Sharp bounds for general commutators on weighted Lebesgue spaces

2010/02/11 by Chung, Daewon, Pereyra, Cristina, Perez, Carlos
#42B20 #42B25 #46B70 #47B38 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1002.2396

Abstract

We show that if an operator T is bounded on weighted Lebesgue space L²(w) and obeys a linear bound with respect to the A₂ constant of the weight, then its commutator [b,T] with a function b in BMO will obey a quadratic bound with respect to the A₂ constant of the weight. We also prove that the kth-order commutator T^k_b=[b,T^k-1_b] will obey a bound that is a power (k+1) of the A₂ constant of the weight. Sharp extrapolation provides corresponding L^p(w) estimates. The results are sharp in terms of the growth of the operator norm with respect to the A_p constant of the weight for all 1

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