Boundedness of the Weyl fractional integral on one-sided weighted Lebesgue and Lipschitz spaces

In this paper we introduce the one-sided weighted spaces L−w (β), −1 <β< 1. The purpose of this definition is to obtain an extension of the Weyl fractional integral operator I+α from Lp w into a suitable weighted space. Under certain condition on the weight w, we have that L−w (0) coincides with the dual of the Hardy space H1 −(w). We prove for 0 <β< 1, that L− w (β) consists of all functions satisfying a weighted Lipschitz condition. In order to give another characterization of L− w (β), 0 ≤ β < 1, we also prove a one-sided version of John-Nirenberg Inequality. Finally, we obtain necessary and sufficient conditions on the weight w for the boundedness of an extension of I+ α from Lp w into L− w (β), −1 <β< 1, and its extension to a bounded operator from L− w (0) into L− w (α)

Proof of an extension of E. Sawyer's conjecture about weighted mixed weak-type estimates

We show that if $v\in A_\infty$ and $u\in A_1$, then there is a constant $c$ depending on the $A_1$ constant of $u$ and the $A_{\infty}$ constant of $v$ such that $$\Big\|\frac{ T(fv)} {v}\Big\|_{L^{1,\infty}(uv)}\le c\, \|f\|_{L^1(uv)},$$ where $T$ can be the Hardy-Littlewood maximal function or any Calder\'on-Zygmund operator. This result was conjectured in [IMRN, (30)2005, 1849--1871] and constitutes the most singular case of some extensions of several problems proposed by E. Sawyer and Muckenhoupt and Wheeden. We also improve and extends several quantitative estimates.Juan de la Cierva - Formación 2015 FJCI- 2015-2454

Weighted mixed weak-type inequalities for multilinear operators

In this paper we present a theorem that generalizes Sawyer's classic result about mixed weighted inequalities to the multilinear context. Let $\vec{w}=(w_1,...,w_m)$ and $\nu = w_1^\frac{1}{m}...w_m^\frac{1}{m}$, the main result of the paper sentences that under different conditions on the weights we can obtain $$\Bigg\| \frac{T(\vec f\,)(x)}{v}\Bigg\|_{L^{\frac{1}{m}, \infty}(\nu v^\frac{1}{m})} \leq C \ \prod_{i=1}^m{\|f_i\|_{L^1(w_i)}},$$ where $T$ is a multilinear Calder\'on-Zygmund operator. To obtain this result we first prove it for the $m$-fold product of the Hardy-Littlewood maximal operator $M$, and also for $\mathcal{M}(\vec{f})(x)$: the multi(sub)linear maximal function introduced in [LOPTT]. As an application we also prove a vector-valued extension to the mixed weighted weak-type inequalities of multilinear Calder\'on-Zygmund operators.Juan de la Cierva-Formaci\'on 2015 FJCI-2015-2454

A note on generalized Fujii-Wilson conditions and BMO spaces

In this note we generalize the definition of the Fujii-Wilson condition providing quantitative characterizations of some interesting classes of weights, such as A∞, A∞weak and Cp, in terms of BMO type spaces suited to them. We will provide as well some self improvement properties for some of those generalized BMO spaces and some quantitative estimates for Bloom’s BMO type spaces

Boundedness of the Weyl fractional integral on one-sided weighted Lebesgue and Lipschitz spaces

In this paper we introduce the one-sided weighted spaces L−w (β), −1 <β< 1. The purpose of this definition is to obtain an extension of the Weyl fractional integral operator I+α from Lp w into a suitable weighted space. Under certain condition on the weight w, we have that L−w (0) coincides with the dual of the Hardy space H1 −(w). We prove for 0 <β< 1, that L− w (β) consists of all functions satisfying a weighted Lipschitz condition. In order to give another characterization of L− w (β), 0 ≤ β < 1, we also prove a one-sided version of John-Nirenberg Inequality. Finally, we obtain necessary and sufficient conditions on the weight w for the boundedness of an extension of I+ α from Lp w into L− w (β), −1 <β< 1, and its extension to a bounded operator from L− w (0) into L− w (α)

An interpolation theorem between one-sided Hardy spaces

In this paper we generalize an interpolation result due to J.-O. Strömberg and A. Torchinsky to the case of one-sided Hardy spaces. This generalization is important in the study of the weak type (1,1) for lateral strongly singular operators. We shall need an atomic decomposition in which for every atom there exists another atom supported contiguously at its right. In order to obtain this decomposition we have developed a rather simple technique to break up an atom into a sum of others atoms. © 2006 by Institut Mittag-Leffler. All rights reserved.Fil:Ombrosi, S. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina.Fil:Testoni, R. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina

Weighted Lorentz spaces: Sharp mixed A<inf>p</inf> − A<inf>∞</inf> estimate for maximal functions

We prove the sharp mixed Ap−A∞ weighted estimate for the Hardy-Littlewood maximal function in the context of weighted Lorentz spaces, namely [Formula presented] where [Formula presented]. Our method is rearrangement free and can also be used to bound similar operators, even in the two-weight setting. We use this to also obtain new quantitative bounds for the strong maximal operator and for M in a dual setting