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.

Continuous filtrations

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

,, and for all

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.

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.