arXiv subjects

Francisco Gonçalves

Publications and source records attributed to Francisco Gonçalves.

3 recordsLinked to original sources

arXiv2026

Sharp mixed $A_p$-$A_\infty$ estimates for sparse operators on filtered and nonhomogeneous measure spaces

We prove mixed $A_p$-$A_\infty$ estimates for sparse operators in two non-doubling settings. In the first setting, we consider sparse operators defined using stopping times in continuous time. We obtain both strong- and weak-type bounds with the same powers of the weight characteristics as in the classical setting. Some of our weak-type bounds are even new for the dyadic filtration on $\mathbb R^d$ and, in particular, imply a sharp weak-type $(2,2)$ estimate for Rubio de Francia square functions, solving a problem left open by Garg, Roncal and Shrivastava [J. Geom. Anal., 31:748-771, 2021]. In the second setting, we consider dyadic sparse forms in which distinct cubes may interact, provided their dyadic distance is bounded. We obtain strong-type bounds with the same powers of the weight characteristics as in the classical setting. As a one-dimensional application, we obtain strong-type bounds for Haar shifts over balanced non-doubling measures, answering a quantitative question posed by Conde-Alonso, Pipher, and Wagner [Math. Ann., 391:2209-2253, 2025].

math.CA↗

arXiv2026

Bounds for the maximal and Riesz potential operators with variable fractionality

We prove $L^{p(\cdot)}$-to-$L^{q(\cdot)}$ bounds for variable versions of the fractional maximal $M^{α(\cdot)}$ and Riesz potential $I^{α(\cdot)}$ operators. The changing fractionality in these operators is given by averaging the function $α(\cdot)$ over balls. The bounds for $M^{α(\cdot)}$ are in terms of a three-exponent Muckenhoupt condition relating $p(\cdot),q(\cdot),$ and $α(\cdot)$, while the bounds for $I^{α(\cdot)}$ are in terms of the boundedness of $M^{α(\cdot)}$ and a packing condition on $α(\cdot).$ These bounds hold under Hardy--Littlewood maximal function boundedness and Muckenhoupt conditions on the individual exponents $p(\cdot),q(\cdot),α(\cdot).$ The proofs are based on an adaptation of sparse domination to variable fractionality and an embedding into variable sequential spaces.

math.CA↗

arXiv2025

Gravity water waves over constant vorticity flows: From laminar flows to touching waves

In a recent paper, Hur & Wheeler [J. Differential Equations, 338:572-590, 2022] proved the existence of periodic steady water waves over an infinitely deep, two-dimensional and constant vorticity flow under the influence of gravity. These solutions include overhanging wave profiles, some of which exhibit surfaces that touch at a point and thereby enclose a bubble of air. We extend these results by formulating a problem that encompasses both infinitely deep and finitely deep flows, and by proving the existence of a continuous curve of water waves that connects a laminar flow to a touching wave for fixed, nonzero gravity. This implies the existence of a wave profile featuring a vertical tangent at a point, which is not overhanging, and is referred to as a breaking wave. We also study the behaviour of critical layers, which are points where the horizontal velocity vanishes, near the surface. In particular, this result holds for arbitrary vorticity.

math.AP↗