In this paper we provide quantitative Bloom type estimates for iterated commutators of fractional integrals improving and extending results from [15]. We give new proofs for those inequalities relying upon a new sparse domination that we provide as well in this paper and also in techniques developed in the recent paper [22]. We extend as well the necessity established in [15] to iterated commutators providing a new proof. As a consequence of the preceding results we recover the one weight estimates in [7, 1] and establish the sharpness in the iterated case. Our result provides as well a new characterization of the BMO space.

We provide a characterization of BMO in terms of endpoint boundedness of commutators of singular integrals. In particular, in one dimension, we show that b BMO ≂ B, where B is the best constant in the endpoint L log L modular estimate for the commutator [H, b]. We provide a similar characterization of the space BMO in terms of endpoint boundedness of higher order commutators of the Hilbert transform. In higher dimension we give the corresponding characterization of BMO in terms of the first order commutators of the Riesz transforms. We also show that these characterizations can be given in terms of commutators of more general singular integral operators of convolution type.2010 Mathematics Subject Classification. Primary 42B20, Secondary: 42B25.

No abstract

In this paper we provide quantitative Bloom type estimates for iterated commutators of fractional integrals improving and extending results from [15]. We give new proofs for those inequalities relying upon a new sparse domination that we provide as well in this paper and also in techniques developed in the recent paper [22]. We extend as well the necessity established in [15] to iterated commutators providing a new proof. As a consequence of the preceding results we recover the one weight estimates in [7, 1] and establish the sharpness in the iterated case. Our result provides as well a new characterization of the BMO space.

No abstract