\partial }}_z^{\alpha }g(a)\mathrm{\mid }\le \alpha !M{\prod }_{j<m}{r}_j^{-{\alpha }_j}$ (Cauchy estimates for mixed derivatives on a polydisc).

Holomorphic functions on open subsets of ${\mathbb{C}}^m$ are continuous (Holomorphic functions of several variables are smooth and their complex derivatives are holomorphic).

Multi-index notation and ${\mathrm{\partial }}^{0}f=f$ use the convention of $C^k$ maps and multi-index derivative notation in Euclidean space.

Under ${\mathrm{A}\mathrm{C}}_{\omega }$, complex $L^2(\mathrm{\Omega })$ with its quotient norm is a Hilbert space, so the norm satisfies the triangle inequality and convergence implies the Cauchy property ($L^2$ with the integral pairing is a Hilbert space, Complex Lp classes and Euclidean test-function conventions).

If $0\le g\le h$ are measurable, then $\int g\le \int h$ (Monotonicity and nonnegative homogeneity of the nonnegative integral).

In a metric space, distance to a nonempty set is $1$-Lipschitz and hence continuous by the $\epsilon$-$\delta$ definition (Metric space: $d(x,y)=0$ iff $x=y$, symmetry, and the triangle inequality; pseudometric and ultrametric, $\mathrm{\mid }d(x,A)-d(y,A)\mathrm{\mid }\le d(x,y)$, so the distance to a fixed nonempty set is $1$-Lipschitz, Continuity of a map between metric spaces, at a point and globally, in the $\epsilon$-$\delta$ form).

In a metric space, openness gives a ball about each point (The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement, Open ball, closed ball and sphere in a metric space).

A compact subset is compact in its subspace metric, and a continuous real-valued function on a nonempty compact metric space attains its minimum (A subset of a metric space is open in the subspace metric exactly when it is the trace of an open set of the ambient space, and it is compact as a metric space in its own right exactly when every family of open subsets of the ambient space covering it, indexed or not, has finitely many members covering it, A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value).

Every Euclidean closed ball of positive radius in ${\mathbb{R}}^{2m}$ is compact (For $n\ge 1$, every Euclidean closed ball and every Euclidean sphere of positive radius is compact).

A Cauchy sequence in $\mathbb{C}$ converges (The complex plane is complete, and convergence is equivalent to convergence of real and imaginary parts).

A locally uniform limit of holomorphic functions is holomorphic with locally uniform convergence of every complex derivative (Locally uniform limits of holomorphic functions are holomorphic, with locally uniform convergence of all derivatives).

For nonnegative measurable functions, Fatou's lemma gives $\int \mathrm{lim\hspace{0.17em}inf}{g}_n\le \mathrm{lim\hspace{0.17em}inf}\int g_n$ (Fatou's lemma).

Sums and scalar multiples of holomorphic functions are holomorphic (Sums, products and nonvanishing quotients of holomorphic functions are holomorphic).

The only choice principle is the Axiom of Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). The Hardy boundary theorem, Hardy-space and torus conventions, $L^2$ Hilbert structure, and Riesz/Szegő construction below are used under this hypothesis; no full Axiom of Choice or arbitrary-index selection is used.

The unit circle is parametrized by $\zeta =e^{2\pi it}$, normalized Haar measure is $dm=dt$, and $m(\mathbb{T})=1$ (The one-dimensional torus and its normalized Haar integral).

$H^p(\mathbb{D})$ is defined by the supremum of the radial $L^p$ means; in particular $H^2(\mathbb{D})$ is a normed complex vector space and $H^2(\mathbb{D})\subseteq H^1(\mathbb{D})$ (Analytic Hardy spaces on the unit disc, Radial p-means of a holomorphic function are nondecreasing).

For $f\in H^2(\mathbb{D})$, the Fatou boundary theorem gives $f^{\ast }\in L^2(\mathbb{T},m)$, $\mathrm{\parallel }{f}_r-f^{\ast }{\mathrm{\parallel }}_2\to 0$ as $r\uparrow 1$, and $\mathrm{\parallel }{f}^{\ast }{\mathrm{\parallel }}_2=\mathrm{\parallel }f{\mathrm{\parallel }}_{H^2}$ (Fatou's boundary theorem for analytic Hardy spaces).

For $f\in H^1(\mathbb{D})$, its boundary function satisfies the Cauchy representation $f(w)=\frac{1}{2\pi i}{\oint }_{\mathbb{T}}\frac{f^{\ast }(\zeta )}{\zeta -w}d\zeta ={\int }_{\mathbb{T}}\frac{f^{\ast }(\zeta )}{1-w\bar{\zeta }}dm(\zeta )$ (Cauchy representation of an $H^1$ function from its boundary values).

$L^2(\mathbb{T},m)$ with $⟨g,h⟩=\int g\bar{h}dm$ is a complex Hilbert space, the pairing is linear in its first variable, and it satisfies Cauchy–Schwarz ($L^2$ with the integral pairing is a Hilbert space).

The Hardy boundary space in the Szegő definition is the $L^2$ closure of traces from $\mathcal{O}(\mathbb{D})\cap C(\bar{\mathbb{D}})$ (The Hardy boundary space, Szegő projection and Szegő kernel on a smoothly bounded domain).

Every bounded linear functional on a complex Hilbert space has a unique representing vector $y$ with $L(x)=⟨x,y⟩$ under the first-variable-linear convention (Riesz representation for Hilbert spaces).

A function on an open subset of $\mathbb{C}$ is holomorphic when it is complex differentiable at each point (Complex differentiability at a point, the complex derivative, holomorphic functions, and entire functions).

Complex differentiability implies continuity (Complex differentiability at a point implies continuity there).

Conjugation is involutive and $z\bar{z}=\mathrm{\mid }z{\mathrm{\mid }}^2$, so $\mathrm{\mid }\bar{z}\mathrm{\mid }=\mathrm{\mid }z\mathrm{\mid }$; modulus is multiplicative and subadditive, hence the reverse triangle inequality follows (Conjugation is an involutive real-field automorphism, $z\bar{z}=\mathrm{\mid }z{\mathrm{\mid }}^2$, and modulus is definite, multiplicative, and subadditive).

The metric on $\mathbb{C}$ is the Euclidean metric on ${\mathbb{R}}^2$, so $\bar{\mathbb{D}}$ is a compact Euclidean closed ball (The Euclidean metric, convergence, Cauchy sequences, and continuity on the complex plane, For $n\ge 1$, every Euclidean closed ball and every Euclidean sphere of positive radius is compact).

A continuous real-valued function on a nonempty compact metric space is bounded and attains its maximum (A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value).

A rational function is holomorphic wherever its denominator is nonzero (Complex polynomials are entire with the power-rule derivative, and rational functions are holomorphic wherever their denominator is nonzero).

Every Cauchy sequence in $\mathbb{C}$ converges (The complex plane is complete, and convergence is equivalent to convergence of real and imaginary parts).

A locally uniform limit of holomorphic functions is holomorphic (Locally uniform limits of holomorphic functions are holomorphic, with locally uniform convergence of all derivatives).