Euskal Herriko Unibertsitatea and Ikerbasque
Commutators of singular integrals with BMO functions
Commutators of singular integral operators with BMO functions were introduced in the seventies by Coifman–Rochberg and Weiss. These are very interesting operators for many reasons and their study became a classical topic in modern harmonic analysis. One reason of this interest is due to the fact that they are more singular than Calderón–Zygmund operators. This idea can be expressed in many ways. In this lecture we plan to give three reasons that explain this “bad” behavior. One of them is related to a sharp weighted L2 estimate with respect to A2 weights but the novelty is that the bound in terms of the A2 constant of the weight is quadratic and no better while in the case of singular integrals it is simply linear. The second reason is the fact that there is an appropriate local sub-exponential decay which in the case of singular integrals is of exponential type instead. The third reason is related to the fact that commutators are controlled by iterations of the maximal function with a sharp new A∞ constant. Pieces of the lecture are part of joint works with D. Chung and C. Pereyra, with C. Ortiz and E. Rela, with T. Luque and E. Rela, and with T. Hytönen.