Christ–Kiselev maximal inequality

In mathematics, the Christ–Kiselev maximal inequality is a maximal inequality for filtrations, named for mathematicians Michael Christ and Alexander Kiselev.[1]

Continuous filtrations

A continuous filtration of is a family of measurable sets such that

  1. , , and for all (stratific)
  2. (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

  1. M. 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.CS1 maint: archived copy as title (link)
  2. Chapter 9 - Harmonic Analysis "Archived copy" (PDF). Archived from the original (PDF) on 2014-05-13. Retrieved 2014-05-12.CS1 maint: archived copy as title (link)


This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.