Weak Type Estimates for Singular Integrals Related to a Dual Problem of Muckenhoupt–Wheeden

Abstract

Abstract.A well known open problem of Muckenhoupt-Wheeden says that any Calderón-Zygmund singular integral operator T is of weak type (1, 1) with respect to a couple of weights (w, M w). In this paper we consider a somewhat "dual" problem:We prove a weaker version of this inequality with M 3 w instead of M w. Also we study a related question about the behavior of the constant in terms of the A 1 characteristic of w.

Quadratic $A^1$ bounds for commutators of singular integrals with BMO functions

“…Lerner,S. Ombrosy and C. Pérez,in [LOPe1] and [LOPe3] (see also [LOPe2] for a dual problem). Indeed, this result was extended to any 1 < p < ∞ and to any Calderón-Zygmund operator.…”

Sharp weighted bounds involvingA∞

“…It seems that most probably the L log L result (1.13) is the best possible. On the other hand, in [Lerner et al 2009b], a sort of "dual" estimate to the last bound was found, which is also of interest for related matters:…”

Exponential decay estimates for singular integral operators

“…We mention here in passing that this optimal weighted dependence, called the A 2 conjecture, has been proved recently and by different means by T. Hytönen in [18] (see also [17], [15] and [16] for a further improvement and the recent work [22] for a very interesting simplication of the proof of the A 2 conjecture). On the other hand, this exponential decay (1.4) has been a crucial step in deriving corresponding sharp A 1 estimate in [25], [27]. Our point of view is different and has been motivated by an improved version of inequality (1.4) due to Karagulyan [21]: (1.5) |{x ∈ Q : T * f (x) > tM f (x)}| ≤ ce −αt |Q|, t > 0 However, it is not clear that the proof can be adapted to other situations.…”