This article may be too technical for most readers to understand. Please to make it understandable to non-experts, without removing the technical details.(June 2014) (Learn how and when to remove this template message)
A continuous filtration of is a family of measurable sets such that
, , and for all (stratific)
(continuity)
For example, with measure that has no pure points and
is a continuous filtration.
Continuum version[]
Let and suppose is a bounded linear operator for finite. Define the Christ–Kiselev maximal function
where . Then is a bounded operator, and
Discrete version[]
Let , and suppose is a bounded linear operator for finite. Define, for ,
and . Then is a bounded operator.
Here, .
The discrete version can be proved from the continuum version through constructing .[2]
Applications[]
The Christ–Kiselev maximal inequality has applications to the Fourier transform and convergence of Fourier series, as well as to the study of Schrödinger operators.[1][2]
References[]
^ abM. Christ, A. Kiselev, Maximal functions associated to filtrations. J. Funct. Anal. 179 (2001), no. 2, 409--425. "Archived copy"(PDF). Archived from the original(PDF) on 2014-05-14. Retrieved 2014-05-12.{{cite web}}: CS1 maint: archived copy as title (link)
^ abChapter 9 - Harmonic Analysis "Archived copy"(PDF). Archived from the original(PDF) on 2014-05-13. Retrieved 2014-05-12.{{cite web}}: CS1 maint: archived copy as title (link)
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Mathematical analysis
Inequalities
Measure theory
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Wikipedia articles that are too technical from June 2014