Transformation law of the Bergman kernel under a biholomorphism

The only choice principle assumed is ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). Under it, the Bergman spaces are Hilbert spaces of unique holomorphic representatives with first-variable-linear inner products, Riesz sections, and reproducing kernels; the real change-of-variables supplier and Hilbert-adjoint definition also use only ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Bergman space $A^2(\mathrm{\Omega })$ and the Bergman kernel, Reproducing property, Bergman projection and the extremal characterization, A C^1 diffeomorphism satisfies the change-of-variables formula for L^1 functions, The Hilbert-space adjoint of a bounded operator).

A biholomorphism and its inverse are holomorphic maps. Their components are holomorphic scalar functions, hence smooth in real coordinates, so under ${\mathbb{C}}^m\cong {\mathbb{R}}^{2m}$ both are real $C^1$ maps and $F$ is a $C^1$ diffeomorphism (Biholomorphic maps between open sets in ${\mathbb{C}}^m$, A map into ${\mathbb{C}}^n$ is holomorphic exactly when each of its components is, Holomorphic functions of several variables are smooth and their complex derivatives are holomorphic, $C^k$ Euclidean maps and diffeomorphisms, Complex $m$-space and its real coordinate dictionary).

The entries of the complex Jacobian matrix are the component derivatives ${\mathrm{\partial }}_{z_k}{F}_j$, which are holomorphic; its determinant is a finite sum of products of these entries, so $J_F$ is holomorphic. Composition of holomorphic maps is holomorphic (Holomorphic maps ${\mathbb{C}}^m\to {\mathbb{C}}^n$ and the complex Jacobian matrix, A map into ${\mathbb{C}}^n$ is holomorphic exactly when each of its components is, Holomorphic functions of several variables are smooth and their complex derivatives are holomorphic, Sums, products and nonvanishing quotients of holomorphic functions are holomorphic, The composite of holomorphic maps is holomorphic and its complex Jacobian is the product).

The complex Jacobian determinant is multiplicative under composition. Applying this to $F^{-1}\circ F=\mathrm{id}$ gives $J_{F^{-1}}(F(z))J_F(z)=1$, hence $J_F(z)\mathrm{\ne }0$ (The complex Jacobian determinant of a composite of equidimensional holomorphic maps is the product).

For a $\mathbb{C}$-linear map with complex determinant $J$, the real determinant under ${\mathbb{C}}^m\cong {\mathbb{R}}^{2m}$ is $\mathrm{\mid }J{\mathrm{\mid }}^2$ (The real Jacobian determinant of a complex-linear automorphism is the squared modulus of its complex determinant).

For the real $C^1$ diffeomorphism underlying $F$, Lebesgue change of variables gives ${\int }_{{\mathrm{\Omega }}^{\mathrm{\prime }}}q(\zeta )d{\lambda }_{2m}(\zeta )={\int }_{\mathrm{\Omega }}q(F(z))\mathrm{\mid }{\det }_{\mathbb{R}}DF(z)\mathrm{\mid }d{\lambda }_{2m}(z)$ for every complex $q\in L^1({\mathrm{\Omega }}^{\mathrm{\prime }})$. The Bergman measures are restrictions of this Lebesgue measure (A C^1 diffeomorphism satisfies the change-of-variables formula for L^1 functions, The Bergman space $A^2(\mathrm{\Omega })$ and the Bergman kernel, Complex $m$-space and its real coordinate dictionary).

The Bergman section $k_z=K_{\mathrm{\Omega }}(\cdot ,z)$ satisfies $f(z)=⟨f,k_z⟩$; the pairing is linear in its first variable (The Bergman space $A^2(\mathrm{\Omega })$ and the Bergman kernel, Reproducing property, Bergman projection and the extremal characterization).

For a bounded linear operator $T:H\to K$ between Hilbert spaces, its adjoint is characterized by $⟨Tx,y{⟩}_K=⟨x,T^{\ast }y{⟩}_H$. An isometry is bounded with bound $1$ (The Hilbert-space adjoint of a bounded operator, A bounded linear operator between normed spaces).

If $g,h\in A^2({\mathrm{\Omega }}^{\mathrm{\prime }})$, then $g\bar{h}\in L^1({\mathrm{\Omega }}^{\mathrm{\prime }})$ by Cauchy–Schwarz (Cauchy–Schwarz: $\mathrm{\mid }⟨x,y⟩\mathrm{\mid }\le \mathrm{\parallel }x\mathrm{\parallel }\mathrm{\parallel }y\mathrm{\parallel }$, with equality exactly for dependent pairs).