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 $(M,\mu )$ is a family of measurable sets $\{A_{\alpha }{\}}_{\alpha \in \mathbb{ℝ}}$ such that

1. $A_{\alpha }\nearrow M$, $\bigcap _{\alpha \in \mathbb{ℝ}}{A}_{\alpha }=\mathrm{\varnothing }$, and $\mu (A_{\beta }\setminus A_{\alpha })<\mathrm{\infty }$ for all $\beta >\alpha$ (stratific)
2. $\underset{\epsilon \to 0^{+}}{\lim }\mu (A_{\alpha +\epsilon }\setminus A_{\alpha })=\underset{\epsilon \to 0^{+}}{\lim }\mu (A_{\alpha }\setminus A_{\alpha +\epsilon })=0$ (continuity)

For example, $\mathbb{ℝ}=M$ with measure $\mu$ that has no pure points and

$A_{\alpha }:=\{\begin{array}{ll}\{|x|\le \alpha \},& \alpha >0,\\ \mathrm{\varnothing },& \alpha \le 0.\end{array}$

is a continuous filtration.

Continuum version

Let $1\le p<q\le \mathrm{\infty }$ and suppose $T:L^p(M,\mu )\to L^q(N,\nu )$ is a bounded linear operator for $\sigma -$finite $(M,\mu ),(N,\nu )$. Define the Christ–Kiselev maximal function

$T^{*}f:=\underset{\alpha }{\sup }|T(f{\chi }_{\alpha })|,$

where ${\chi }_{\alpha }:={\chi }_{A_{\alpha }}$. Then $T^{*}:L^p(M,\mu )\to L^q(N,\nu )$ is a bounded operator, and

$\Vert T^{*}f{\Vert }_q\le 2^{-(p^{-1}-q^{-1})}(1-2^{-(p^{-1}-q^{-1})}{)}^{-1}\Vert T\Vert \Vert f{\Vert }_p.$

Discrete version

Let $1\le p<q\le \mathrm{\infty }$, and suppose $W:{\ell }^p(\mathbb{\mathbb{Z}})\to L^q(N,\nu )$ is a bounded linear operator for $\sigma -$finite $(M,\mu ),(N,\nu )$. Define, for $a\in {\ell }^p(\mathbb{\mathbb{Z}})$,

$({\chi }_{n}a):=\{\begin{array}{ll}{a}_k,& |k|\le n\\ 0,& \text{otherwise}.\end{array}$

and $\underset{n\in {\mathbb{\mathbb{Z}}}^{\ge 0}}{\sup }|W({\chi }_{n}a)|=:W^{*}(a)$. Then $W^{*}:{\ell }^p(\mathbb{\mathbb{Z}})\to L^q(N,\nu )$ is a bounded operator.

Here, $A_{\alpha }=\{\begin{array}{ll}[-\alpha ,\alpha ],& \alpha >0\\ \mathrm{\varnothing },& \alpha \le 0\end{array}$.

The discrete version can be proved from the continuum version through constructing $T:L^p(\mathbb{ℝ},dx)\to L^q(N,\nu )$.

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.