S-functions from g-functions

Abstract

(The statements of the results in this section are complicated. But, these results will not be used in later sections. You can skip this section.) In this section we will show that L ^q-norms of Sψδf are essentially independent of the choice of ψ and S ≥ 0 if ψ satisfies certain conditons. The point is the fact that this includes the case δ = 0. (The “q-function” in the title means Sψ0 f) We write

$$ \chi \left( x \right) = \chi _{B\left( {0,1} \right)} \left( x \right). $$

(1)

For the definition of Sψ,δm recall Definition 3.6. In the notation M(f * (φ)t)(x), the convolution * and the maximal operator M are taken with respect to the variable x ∈ R ⁿ, with t ∈ (0, +∞) fixed.