The following iterated commutators $T_{\ast ,\Pi b}$ of the maximal operator for multilinear singular integral operators and $I_{\alpha ,\Pi b}$ of the multilinear fractional integral operator are introduced and studied: $T_{\ast ,\Pi b}\left(f⃗\right)\left(x\right)=su{p}_{\delta >0}\left|[b₁,[b₂,\dots {[b_{m-1},[bₘ,T_{\delta }]ₘ]}_{m-1}\cdots ]₂]₁\left(f⃗\right)\left(x\right)\right|$, $I_{\alpha ,\Pi b}\left(f⃗\right)\left(x\right)=[b₁,[b₂,\dots {[b_{m-1},[bₘ,I_{\alpha }]ₘ]}_{m-1}\cdots ]₂]₁\left(f⃗\right)\left(x\right)$, where $T_{\delta }$ are the smooth truncations of the multilinear singular integral operators and $I_{\alpha }$ is the multilinear fractional integral operator, $b_i\in BMO$ for i = 1,…,m and f⃗ = (f1,…,fm). Weighted strong and L(logL) type end-point estimates for the above iterated commutators associated with two classes of multiple weights,...

An improved multiple Cotlar inequality is obtained. From this result, weighted norm inequalities for the maximal operator of a multilinear singular integral including weak and strong estimates are deduced under the multiple weights constructed recently.

Many problems in analysis are described as weighted norm inequalities that have given rise to different classes of weights, such as $A_p$-weights of Muckenhoupt and $B_p$-weights of Ariño and Muckenhoupt. Our purpose is to show that different classes of weights are related by means of composition with classical transforms. A typical example is the family $M_p$ of weights w for which the Hardy transform is $L_p\left(w\right)$-bounded. A $B_p$-weight is precisely one for which its Hardy transform is in $M_p$, and also a weight whose indefinite...

We characterize geometric properties of a family of approach regions by means of analytic properties of the class of weights related to the boundedness of the maximal operator associated with this family.

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.