The boundedness, compactness and closedness of the range of weighted composition operators acting on weighted Lorentz spaces L(p,q,wdμ) for 1 < p ≤ ∞, 1 ≤ q ≤ ∞ are characterized.

Let $M_g^{+}$ be the maximal operator defined by $M_g^{+}⨍\left(x\right)=su{p}_{h>0}({ʃ}_x^{x+h}\left|⨍\right|g)/\left({ʃ}_x^{x+h}g\right)$, where g is a positive locally integrable function on ℝ. Let Φ be an N-function such that both Φ and its complementary N-function satisfy ${\Delta }_2$. We characterize the pairs of positive functions (u,ω) such that the weak type inequality $u\left(x\in ℝ|M_g^{+}⨍\left(x\right)>\lambda \right)\le C/\left(\Phi \left(\lambda \right)\right){ʃ}_{ℝ}\Phi \left(\right|⨍\left|\right)\omega$ holds for every ⨍ in the Orlicz space $L_{\Phi }\left(\omega \right)$. We also characterize the positive functions ω such that the integral inequality ${ʃ}_{ℝ}\Phi \left(\right|M_g^{+}⨍\left|\right)\omega \le {ʃ}_{ℝ}\Phi \left(\right|⨍\left|\right)\omega$ holds for every $⨍\in L_{\Phi }\left(\omega \right)$. Our results include some already obtained for functions in $L^p$ and yield as consequences...

Necessary and sufficient conditions are given on the weights t, u, v, and w, in order for ${\Phi }_2^{-1}(ʃ{\Phi }_2\left(w\left(x\right)\right|Tf\left(x\right)\left|\right)t\left(x\right)dx)\le {\Phi }_1^{-1}(ʃ{\Phi }_1\left(Cu\left(x\right)\right|f\left(x\right)\left|\right)v\left(x\right)dx)$ to hold when ${\Phi }_1$ and ${\Phi }_2$ are N-functions with ${\Phi }_2\circ {\Phi }_1^{-1}$ convex, and T is the Hardy operator or a generalized Hardy operator. Weak-type characterizations are given for monotone operators and the connection between weak-type and strong-type inequalities is explored.

We discuss the characterization of the inequality (RN+ fq u)1/q C (RN+ fp v )1/p,   0<q, p <, for monotone functions $f\ge 0$ and nonnegative weights $u$ and $v$ and $N\ge 1$. We prove a new multidimensional integral modular inequality for monotone functions. This inequality generalizes and unifies some recent results in one and several dimensions.

Let X be a homogeneous space and let E be a UMD Banach space with a normalized unconditional basis ${\left(e_j\right)}_{j\ge 1}$. Given an operator T from $L_c^{\infty }\left(X\right)$ to L¹(X), we consider the vector-valued extension T̃ of T given by $T̃\left({\sum }_j{f}_j{e}_j\right)={\sum }_{j}T\left(f_j\right)e_j$. We prove a weighted integral inequality for the vector-valued extension of the Hardy-Littlewood maximal operator and a weighted Fefferman-Stein inequality between the vector-valued extensions of the Hardy-Littlewood and the sharp maximal operators, in the context of Orlicz spaces. We give sufficient...

We give a characterization of the weights (u,w) for which the Hardy-Littlewood maximal operator is bounded from the Orlicz space L_Φ(u) to L_Φ(w). We give a characterization of the weight functions w (respectively u) for which there exists a nontrivial u (respectively w > 0 almost everywhere) such that the Hardy-Littlewood maximal operator is bounded from the Orlicz space L_Φ(u) to L_Φ(w).