Kolmogorov Averages and Approximate Identities

Abstract

We study the action of Kolmogorov-type nonlinear averaging operators of the form V ⁻¹ AV on smooth functions. Here, A runs through a family of convolution operators A ^[K] _ε , ε>0, generated by a single kernel K∈L ¹(ℝⁿ) in the usual way and forming an “approximate identity” as ε→0, while V is a superposition map given by Vf=v○f, with a monotone continuous function v. The main result characterizes the kernels K with the property that the natural estimate

$$\|V^{-1}A^{[K]}_{\varepsilon }Vf-f\|_{\infty}\le {\mathrm {const}}\cdot\|f\|_{\varLambda _{\omega }}\cdot \omega (\varepsilon )$$

holds for all admissible functions f in the Lipschitz space Λ _ω , associated with a majorant ω. Namely, it is shown that for fairly general (locally unbounded) functions v, the kernels in question must have compact support. Moreover, the same conclusion is already implied by various weak versions of the above estimate (by infinitely weak ones, in a sense), even though the phenomenon has its limits.