A sparse approach to mixed weak type inequalities
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Mathematische Zeitschrift
A sparse approach to mixed weak type inequalities Marcela Caldarelli1 · Israel P. Rivera-Ríos1,2 Received: 20 December 2018 / Accepted: 6 November 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2019
Abstract In this paper we provide some quantitative mixed weak-type estimates assuming conditions that imply that uv ∈ A∞ for Calderón–Zygmund operators, rough singular integrals and commutators. The main novelty of this paper lies in the fact that we rely upon sparse domination results, pushing an approach to endpoint estimates that was introduced in Domingo-Salazar et al. (Bull Lond Math Soc 48(1):63–73, 2016) and extended in Lerner et al. (Adv Math 319:153–181, 2017) and Li et al. (J Geom Anal, 2018).
1 Introduction and main results In [27], Muckenhoupt and Wheeden, introduced a new type of weak type inequality, that we call mixed weak-type inequality, that consists in considering a perturbation of the Hardy– Littlewood maximal operator with an A p weight. Their result was the following Theorem A Let w ∈ A1 then |{x ∈ R : w(x)M f (x) > t}| ≤ cw
1 t
R
| f |w(x)d x.
Although this kind of estimate may seem not very different to the standard one, the perturbation caused by having the weight inside the level set makes it way harder to be settled, in contrast with the analogous case of strong type estimates. Furthermore, w ∈ A1 is no longer a necessary condition for this endpoint estimate to hold (see [27, Section 5]). Later on, Sawyer [33], motivated by the possibility of providing a new proof of Muckenhoupt’s theorem, obtained the following result.
I. P. Rivera-Ríos is supported by CONICET PIP 11220130100329CO.
B
Israel P. Rivera-Ríos [email protected] Marcela Caldarelli [email protected]
1
Departamento de Matemática, Universidad Nacional del Sur, Alem 1253, Bahía Blanca, Argentina
2
INMABB, CONICET, Alem 1253, Bahía Blanca, Argentina
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M. Caldarelli, I. P. Rivera-Ríos
Theorem B Let u, v ∈ A1 then M( f v)(x) 1 uv x ∈R : | f |u(x)v(x)d x. (1.1) >t ≤ cu,v v(x) t R Sawyer also conjectured that (1.1) should hold as well for the Hilbert transform. CruzUribe et al. [7] generalized (1.1) to higher dimensions and actually proved that Sawyer’s conjecture holds for Calderón–Zygmund operators via the following extrapolation argument. Theorem C Assume that for every w ∈ A∞ and some 0 < p < ∞, T f L p (w) ≤ cw G f L p (w) . Then for every u ∈ A1 and every v ∈ A∞ T f L 1,∞ (uv) ≤ cu,v,n G f L 1,∞ (uv) . The conditions on the weights in that extrapolation result lead them to conjecture that (1.1), and consequently the corresponding estimate for Calderón–Zygmund operators should hold as well with u ∈ A1 and v ∈ A∞ . That conjecture was positively answered recently in [25] where several quantitative estimates were provided as well. At this point we would like to mention, as well, a recent generalization for Orlicz maximal operators provided in [2]. In [7], besides the aforementioned results, it was shown that (1.1) holds if u ∈ A1 and v ∈
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