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] A continuous filtration of

is a family of measurable sets

μ

that has no pure points and is a continuous filtration.

, μ ) →

, ν )

is a bounded linear operator for

finite

, μ ) , (

Define the Christ–Kiselev maximal function

α

α

α

α

, μ ) →

, ν )

is a bounded operator, and

ℓ

, ν )

is a bounded linear operator for

finite

, μ ) , (

, ν )

Define, for

ℓ

ℓ

is a bounded operator.

The discrete version can be proved from the continuum version through constructing

[2] 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.