Calderón-Zygmund theory for operator-valued kernels (original) (raw)

A new quantitative two weight theorem for the Hardy-Littlewood maximal operator

Proceedings of the American Mathematical Society, 2015

A quantitative two weight theorem for the Hardy-Littlewood maximal operator is proved improving the known ones. As a consequence a new proof of the main results in [HP] and [HPR12] is obtained which avoids the use of the sharp quantitative reverse Holder inequality for A∞ proved in those papers. Our results are valid within the context of spaces of homogeneous type without imposing the nonempty annuli condition.

Weighted weak type integral inequalities for the Hardy-Littlewood maximal operator

Israel Journal of Mathematics, 1989

In this paper we characterize the pairs of weights (u, w) for which the Hardy-Littlewood maximal operator M satisfies a weak type integral inequality of the form C f~,':,,axl>a) udx _-<~-~ fr ~lfl)wdx, with C independent of f and 2 > 0, where ~ is an N-function. Moreover, for a given weight w, a necessary and sufficient condition is found for the existence of a positive weight u such that M satisfies an integral inequality as above. Lastly, in the case u = w, we notice that the conclusion of the extrapolation theorem given by J. L. Rubio de Francia, which appeared in Am. J. Math. 106 (1984), can be strengthened to Orlicz spaces.

Singular integral operators and maximal functions with Hardy space kernels

TURKISH JOURNAL OF MATHEMATICS, 2021

In this paper, we study singular integrals along compound curves with Hardy space kernels. We introduce a class of bidirectional generalized Hardy Littlewood maximal functions. We prove that the considered singular integrals and the maximal functions are bounded on L p , 1 < p < ∞ provided that the compound curves are determined by generalized polynomials and convex increasing functions. The obtained results offer L p estimates that are not only new but also they generalize as well as improve previously known results.

Restricted weak type inequalities for the one-sided Hardy-Littlewood maximal operators in higher dimensions

2021

In 1986 ([10]) E. Swayer started the theory of one-sided weights. Namely, he introduced the class of weights Ap and showed that this class is necessary and sufficient for the weighted boundedness of the one-side Hardy-Littlewood maximal function. Some extensions and generalizations were given consequently in the articles [5], [6] and [7], among others. In [9] the author characterizes the functions w for which the one-sided Hardy-Littlewood maximal operator

Maximal, potential and singular operators

2011

We consider local "complementary" generalized Morrey spaces ∁ M p(•),ω {x 0 } (Ω) in which the p-means of function are controlled over Ω\B(x 0 , r) instead of B(x 0 , r) , where Ω ⊂ R n is a bounded open set, p(x) is a variable exponent, and no monotonicity type conditio is imposed onto the function ω(r) defining the "complementary" Morrey-type norm. In the case where ω is a power function, we reveal the relation of these spaces to weighted Lebesgue spaces. In the general case we prove the boundedness of the Hardy-Littlewood maximal operator and Calderon-Zygmund singular operators with standard kernel, in such spaces. We also prove a Sobolev type ∁ M p(•),ω {x 0 } (Ω) → ∁ M q(•),ω {x 0 } (Ω)-theorem for the potential operators I α(•) , also of variable order. In all the cases the conditions for the boundedness are given it terms of Zygmund-type integral inequalities on ω(r) , which do not assume any assumption on monotonicity of ω(r) .

Multiple-weighted norm inequalities for maximal multi-linear singular integrals with non-smooth kernels

Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 2011

We obtain weighted norm inequalities for maximal truncated operators of multi-linear singular integrals with non-smooth kernels in the sense of Duong et al. This class of operators extends the class of multi-linear Calderón-Zygmund operators introduced by Coifman and Meyer and includes the higher-order commutators of Calderón. The weighted norm inequalities obtained in this work are with respect to the new class of multiple weights of Lerner et al. The key ingredient in the proof is the introduction of a new multi-sublinear maximal operator that plays the role of the Hardy-Littlewood maximal function in a version of Cotlar's inequality. As applications of these results, new weighted estimates for the mth order Calderón commutators and their maximal counterparts are deduced.