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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].

##### MSC:
 58C35 Integration on manifolds; measures on manifolds