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

1. [math]\displaystyle{ A_\alpha\nearrow M }[/math], [math]\displaystyle{ \bigcap_{\alpha\in\mathbb{R}}A_\alpha=\emptyset }[/math], and [math]\displaystyle{ \mu(A_\beta\setminus A_\alpha)\lt \infty }[/math] for all [math]\displaystyle{ \beta\gt \alpha }[/math] (stratific)
2. [math]\displaystyle{ \lim_{\varepsilon\to0^+}\mu(A_{\alpha+\varepsilon}\setminus A_\alpha)=\lim_{\varepsilon\to0^+}\mu(A_\alpha\setminus A_{\alpha+\varepsilon})=0 }[/math] (continuity)

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

[math]\displaystyle{ A_\alpha:=\begin{cases}\{|x|\le\alpha\},&\alpha\gt 0, \\ \emptyset,&\alpha\le0. \end{cases} }[/math]

is a continuous filtration.

Continuum version

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

[math]\displaystyle{ T^*f:=\sup_\alpha|T(f\chi_\alpha)|, }[/math]

where [math]\displaystyle{ \chi_\alpha:=\chi_{A_\alpha} }[/math]. Then [math]\displaystyle{ T^*:L^p(M,\mu)\to L^q(N,\nu) }[/math] is a bounded operator, and

[math]\displaystyle{ \|T^*f\|_q\le2^{-(p^{-1}-q^{-1})}(1-2^{-(p^{-1}-q^{-1})})^{-1}\|T\|\|f\|_p. }[/math]

Discrete version

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

[math]\displaystyle{ (\chi_n a):=\begin{cases}a_k,&|k|\le n\\0,&\text{otherwise}.\end{cases} }[/math]

and [math]\displaystyle{ \sup_{n\in\mathbb{Z}^{\ge0}}|W(\chi_na)|=:W^*(a) }[/math]. Then [math]\displaystyle{ W^*:\ell^p(\mathbb{Z})\to L^q(N,\nu) }[/math] is a bounded operator.

Here, [math]\displaystyle{ A_\alpha=\begin{cases}[-\alpha,\alpha],&\alpha\gt 0\\\emptyset,&\alpha\le0\end{cases} }[/math].

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

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.