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

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

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

${\displaystyle A_{\alpha }:={\begin{cases}\{|x|\leq \alpha \},&\alpha >0,\\\emptyset ,&\alpha \leq 0.\end{cases}}}$

is a continuous filtration.

Continuum version

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

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

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

${\displaystyle \|T^{*}f\|_{q}\leq 2^{-(p^{-1}-q^{-1})}(1-2^{-(p^{-1}-q^{-1})})^{-1}\|T\|\|f\|_{p}.}$

Discrete version

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

${\displaystyle (\chi _{n}a):={\begin{cases}a_{k},&|k|\leq n\\0,&{\text{otherwise}}.\end{cases}}}$

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

Here, ${\displaystyle A_{\alpha }={\begin{cases}[-\alpha ,\alpha ],&\alpha >0\\\emptyset ,&\alpha \leq 0\end{cases}}}$.

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