Recently, F. Nazarov, S. Treil and A. Volberg (and independently X. Tolsa) have extended the classical theory of Calderón-Zygmund operators to the context of a non-homogeneous space (X,d,μ) where, in particular, the measure μ may be non-doubling. In the present work we study weighted inequalities for these operators. Specifically, for 1 < p < ∞, we identify sufficient conditions for the weight on one side, which guarantee the existence of another weight in the other side, so that the...

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 }_{}$

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 $$K(x,y)=k_1(x-A_{1}y)\cdots k_m\left(x-A_{m}y\right),$$$$k_i\left(x\right)=...$$

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.

Let µ be a Borel measure on Rd which may be non doubling. The only condition that µ must satisfy is µ(B(x, r)) ≤ Crn for all x ∈ Rd, r > 0 and for some fixed n with 0 < n ≤ d. In this paper we introduce a maximal operator N, which coincides with the maximal Hardy-Littlewood operator if µ(B(x, r)) ≈ rn for x ∈ supp(µ), and we show that all n-dimensional Calderón-Zygmund operators are bounded on Lp(w dµ) if and only if N is bounded on Lp(w dµ), for a fixed p ∈ (1, ∞). Also, we prove...

Let μ be a nonnegative Radon measure on ${ℝ}^d$ which satisfies μ(B(x,r)) ≤ Crⁿ for any $x\in {ℝ}^d$ and r > 0 and some positive constants C and n ∈ (0,d]. In this paper, some weighted norm inequalities with $A_p^{\varrho }\left(\mu \right)$ weights of Muckenhoupt type are obtained for maximal singular integral operators with such a measure μ, via certain weighted estimates with $A_{\infty }^{\varrho }\left(\mu \right)$ weights of Muckenhoupt type involving the John-Strömberg maximal operator and the John-Strömberg sharp maximal operator, where ϱ,p ∈ [1,∞).

In this paper we establish weighted norm inequalities for singular integral operators with kernel satisfying a variant of the classical Hörmander's condition.

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 establish weighted sharp maximal function inequalities for a linear operator associated to a singular integral operator with non-smooth kernel. As an application, we obtain the boundedness of a commutator on weighted Lebesgue spaces.

We consider the $A_1$-weights and prove the weighted weak type (1,1) inequalities for certain oscillatory singular integrals.

We prove some weighted weak type (1,1) inequalities for certain singular integrals and Littlewood-Paley functions.