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.M. Christ, A. Kiselev, Maximal functions associated to filtrations. J. Funct. Anal. 179 (2001), no. 2, 409--425.

Continuous filtrations

A continuous filtration of $(M,\mu)$ is a family of measurable sets $\_$ such that
# $A_\alpha\nearrow M$, $\bigcap_A_\alpha=\emptyset$, and $\mu(A_\beta\setminus A_\alpha)<\infty$ for all $\beta>\alpha$ (stratific)
# $\lim_\mu(A_\setminus A_\alpha)=\lim_\mu(A_\alpha\setminus A_)=0$ (continuity)

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

: $A_\alpha:=\begin\,&\alpha>0, \\ \emptyset,&\alpha\le0. \end$

is a continuous filtration.

Continuum version

Let 1\le p 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 $\chi_\alpha:=\chi_$. Then $T^*:L^p(M,\mu)\to L^q(N,\nu)$ is a bounded operator, and

$\, T^*f\, _q\le2^(1-2^)^\, T\, \, f\, _p.$

Discrete version

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

: $(\chi_n a):=\begina_k,&, k, \le n\\0,&\text.\end$

and $\sup_, W(\chi_na), =:W^*(a)$. Then $W^*:\ell^p(\mathbb)\to L^q(N,\nu)$ is a bounded operator.

Here, A_\alpha=\begin \alpha,\alpha&\alpha>0\\\emptyset,&\alpha\le0\end.

The discrete version can be proved from the continuum version through constructing $T:L^p(\mathbb,dx)\to L^q(N,\nu)$.Chapter 9 - Harmonic Analysis

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.

References

{{DEFAULTSORT:Christ-Kiselev maximal inequality
Mathematical analysis
Inequalities (mathematics)
Measure theory