Applications of the Radon transform in the theory of plurisubharmonic functions (Q2386921)

This article is devoted to the problem of representability of plurisubharmonic functions by a potential. For a domain \(\Omega\subset\mathbb{C}^n\), we denote by \(H(\Omega)\), \(SH(\Omega)\) and \(PSH(\Omega)\) the spaces of holomorphic, subharmonic, plurisubharmonic functions in \(\Omega\), respectively. We denote by \(D(\Omega)\) the space of smooth \(\mathbb{C}\)-valued, finite functions whose supports are compacts contained in \(\Omega\). Next, \(\mathcal{D}^{(n-1,n-1)}( \Omega)\) stands for the space of forms of degree \((n-1,n-1)\) with coefficients from \(D(\Omega)\). Let \(\mathcal{F}\) be a certain family of functions holomorphic in \(\Omega\). Then by the Radon \(\mathcal{F}\)-transform given by \(\varphi\in\mathcal{D}^{(n-1,n-1)}( \Omega)\) of the view \(\varphi=\sum\limits_{i,j=1}^n\varphi_{i,j}\wedge\omega_{i,j}\) is called the function \[ \widehat{\varphi}(P)=\langle\ln| P| ,dd^c\varphi\rangle=\int\ln| P(z)| \sum_{i,j=1}^n4\frac{\partial^2\varphi_{i,j}(z)}{\partial z_i\partial\bar{z}_j}\,d\omega_{2n}(z),> P\in\mathcal{F}\,. \] Let \(\Omega\) be a pseudoconvex domain. We denote by \(\mathcal{P}(\Omega )\) a set of equivalence classes of functions from \(PSH(\Omega)\), assuming that the two functions are equivalent if their difference is pluriharmonic in \(\Omega\). Let us consider the set \(H\) consisting of functions analytic in \(\Omega\) and satisfying the following conditions: \[ | f(z)| \leq\exp| z| ,> f\equiv 0,> D_f=\{z\in\Omega \mid f(z)=0\}\neq\emptyset\,. \] Let us define on \(H\) the topology induced from \(H(\Omega)\). Denote by \(C(H)\) the space of real-valued functions continuous on \(H\). Then the Radon \(H\)-transform of forms from \(M\) creates a subspace in \(C(H)\) which we denote by \(R\). Further, let \(\widetilde{R}\) be a subspace in \(C(H)\), formed by functions majorized by functions from \(R\), i.e., \(\psi\in\widetilde{R}\) if and only if a function \(\widehat{\varphi}\in R\) exists such that \(| \psi| \leq\widehat{\varphi}\). We introduce the set \(F(\Omega)\) of equivalence classes of positive functionals on \(\tilde{R}\), assuming that functionals coinciding on \(R\) are equivalent. The basic result of this article is the following. If \(\Omega\) is a Runge domain such that the factor-group \(H^2(\Omega,\mathbb{C})\) of \(d\)-closed forms of the second degree by \(d\)-exact forms is trivial, then the spaces \(\mathcal{P}(\Omega )\) and \(F(\Omega )\) are linear isomorphic. This result can be treated as an analog of the Riesz representation of subharmonic functions of one variable, which establishes a linear isomorphism between the set of subharmonic functions an the set of positive measures. This result gives us also the possibility to use in the investigation, for example, of questions of plurisubharmonic continuation, the technique of the theory of ordinary positive functionals on the space of continuous functions, which is more developed than the apparatus of the theory of positive closed flows on spaces of differential forms.