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

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

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

is a continuous filtration.

Continuum version

Let $1\leq p<q\leq \infty$ 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

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

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

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

Discrete version

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

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

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

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

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

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.