Christ–Kiselev maximal inequality

Continuous fibrations

A continuous fibration of ( M , μ ) is a family of measurable sets { A α } α ∈ R such that

1. A α ↗ M , ⋂ α ∈ R A α = ∅ , and μ ( A β ∖ A α ) < ∞ for all β > α (stratific)
2. lim ε → 0 + μ ( A α + ε ∖ A α ) = lim ε → 0 + μ ( A α ∖ A α + ε ) = 0 (continuity)

For example, R = M with measure μ that has no pure points and

is a continuous fibration.

Continuum version

Let 1 ≤ p < q ≤ ∞ and suppose T : L p ( M , μ ) → L q ( N , ν ) is a bounded linear operator for σ − finite ( M , μ ) , ( N , ν ) . Define the Christ–Kiselev maximal function

T ∗ f := sup α | T ( f χ α ) | ,

where χ α := χ A α . Then T ∗ : L p ( M , μ ) → L q ( N , ν ) is a bounded operator, and

∥ T ∗ f ∥ q ≤ 2 − ( p − 1 − q − 1 ) ( 1 − 2 − ( p − 1 − q − 1 ) ) − 1 ∥ T ∥ ∥ f ∥ p .

Discrete version

Let 1 ≤ p < q ≤ ∞ , and suppose W : ℓ p ( Z ) → L q ( N , ν ) is a bounded linear operator for σ − finite ( M , μ ) , ( N , ν ) . Define, for a ∈ ℓ p ( Z ) ,

and sup n ∈ Z ≥ 0 | W ( χ n a ) | =: W ∗ ( a ) . Then W ∗ : ℓ p ( Z ) → L q ( N , ν ) is a bounded operator.

Here, A α = { [ − α , α ] , α > 0 ∅ , α ≤ 0 .

The discrete version can be proved from the continuum version through constructing T : L p ( R , d x ) → L q ( N , ν ) .

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.