Three Observations on Commutators of Singular Integral Operators with BMO Functions

Abstract

Three observations on commutators of Singular Integral Operators with BMO functions are exposed, namely Section 2 The already known subgaussian local decay for the commutator, namelyis sharp, since it cannot be better than subgaussian. Section 3 It is not possible to obtain a pointwise control of the commutator by a finite sum of sparse operators defined by L log L averages. Section 4 Motivated by the conjugation method for commutators, it is shown the failure of the following endpoint estimate, if w ∈ A p \ A 1 then

A_1 theory of weights for rough homogeneous singular integrals and commutators

“…As an application of this result, a commutator estimate with BMO functions can be derived by means of the conjugation method as introduced in [6, p. 621] (see also [1,8,35]). Indeed, recall that given a linear operator T and b…”

Sparse and weighted estimates for generalized Hörmander operators and commutators

“…Now for the commutator and the iterated commutator we use the conjugation method (See [4,3,39] for more details about this method). We recall that…”

$A_1$ theory of weights for rough homogeneous singular integrals and commutators

“…As an application of this result, a commutator estimate with BMO functions can be derived by means of the conjugation method as introduced in [5, p. 621] (see also [1,7,34]). Indeed, recall that given a linear operator T and b ∈ BMO the commutator of Coifman-Rochberg-Weiss [b, T ] is defined as…”