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Uniform estimates of volumes. (English. Russian original) Zbl 0967.58007
Proc. Steklov Inst. Math. 221, 214-220 (1998); translation from Tr. Mat. Inst. Steklova 221, 225-231 (1998).
Consider a real analytical function \(f\) defined in some neighborhood \(B\subset\mathbb C^n\) of the point \(0\), \(f: (B,B\cap \mathbb R^n)\to (\mathbb C,\mathbb R)\), \(f(0)=df(0)=0,\) and a small enough neighborhood \(A\) of the point \(0\in\mathbb R^n.\) The measure of the set \(A_\delta=\{x\in A:-\delta\leq f(x)\leq \delta\}\) admits the asymptotic decomposition at \(\delta\to 0\): \(b\delta^{-\gamma}|\ln\delta|^q+\dots\); \(b\neq 0\), \(\gamma<0\), \(q\geq 0\) is an integer. The pair \((\gamma,q)\) refers to as individual volume exponent (IVE).
Let \(\varphi\) be a smooth function with support in a small enough neighborhood of \(0\in\mathbb R^n\), \(\varphi(0)\neq 0.\) The integral over \(A_\delta\) of the kind \(\int_{A_\delta}\varphi dx\) is called the volume with the phase \(f\) and amplitude \(\varphi.\)
In \(\mathbb R^2,\) the author proves the uniform estimate for the volumes (areas) with IVE (uniformity is taken on a small disturbance of the phase \(f\)). In \(\mathbb R^3\), the uniform estimates of volumes coincide with IVE for all parabolic and hyperbolic germs. Their IVEs are equal to \((-1,1)\) and \((-1,2)\) respectively. For all 14 exclusive unimodal families, the uniform estimates of volumes are valid with IVEs which coincide with exponents of oscillation taken with the corresponding multiplicities [V. N. Karpushkin, Usp. Mat. Nauk. 38, No. 5, 128 (1983)]. On the basis of the uniform estimates of volumes in \(\mathbb R^2,\) the uniform estimates of volumes in \(\mathbb R^n\) are obtained.
For the entire collection see [Zbl 0926.00021].

58C35 Integration on manifolds; measures on manifolds