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Algebraic estimates, stability of local zeta functions, and uniform estimates for distribution functions. (English) Zbl 0995.11065

The paper is devoted to the study of local zeta functions, i.e. to integrals of the form \[ \int_{B^{(n)}} |f(z)|^{-\delta} dV_1\cdots DV_n , \] where \(B^{(n)}\) is a polydisk in \({\mathbb C}^n\), \(f(z) = \bigl(f_j(z)\bigr)_{j=1,\dots,J}\) is a \(J\)-dimensional vector holomorphic function on \(B^{(n)}\) and \(|f(z)|^2 = \sum_{j=1}^{J} |f_j(z)|^2\). The authors study the stability for the finiteness of these integrals under deformations of the function \(f(z)\). In their main result they consider a \(J\)-vector of holomorphic functions \(g(z,c)\) on a polydisk \(B^{(n)}\times B^{(1)}\) and assume that \(\int_{B^{(n)}}|g(z,0)|^{-\delta}D V_1\cdots d V_n < \infty\). It is proved that there exists a smaller polydisk \(B^{\prime (n)}\times B^{\prime (1)}\) such that the function \(c\rightarrow \int_{B^{\prime (n)}}|g(z,0)|^{-\delta}D V_1\cdots d V_n\) is finite and continuous for \(c\in B^{\prime (1)}\).
Results of other authors are deduced as corollaries, for example the theorem of B. Lichtin [Adv. Stud. Pure Math. 8, 241-272 (1987; Zbl 0615.32007)].

MSC:

11S40 Zeta functions and \(L\)-functions
32A55 Singular integrals of functions in several complex variables

Citations:

Zbl 0615.32007
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