Mikel Flórez-Amatriain

arXiv2025

$L^p$-estimates for singular integral operators along codimension one subspaces

In this paper we study maximal directional singular integral operators in $ \mathbb{R}^n $ given by a Hörmander--Mihlin multiplier on an $ (n-1)$-dimensional subspace and acting trivially in the perpendicular direction. The subspace is allowed to depend measurably on the first $ n-1 $ variables of $ \mathbb{R}^n $. Assuming the subspace to be non degenerate in the sense that it is away from a cone around $e_n$ and the function $ f $ to be frequency supported in a cone away from $ \mathbb{R}^{n-1} $, we prove $ L^p $-bounds for these operators for $ p > 3/2 $. If we assume, additionally, that $ \widehat{f} $ is supported in a single frequency band, we are able to extend the boundedness range to $ p >1 $. The non-degeneracy assumption cannot in general be removed, even in the band-limited case.

math.CA↗

arXiv2024

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

Let $H_ωf$ be the Fourier restriction of $f\in L^2(\mathbb{R})$ to an interval $ω\subset \mathbb{R}$. If $Ω$ is an arbitrary collection of pairwise disjoint intervals, the square function of $\{H_ωf: ω\in Ω\}$ is termed the Rubio de Francia square function $T^Ω$. This article proves a pointwise bound for $T^Ωf$ by a sparse operator involving local $L^2$-averages. A pointwise bound for the smooth version of $T^Ω$ 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 $T^Ω$. 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 $T^Ω$ 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 $T^Ω$.

math.CA↗