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) $$\lim_{\varepsilon\to0^+}\mu(A_{\alpha+\varepsilon}\setminus A_\alpha)=\lim_{\varepsilon\to0^+}\mu(A_\alpha\setminus A_{\alpha+\varepsilon})=0$$ (continuity)

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


 * $$A_\alpha:=\begin{cases}\{|x|\le\alpha\},&\alpha>0, \\ \emptyset,&\alpha\le0. \end{cases}$$

is a continuous filtration.

Continuum version
Let $$1\le p<q\le\infty$$ 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

$$\|T^*f\|_q\le2^{-(p^{-1}-q^{-1})}(1-2^{-(p^{-1}-q^{-1})})^{-1}\|T\|\|f\|_p.$$

Discrete version
Let $$1\le p0\\\emptyset,&\alpha\le0\end{cases}$$.

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.