Exponential estimates for plurisubharmonic functions
Let \(X\) be a complex manifold of dimension \(k\). A positive measure \(\mu\) on \(X\) is said to be locally moderate if, for any open set \(U\subset X\), any compact set \(K\subset U\), and any compact family \(\mathcal F\) of plurisubharmonic functions (p.s.h. for short) on \(U\), there are constants \(\alpha>0\) and \(c>0\) such that \(\int_Ke^{-\alpha\psi}\,d\mu\leq c\)\ for \(\psi\in\mathcal F\). Fix a smooth strictly positive (1.1)-form \(\omega\) on \(X\). A positive closed current \(S\) of bidegree \((p,p)\) on \(X\) is said to be locally moderate if its trace measure \(\sigma_S\):=\(S\wedge\omega^{k-p}\) is locally moderate. If \(u\) is a continuous real-valued function on the support of \(S\), then the multiplication \(uS\) defines a current on \(X\), so the current \(dd^c(uS)\) is also well-defined. The function \(u\) is \(S\)-p.s.h. if \(dd^c(uS)\) is a positive current. One of the main results of the paper is: Theorem 1.1. Let \(S\) be a locally moderate positive closed \((p,p)\)-current on \(X\). If \(u\) is a Hölder continuous \(S\)-p.s.h. function, then \(dd^c(uS)\) is locally moderate. In particular, if \(R\) is a positive closed (1,1)-current with Hölder continuous local potentials, then \(R\wedge S\) is locally moderate. This yields the following Corollary 1.2. Let \(u\) be a Hölder continuous p.s.h. function on \(X\). Then the Monge-Ampère currents \((dd^cu)^p\), \(1\leq p\leq k\), are locally moderate. The authors apply their result to dynamics. Given a non-invertible holomorphic endomorphism \(f\) of the projective space \(\mathbb P^k\) that is induced by a homogeneous polynomial endomorphism of degree \(d\geq 2\) on \(\mathbb C^{k+1}\), let \(T:=\lim_{n\to\infty}d^{-n}(f^n)^*(\omega_{FS})\), where \(\omega_{FS}\) is the normalized Fubini-Study form. Then \(T\) is a positive closed (1,1)-current of mass 1 on \(\mathbb P^k\) and one can define \(T^p\):=\(T\wedge\dots\wedge T\) (\(p\) times). The currents \(T^p\) are the Green currents associated to \(f\) and \(\mu:=T^k\) is the Green measure. Corollary 1.3. The Green currents and the Green measure associated to a non-invertible holomorphic endomorphism \(f\) of \(\mathbb P^k\) are locally moderate. As an application, the authors obtain several stochastic properties of dynamical systems associated to \(f\), e.g., the exponential decay of correlations, the central limit theorem and the large deviations theorem.
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