[proofplan] The argument is a synthesis of three Bell-theoretic inputs. First, Condition R is unpacked as global Sobolev regularity for the Bergman projection on each bounded smoothly bounded pseudoconvex domain. Second, the Bell-Ligocka boundary [extension theorem](/theorems/59), a previously established result about biholomorphisms between Condition R domains, transfers this projection regularity to smooth boundary extensions of the biholomorphism and its inverse. Third, Bell's kernel criterion identifies this same projection regularity with smooth extendability of the Bergman projection kernel, namely the Bergman kernel representing the projection, on the product closure away from the boundary diagonal. [/proofplan] [step:Convert Condition R into global regularity of the Bergman projections] Let \begin{align*} P_{\Omega}:L^2(\Omega)&\to A^2(\Omega), & P_{\widetilde{\Omega}}:L^2(\widetilde{\Omega})&\to A^2(\widetilde{\Omega}) \end{align*} denote the Bergman projections, where $A^2(\Omega)$ and $A^2(\widetilde{\Omega})$ are the closed subspaces of square-integrable holomorphic functions. For an integer $k\geq 0$, let $W^{k,2}(\Omega)$ denote the [Sobolev space](/page/Sobolev%20Space) of functions on $\Omega$ whose distributional derivatives $D^\alpha u$ of order $|\alpha|\leq k$ belong to $L^2(\Omega)$; define $W^{k,2}(\widetilde{\Omega})$ analogously. For this proof, Condition R for $\Omega$ means the following global regularity property: for every integer $s\geq 0$ there is an integer $m_{\Omega}(s)\geq 0$ such that \begin{align*} P_{\Omega}:W^{m_{\Omega}(s),2}(\Omega)&\to W^{s,2}(\Omega) \end{align*} is continuous. Equivalently, $P_{\Omega}$ maps $C^{\infty}(\overline{\Omega})$ into $C^{\infty}(\overline{\Omega})$ with the corresponding Fréchet-space continuity. Since $\widetilde{\Omega}$ also satisfies Condition R, for every integer $s\geq 0$ there is an integer $m_{\widetilde{\Omega}}(s)\geq 0$ such that \begin{align*} P_{\widetilde{\Omega}}:W^{m_{\widetilde{\Omega}}(s),2}(\widetilde{\Omega})&\to W^{s,2}(\widetilde{\Omega}) \end{align*} is continuous. The projection-regularity hypotheses needed below are therefore satisfied: both domains are bounded, smoothly bounded, pseudoconvex, and their Bergman projections satisfy Condition R in this Sobolev sense. [guided] We first translate the hypothesis into the analytic regularity statement used by Bell theory. The map \begin{align*} P_{\Omega}:L^2(\Omega)&\to A^2(\Omega) \end{align*} is the [orthogonal projection](/theorems/437) from square-integrable functions onto square-integrable holomorphic functions. In this theorem, Condition R is not being used as an undefined slogan; it is the global Sobolev regularity of the Bergman projection. For an integer $k\geq 0$, the symbol $W^{k,2}(\Omega)$ denotes the [Sobolev space](/page/Sobolev%20Space) of functions on $\Omega$ whose distributional derivatives up to order $k$ are square-integrable; the space $W^{k,2}(\widetilde{\Omega})$ is defined in the same way on $\widetilde{\Omega}$. Thus Condition R for $\Omega$ says that for each target order $s\geq 0$ there is an input order $m_{\Omega}(s)\geq 0$ with \begin{align*} P_{\Omega}:W^{m_{\Omega}(s),2}(\Omega)&\to W^{s,2}(\Omega) \end{align*} continuous. Equivalently, $P_{\Omega}$ preserves smoothness up to the boundary. Since the theorem assumes Condition R for $\widetilde{\Omega}$ as well, for each integer $s\geq 0$ there is an integer $m_{\widetilde{\Omega}}(s)\geq 0$ with \begin{align*} P_{\widetilde{\Omega}}:W^{m_{\widetilde{\Omega}}(s),2}(\widetilde{\Omega})&\to W^{s,2}(\widetilde{\Omega}) \end{align*} continuous. These are exactly the projection-regularity hypotheses used in the Bell-Ligocka boundary [extension theorem](/theorems/59), together with boundedness, smooth boundary, and pseudoconvexity. [/guided] [/step] [step:Apply Bell's biholomorphic extension theorem to obtain smooth boundary values] Let \begin{align*} F:\Omega&\to\widetilde{\Omega} \end{align*} be the given biholomorphism, and let \begin{align*} G:\widetilde{\Omega}&\to\Omega \end{align*} be its holomorphic inverse. We use the Bell-Ligocka boundary [extension theorem](/theorems/59) in the following form: if $D$ and $D'$ are bounded smoothly bounded pseudoconvex domains satisfying Condition R in the Sobolev sense above and $H:D\to D'$ is biholomorphic, then $H$ extends to a $C^{\infty}$ map $\overline{D}\to\overline{D'}$. This is a one-directional [extension theorem](/theorems/59) for biholomorphisms; it does not include the kernel equivalence asserted later in the present criterion. Applying it with $D=\Omega$, $D'=\widetilde{\Omega}$, and $H=F$ gives a map \begin{align*} \overline{F}:\overline{\Omega}&\to\overline{\widetilde{\Omega}} \end{align*} of class $C^{\infty}$. Applying it with $D=\widetilde{\Omega}$, $D'=\Omega$, and $H=G$ gives a map \begin{align*} \overline{G}:\overline{\widetilde{\Omega}}&\to\overline{\Omega} \end{align*} of class $C^{\infty}$. The theorem applies in both directions because both domains are smoothly bounded pseudoconvex domains satisfying Condition R, and both $F$ and $G$ are biholomorphic. [guided] The regularity result now used is the Bell-Ligocka boundary [extension theorem](/theorems/59). Its precise input is a pair of bounded smoothly bounded pseudoconvex domains $D$ and $D'$ satisfying Condition R in the Sobolev projection sense stated above, together with a biholomorphic map $H:D\to D'$. Its conclusion is that $H$ extends to a $C^{\infty}$ map $\overline{D}\to\overline{D'}$. This theorem is narrower than the present criterion: it supplies smooth boundary values for a biholomorphism, while the present theorem also records invertibility on the closures and the equivalent kernel formulation of Condition R. We verify the hypotheses for $F$. The domains $\Omega$ and $\widetilde{\Omega}$ are smoothly bounded and pseudoconvex by hypothesis, both satisfy Condition R by hypothesis, and \begin{align*} F:\Omega&\to\widetilde{\Omega} \end{align*} is biholomorphic. Hence the Bell-Ligocka theorem gives a $C^{\infty}$ extension \begin{align*} \overline{F}:\overline{\Omega}&\to\overline{\widetilde{\Omega}}. \end{align*} Let \begin{align*} G:\widetilde{\Omega}&\to\Omega \end{align*} be the inverse biholomorphism. The same hypotheses hold with the two domains interchanged, so applying the Bell-Ligocka theorem to $G$ gives a $C^{\infty}$ extension \begin{align*} \overline{G}:\overline{\widetilde{\Omega}}&\to\overline{\Omega}. \end{align*} This is where pseudoconvexity and Condition R are used: they are the analytic assumptions that make the Bergman projection regular enough for the Bell-Ligocka [extension theorem](/theorems/59) to force boundary smoothness of biholomorphisms. [/guided] [/step] [step:Show the two boundary extensions are inverse maps] Let $\operatorname{id}_{\Omega}:\Omega\to\Omega$, $\operatorname{id}_{\widetilde{\Omega}}:\widetilde{\Omega}\to\widetilde{\Omega}$, $\operatorname{id}_{\overline{\Omega}}:\overline{\Omega}\to\overline{\Omega}$, and $\operatorname{id}_{\overline{\widetilde{\Omega}}}:\overline{\widetilde{\Omega}}\to\overline{\widetilde{\Omega}}$ denote the corresponding identity maps. On $\Omega$ we have $G\circ F=\operatorname{id}_{\Omega}$, and on $\widetilde{\Omega}$ we have $F\circ G=\operatorname{id}_{\widetilde{\Omega}}$. Since $\Omega$ is dense in $\overline{\Omega}$ and $\widetilde{\Omega}$ is dense in $\overline{\widetilde{\Omega}}$, continuity of $\overline{F}$ and $\overline{G}$ gives \begin{align*} \overline{G}\circ\overline{F}&=\operatorname{id}_{\overline{\Omega}}, & \overline{F}\circ\overline{G}&=\operatorname{id}_{\overline{\widetilde{\Omega}}}. \end{align*} Thus $\overline{F}$ is a bijection with inverse $\overline{G}$. Because both maps are $C^{\infty}$, $\overline{F}:\overline{\Omega}\to\overline{\widetilde{\Omega}}$ is a $C^{\infty}$ diffeomorphism. The extension is unique: if $F_1,F_2:\overline{\Omega}\to\overline{\widetilde{\Omega}}$ are continuous extensions of $F$, then $F_1=F_2$ on the [dense subset](/page/Dense%20Subset) $\Omega$, and continuity implies $F_1=F_2$ on all of $\overline{\Omega}$. [guided] The Bell-Ligocka theorem has given smooth maps on the closures, but we still have to prove that these two smooth maps are inverse maps on the closures. Define the identity maps \begin{align*} \operatorname{id}_{\Omega}:\Omega&\to\Omega, & \operatorname{id}_{\widetilde{\Omega}}:\widetilde{\Omega}&\to\widetilde{\Omega}, \\ \operatorname{id}_{\overline{\Omega}}:\overline{\Omega}&\to\overline{\Omega}, & \operatorname{id}_{\overline{\widetilde{\Omega}}}:\overline{\widetilde{\Omega}}&\to\overline{\widetilde{\Omega}}. \end{align*} Since $G$ is the inverse biholomorphism of $F$, the compositions satisfy \begin{align*} G\circ F&=\operatorname{id}_{\Omega} \end{align*} on $\Omega$, and \begin{align*} F\circ G&=\operatorname{id}_{\widetilde{\Omega}} \end{align*} on $\widetilde{\Omega}$. The point of passing to closures is that these identities persist by density and continuity. The set $\Omega$ is dense in $\overline{\Omega}$, and $\overline{G}\circ\overline{F}:\overline{\Omega}\to\overline{\Omega}$ and $\operatorname{id}_{\overline{\Omega}}:\overline{\Omega}\to\overline{\Omega}$ are continuous maps that agree on the [dense subset](/page/Dense%20Subset) $\Omega$. Therefore they agree on all of $\overline{\Omega}$: \begin{align*} \overline{G}\circ\overline{F}&=\operatorname{id}_{\overline{\Omega}}. \end{align*} The same argument on the [dense subset](/page/Dense%20Subset) $\widetilde{\Omega}\subset\overline{\widetilde{\Omega}}$ gives \begin{align*} \overline{F}\circ\overline{G}&=\operatorname{id}_{\overline{\widetilde{\Omega}}}. \end{align*} Hence $\overline{F}$ is bijective and its inverse is $\overline{G}$. Since both $\overline{F}$ and $\overline{G}$ are $C^{\infty}$ maps, $\overline{F}:\overline{\Omega}\to\overline{\widetilde{\Omega}}$ is a $C^{\infty}$ diffeomorphism. We also record uniqueness of the extension. If $F_1,F_2:\overline{\Omega}\to\overline{\widetilde{\Omega}}$ are continuous extensions of the original biholomorphism $F$, then $F_1=F_2$ on $\Omega$. Since $\Omega$ is dense in $\overline{\Omega}$ and both maps are continuous, equality extends from $\Omega$ to every point of $\overline{\Omega}$. Thus the smooth extension obtained above is unique. [/guided] [/step] [step:Identify Condition R with Bell kernel smoothness off the boundary diagonal] Let \begin{align*} K_{\Omega}:\Omega\times\Omega&\to\mathbb{C}, & K_{\widetilde{\Omega}}:\widetilde{\Omega}\times\widetilde{\Omega}&\to\mathbb{C} \end{align*} be the Bergman kernels of $\Omega$ and $\widetilde{\Omega}$. In Bell's terminology, the phrase Bergman projection kernel means precisely this reproducing kernel viewed as the integral kernel of the Bergman projection. Thus, for each domain $D$ under discussion, the Bergman projection kernel is the map \begin{align*} K_D:D\times D&\to\mathbb{C} \end{align*} representing $P_D$ on the standard dense test class by the Bergman projection formula. Define the boundary diagonal and the off-diagonal product-closure region by \begin{align*} \Delta_{\partial D}&:=\{(p,p):p\in\partial D\}, & \mathcal{O}_{D}&:=(\overline{D}\times\overline{D})\setminus\Delta_{\partial D}. \end{align*} Bell's kernel regularity criterion states that, for a bounded smoothly bounded pseudoconvex domain $D$, the Sobolev projection regularity formulation of Condition R is equivalent to the following kernel condition: for every pair of multi-indices $\alpha$ and $\beta$, the mixed derivative $\partial_z^{\alpha}\partial_{\overline{w}}^{\beta}K_D(z,w)$ extends to a $C^{\infty}$ function on $\mathcal{O}_{D}$. This is exactly the sense in which the Bergman projection kernel extends smoothly away from the unavoidable boundary singularity on the diagonal. The hypotheses of this criterion are satisfied for $D=\Omega$ and for $D=\widetilde{\Omega}$, because both domains are bounded, smoothly bounded, and pseudoconvex. Applying the criterion to both domains gives the kernel formulation stated in the theorem. [guided] It remains to connect the projection formulation of Condition R with the kernel formulation in the statement. The terminology can be a source of ambiguity, so we fix it explicitly. Define \begin{align*} K_{\Omega}:\Omega\times\Omega&\to\mathbb{C}, & K_{\widetilde{\Omega}}:\widetilde{\Omega}\times\widetilde{\Omega}&\to\mathbb{C} \end{align*} to be the Bergman kernels of the two domains. These kernels are also the Bergman projection kernels: they are the integral kernels that represent the Bergman projections $P_{\Omega}$ and $P_{\widetilde{\Omega}}$ on the usual dense test class. Thus the theorem statement's phrase 'Bergman projection kernel' refers to the same object denoted here by $K_D$. For a domain $D$, define the boundary diagonal and the off-diagonal product-closure region by \begin{align*} \Delta_{\partial D}&:=\{(p,p):p\in\partial D\}, & \mathcal{O}_{D}&:=(\overline{D}\times\overline{D})\setminus\Delta_{\partial D}. \end{align*} Bell's kernel regularity criterion says the following precise statement. If $D$ is a bounded smoothly bounded pseudoconvex domain and $K_D:D\times D\to\mathbb{C}$ is its Bergman projection kernel, then $D$ satisfies Condition R in the Sobolev projection sense used above if and only if, for every pair of multi-indices $\alpha$ and $\beta$, the mixed derivative $\partial_z^{\alpha}\partial_{\overline{w}}^{\beta}K_D(z,w)$ extends to a $C^{\infty}$ function on \begin{align*} \mathcal{O}_{D}&=(\overline{D}\times\overline{D})\setminus\{(p,p):p\in\partial D\}. \end{align*} This is exactly the smooth extendability 'away from the unavoidable boundary singularity on the diagonal' mentioned in the theorem statement: the only excluded points are pairs where both variables approach the same boundary point. We now verify the hypotheses before applying the criterion. The domains $\Omega$ and $\widetilde{\Omega}$ are bounded smoothly bounded pseudoconvex domains by the theorem hypotheses, and $K_{\Omega}$ and $K_{\widetilde{\Omega}}$ are their Bergman projection kernels. Therefore Bell's criterion applies to both domains. It identifies the Sobolev Condition R used in the first step with the mixed-derivative smooth extendability of the Bergman projection kernel on the corresponding off-diagonal product-closure region. [/guided] [/step] [step:Conclude the Bell boundary regularity criterion] The preceding steps prove both assertions. The biholomorphism $F$ extends uniquely to a $C^{\infty}$ diffeomorphism $\overline{\Omega}\to\overline{\widetilde{\Omega}}$, and Condition R is equivalent, in Bell's sense, to off-diagonal smooth extendability of the Bergman projection kernel on the product closure away from the unavoidable boundary diagonal singularity. [/step]