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Refined uniform estimates of oscillatory integrals and areas. (Russian) Zbl 0692.47015
Let f: $$R^ 2\to R$$, $$\phi$$ : $$R^ 2\to R$$ and $I(\tau,f,\phi)=\int_{R^ 2}\phi \exp \{i\tau f\},\quad V(\epsilon,f,c,\phi,A)=\int_{R^ 2}\chi \phi,$ where $$\chi$$ is the indicator of the set $$\{x\in A:\quad c-\epsilon \leq f(x)\leq c+\epsilon \}$$ and A is an open set. In the paper some uniform two-term upper estimates for $$| I(\tau,f,\phi)|$$ and $$| V(\epsilon,f,c,\phi,A)|$$ are obtained.
Reviewer: Yu.M.Ryžov

##### MSC:
 47A55 Perturbation theory of linear operators 26D10 Inequalities involving derivatives and differential and integral operators 58J40 Pseudodifferential and Fourier integral operators on manifolds 42B99 Harmonic analysis in several variables
##### Keywords:
oscillatory integral; canonical Maslov’s operator