On a weak type estimate for sparse operators of strong type

We define sparse operators of strong type on abstract measure spaces with ball-bases. Weak and strong type inequalities for such operators are proved.Comment: 9 page

A Fefferman-Stein inequality for the Carleson operator

We provide a Fefferman-Stein type weighted inequality for maximally modulated Calder\'on-Zygmund operators that satisfy \textit{a priori} weak type unweighted estimates. This inequality corresponds to a maximally modulated version of a result of P\'erez. Applying it to the Hilbert transform we obtain the corresponding Fefferman-Stein inequality for the Carleson operator $\mathcal{C}$, that is $\mathcal{C}: L^p(M^{\lfloor p \rfloor +1}w) \to L^p(w)$ for any $1<p<\infty$ and any weight function $w$, with bound independent of $w$. We also provide a maximal-multiplier weighted theorem, a vector-valued extension, and more general two-weighted inequalities. Our proof builds on a recent work of Di Plinio and Lerner combined with some results on Orlicz spaces developed by P\'erez.Comment: Revised version. To appear in Rev. Mat. Ibe

Pointwise localization and sharp weighted bounds for Rubio de Francia square functions

Let $H_\omega f$ be the Fourier restriction of $f\in L^2(\mathbb{R})$ to an interval $\omega\subset \mathbb{R}$. If $\Omega$ is an arbitrary collection of pairwise disjoint intervals, the square function of $\{H_\omega f: \omega \in \Omega\}$ is termed the Rubio de Francia square function $T^\Omega$. This article proves a pointwise bound for $T^\Omega f$ by a sparse operator involving local $L^2$-averages. A pointwise bound for the smooth version of $T^\Omega$ by a sparse square function is also proved. These pointwise localization principles lead to quantified $L^p(w)$, $p>2$ and weak $L^p(w)$, $p\geq 2$ norm inequalities for $T^\Omega$. In particular, the obtained weak $L^p(w)$ norm bounds are new for $p\geq 2$ and sharp for $p>2$. The proofs rely on sparse bounds for abstract balayages of Carleson sequences, local orthogonality and very elementary time-frequency analysis techniques. The paper also contains two results related to the outstanding conjecture that $T^\Omega$ is bounded on $L^2(w)$ if and only if $w\in A_1$. The conjecture is verified for radially decreasing even $A_1$ weights, and in full generality for the Walsh group analogue of $T^\Omega$.Comment: 27 pages; submitted. v2: typos fixed and more details provide

Control of pseudodifferential operators by maximal functions via weighted inequalities

We establish general weighted L 2 inequalities for pseudodifferential operators associated to the Hörmander symbol classes S ρ,δm . Such inequalities allow one to control these operators by fractional “non-tangential” maximal functions and subsume the optimal range of Lebesgue space bounds for pseudodifferential operators. As a corollary, several known Muckenhoupt-type bounds are recovered, and new bounds for weights lying in the intersection of the Muckenhoupt and reverse Hölder classes are obtained. The proof relies on a subdyadic decomposition of the frequency space, together with applications of the Cotlar–Stein almost orthogonality principle and a quantitative version of the symbolic calculus

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