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Konstantin M. Dyakonov 《Constructive Approximation》2009,30(1):17-31
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
ε
[K], ε>0, generated by a single kernel K∈L
1(ℝ
n
) 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
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.
Supported in part by grants MTM2005-08984-C02-02, MTM2006-26627-E and HF2006-0211 from El Ministerio de Educación y Ciencia
(Spain), and by grant 2005-SGR-00611 from DURSI (Generalitat de Catalunya). 相似文献
2.
In Martín et al. (J Funct Anal 252:677–695, 2007) we developed a new method to obtain symmetrization inequalities of Sobolev type for functions in . In this paper we extend our method to Sobolev functions that do not vanish at the boundary.
This paper is in final form and no version of it will be submitted for publication elsewhere. 相似文献
3.
We develop a new method to obtain symmetrization inequalities of Sobolev type. Our approach leads to new inequalities and considerable simplification in the theory of embeddings of Sobolev spaces based on rearrangement invariant spaces. 相似文献
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