In mathematics, the Christ–Kiselev maximal inequality is a maximal inequality for filtrations, named for mathematicians Michael Christ and Alexander Kiselev.M. Christ, A. Kiselev, Maximal functions associated to filtrations. J. Funct. Anal. 179 (2001), no. 2, 409--425.
Continuous filtrations 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 .Chapter 9 - Harmonic Analysis
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.
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