Vector-valued operators, optimal weighted estimates and the Cp condition

Abstract

In this paper some new results concerning the C p classes introduced by Muckenhoupt [28] and later extended by Sawyer [39], are provided. In particular we extend the result to the full range expected p > 0, to the weak norm, to other operators and to their vector-valued extensions. Some of those results rely upon sparse domination results that in some cases we provide as well. We will also provide sharp weighted estimates for vector valued extensions relying on those sparse domination results.

Weighted variation inequalities for singular integrals and commutators

“…We also refer readers to [10] for its application to the quantitative C p estimate for Calderón-Zygmund operators. Now, by Theorem 1.3 and employing the arguments in [11,28], we may obtain: This provides an extension of the corresponding results for singular integrals and commutators in [28,44] to the variation operators. As far as we are concerned, these results are completely new, since no local exponential decay estimates of variation operators have been considered before.…”

“…where A S f (x) = Q∈S |Q| −1 ´Q |f |χ Q (x) and S is a sparse family of dyadic cubes from R n (see [34] for the definition of T ). Since then, there is a good number of literature using sparse domination methods to deal with other operators, see [3,11,14,26,32], and these are far from complete. Now let's turn to the commutators of a linear or sublinear operator T .…”

Sharp Reverse Hölder Inequality for $$C_p$$ Weights and Applications

“…See Section 5 for the precise definitions and for an exposition on the inequality. Recently, inequality (CFI−p) has been shown to hold for a wider variety of operators [5,7].…”

“…For example, the Fefferman-Stein inequality, between the maximal operators of Hardy-Littlewood and of Fefferman-Stein, as can be found in [27], [6] for a quantified version, [17] in the weak-type context. In [7], the authors extended Sawyer's result to a wider class of operators than Calderón-Zygmund operators, including some pseudo-differential operators and oscillatory integrals. Finally, in [5], Sawyer's result was extended to rough singular integrals and sparse forms.…”

Sparse domination for the lattice Hardy–Littlewood maximal operator

“…In the particular case E = ℓ q , the strong-type weighted bound (5.3) together with its sharpness was proved in [6]. After the appearance of this manuscript on arXiv, another manuscript appeared, in which the weighted bounds (5.2) and (5.4) for the lattice maximal operator were deduced independently in the particular case E = ℓ q , see [3,Theorem 2].…”