Two-weight mixed ф-inequalities for the one-sided maximal function
Volume 115 / 1995
Abstract
Suppose u, v, w, and t are weight functions on an appropriate measure space (X,μ), and $Φ_1$, $Φ_2$ are Young functions satisfying a certain relationship. Let T denote an operator to be specified below. The main purpose of this paper is to characterize (i) the strong type mixed Φ-inequality $Φ^{-1}_{2}(ʃ_{X} Φ_{2}(T(fv))wdμ) ≤ Φ^{-1}_{1} (ʃ_X Φ_{1}(Cf)vdμ)$, (ii) the weak type mixed Φ-inequality $Φ^{-1}_2 (ʃ_{|Tf|>λ}$ Φ_{2}(λw)tdμ) ≤ Φ^{-1}_{1} (ʃ_{X} Φ_{1}(Cfu)vdμ)$ and (iii) the extra-weak type mixed Φ-inequality $|{x ∈ X : |Tf(x)| > λ}|_{wdμ} ≤ Φ_{2}Φ^{-1}_{1} (ʃ_{X} Φ_{1}(Cfu/λ)vdμ)$, when T is the one-sided maximal function $M^{+}_{g}$; as well to characterize (iii) for the Fefferman-Stein type fractional maximal operator and the Hardy-type operator.
