We prove weighted inequalities for commutators of one-sided singular integrals (given by a Calder’on-Zygmund kernel with support in $(-\infty ,0)$) with BMO functions. We give the one-sided version of the results in C. Pérez, Sharp estimates for commutators of singular integrals via iterations of the Hardy-Littlewood maximal function, J. Fourier Anal. Appl., vol. 3 (6), 1997, pages 743–756 and C. Pérez, Endpoint estimates for commutators of singular integral operators, J. Funct. Anal., vol 128 (1), 1995, pages...

In this paper we study integral operators of the form $$Tf\left(x\right)=\int |x-a_1{y|}^{-{\alpha }_1}\cdots {|x-a_{m}y|}^{-{\alpha }_m}f\left(y\right)\mathrm{d}y,$$${\alpha }_1+\cdots +{\alpha }_{}$

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...

Weighted inequalities for some square functions are studied. L² results are proved first using the particular structure of the operator and then extrapolation of weights is applied to extend the results to other $L^p$ spaces. In particular, previous results for square functions with rough kernel are obtained in a simpler way and extended to a larger class of weights.

In this paper we study integral operators with kernels $$K(x,y)=k_1(x-A_{1}y)\cdots k_m\left(x-A_{m}y\right),$$$$k_i\left(x\right)=...$$

We prove weighted inequalities for square functions of Littlewood-Paley type defined from a decomposition of the plane into sectors of lacunary aperture and for the maximal function over a lacunary set of directions. Some applications to multiplier theorems are also given.

In [4] P. Jones solved the question posed by B. Muckenhoupt in [7] concerning the factorization of Ap weights. We recall that a non-negative measurable function w on Rn is in the class Ap, 1 < p < ∞ if and only if the Hardy-Littlewood maximal operator is bounded on Lp(Rn, w). In what follows, Lp(X, w) denotes the class of all measurable functions f defined on X for which ||fw1/p||Lp(X) < ∞, where X is a measure space and w is a non-negative measurable function on X.It has recently...

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.

Applying discrete Calderón’s identity, we study weighted multi-parameter mixed Hardy space $H_{\mathrm{mix}}^p(\omega ,{ℝ}^{n_1}\times {ℝ}^{n_2})$. Different from classical multi-parameter Hardy space, this space has characteristics of local Hardy space and Hardy space in different directions, respectively. As applications, we discuss the boundedness on $H_{\mathrm{mix}}^p(\omega ,{ℝ}^{n_1}\times {ℝ}^{n_2})$ of operators in mixed Journé’s class.