Weighted norm inequalities for oscillatory integrals with finite type phases on the line

We obtain two-weighted $L^2$ norm inequalities for oscillatory integral operators of convolution type on the line whose phases are of finite type. The conditions imposed on the weights involve geometrically-defined maximal functions, and the inequalities are best-possible in the sense that they imply the full $L^p(\mathbb{R})\rightarrow L^q(\mathbb{R})$ mapping properties of the oscillatory integrals. Our results build on work of Carbery, Soria, Vargas and the first author.Comment: 22 page

The ℓs\ell^s-boundedness of a family of integral operators on UMD Banach function spaces

We prove the $\ell^s$-boundedness of a family of integral operators with an operator-valued kernel on UMD Banach function spaces. This generalizes and simplifies earlier work by Gallarati, Veraar and the author, where the $\ell^s$-boundedness of this family of integral operators was shown on Lebesgue spaces. The proof is based on a characterization of $\ell^s$-boundedness as weighted boundedness by Rubio de Francia.Comment: 13 pages. Generalization of arXiv:1410.665

Weighted norm inequalities, off-diagonal estimates and elliptic operators. Part I: General operator theory and weights

This is the first part of a series of four articles. In this work, we are interested in weighted norm estimates. We put the emphasis on two results of different nature: one is based on a good-$\lambda$ inequality with two-parameters and the other uses Calder\'on-Zygmund decomposition. These results apply well to singular 'non-integral' operators and their commutators with bounded mean oscillation functions. Singular means that they are of order 0, 'non-integral' that they do not have an integral representation by a kernel with size estimates, even rough, so that they may not be bounded on all $L^p$ spaces for $1 < p < \infty$. Pointwise estimates are then replaced by appropriate localized $L^p-L^q$ estimates. We obtain weighted $L^p$ estimates for a range of $p$ that is different from $(1,\infty)$ and isolate the right class of weights. In particular, we prove an extrapolation theorem ' \`a la Rubio de Francia' for such a class and thus vector-valued estimates.Comment: 43 pages. Series of 4 paper

Polynomial Carleson operators along monomial curves in the plane

We prove $L^p$ bounds for partial polynomial Carleson operators along monomial curves $(t,t^m)$ in the plane $\mathbb{R}^2$ with a phase polynomial consisting of a single monomial. These operators are "partial" in the sense that we consider linearizing stopping-time functions that depend on only one of the two ambient variables. A motivation for studying these partial operators is the curious feature that, despite their apparent limitations, for certain combinations of curve and phase, $L^2$ bounds for partial operators along curves imply the full strength of the $L^2$ bound for a one-dimensional Carleson operator, and for a quadratic Carleson operator. Our methods, which can at present only treat certain combinations of curves and phases, in some cases adapt a $TT^*$ method to treat phases involving fractional monomials, and in other cases use a known vector-valued variant of the Carleson-Hunt theorem.Comment: 27 page

Bounds for the Hilbert transform with matrix A2 weights

International audienceWe establish a new estimate for the Hilbert transform in L^2 space endowed with a matrix weight

Operators of harmonic analysis in weighted spaces with non-standard growth

Last years there was increasing an interest to the so-called function spaces with non-standard growth, known also as variable exponent Lebesgue spaces. For weighted such spaces on homogeneous spaces, we develop a certain variant of Rubio de Francia's extrapolation theorem. This extrapolation theorem is applied to obtain the boundedness in such spaces of various operators of harmonic analysis, such as maximal and singular operators, potential operators, Fourier multipliers, dominants of partial sums of trigonometric Fourier series and others, in weighted Lebesgue spaces with variable exponent. There are also given their vector-valued analogues. (C) 2008 Elsevier Inc. All rights reserved.INTAS [06-1000017-8792]; Center CEMAT, Instituto Superior Tecnico, Lisbon, Portugalinfo:eu-repo/semantics/publishedVersio

Representation of singular integrals by dyadic operators, and the A(2) theorem

This exposition presents a self-contained proof of the A(2) theorem, the quantitatively sharp norm inequality for singular integral operators in the weighted space L-2 (w). The strategy of the proof is a streamlined version of the author's original one, based on a probabilistic Dyadic Representation Theorem for singular integral operators. While more recent non-probabilistic approaches are also available now, the probabilistic method provides additional structural information, which has independent interest and other applications. The presentation emphasizes connections to the David-Journe T(1) theorem, whose proof is obtained as a byproduct. Only very basic Probability is used; in particular, the conditional probabilities of the original proof are completely avoided. (C) 2016 Elsevier GmbH. All rights reserved.Peer reviewe