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 , μ ) {\displaystyle (M,\mu )} is a family of measurable sets { A α } α ∈ R {\displaystyle \{A_{\alpha }\}_{\alpha \in \mathbb {R} }} such that
A α ↗ M {\displaystyle A_{\alpha }\nearrow M} , ⋂ α ∈ R A α = ∅ {\displaystyle \bigcap _{\alpha \in \mathbb {R} }A_{\alpha }=\emptyset } , and μ ( A β ∖ A α ) < ∞ {\displaystyle \mu (A_{\beta }\setminus A_{\alpha })<\infty } for all β > α {\displaystyle \beta >\alpha } (stratific)
lim ε → 0 + μ ( A α + ε ∖ A α ) = lim ε → 0 + μ ( A α ∖ A α + ε ) = 0 {\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, R = M {\displaystyle \mathbb {R} =M} with measure μ {\displaystyle \mu } that has no pure points and
A α := { { | x | ≤ α } , α > 0 , ∅ , α ≤ 0. {\displaystyle A_{\alpha }:={\begin{cases}\{|x|\leq \alpha \},&\alpha >0,\\\emptyset ,&\alpha \leq 0.\end{cases}}}
is a continuous filtration.
Continuum version Let 1 ≤ p < q ≤ ∞ {\displaystyle 1\leq p<q\leq \infty } and suppose T : L p ( M , μ ) → L q ( N , ν ) {\displaystyle T:L^{p}(M,\mu )\to L^{q}(N,\nu )} is a bounded linear operator for σ − {\displaystyle \sigma -} finite ( M , μ ) , ( N , ν ) {\displaystyle (M,\mu ),(N,\nu )} . Define the Christ–Kiselev maximal function
T ∗ f := sup α | T ( f χ α ) | , {\displaystyle T^{*}f:=\sup _{\alpha }|T(f\chi _{\alpha })|,}
where χ α := χ A α {\displaystyle \chi _{\alpha }:=\chi _{A_{\alpha }}} . Then T ∗ : L p ( M , μ ) → L q ( N , ν ) {\displaystyle T^{*}:L^{p}(M,\mu )\to L^{q}(N,\nu )} is a bounded operator, and
‖ T ∗ f ‖ q ≤ 2 − ( p − 1 − q − 1 ) ( 1 − 2 − ( p − 1 − q − 1 ) ) − 1 ‖ T ‖ ‖ f ‖ p . {\displaystyle \|T^{*}f\|_{q}\leq 2^{-(p^{-1}-q^{-1})}(1-2^{-(p^{-1}-q^{-1})})^{-1}\|T\|\|f\|_{p}.}
Discrete version Let 1 ≤ p < q ≤ ∞ {\displaystyle 1\leq p<q\leq \infty } , and suppose W : ℓ p ( Z ) → L q ( N , ν ) {\displaystyle W:\ell ^{p}(\mathbb {Z} )\to L^{q}(N,\nu )} is a bounded linear operator for σ − {\displaystyle \sigma -} finite ( M , μ ) , ( N , ν ) {\displaystyle (M,\mu ),(N,\nu )} . Define, for a ∈ ℓ p ( Z ) {\displaystyle a\in \ell ^{p}(\mathbb {Z} )} ,
( χ n a ) := { a k , | k | ≤ n 0 , otherwise . {\displaystyle (\chi _{n}a):={\begin{cases}a_{k},&|k|\leq n\\0,&{\text{otherwise}}.\end{cases}}}
and sup n ∈ Z ≥ 0 | W ( χ n a ) | =: W ∗ ( a ) {\displaystyle \sup _{n\in \mathbb {Z} ^{\geq 0}}|W(\chi _{n}a)|=:W^{*}(a)} . Then W ∗ : ℓ p ( Z ) → L q ( N , ν ) {\displaystyle W^{*}:\ell ^{p}(\mathbb {Z} )\to L^{q}(N,\nu )} is a bounded operator. Here, A α = { [ − α , α ] , α > 0 ∅ , α ≤ 0 {\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 T : L p ( R , d x ) → L q ( N , ν ) {\displaystyle 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.
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