We obtain sharp weighted L p estimates in the Rubio de Francia extrapolation theorem in terms of the Ap characteristic constant of the weight. Precisely, if for a given 1 < r < ∞ the norm of a sublinear operator on L r (w) is bounded by a function of the Ar characteristic constant of the weight w, then for p > r it is bounded on L p (v) by the same increasing function of the Ap characteristic constant of v, and for p < r it is bounded on L p (v) by the same increasing function of the r−1 p−1 power of the Ap characteristic constant of v. For some operators these bounds are sharp, but not always. In particular, we show that they are sharp for the Hilbert, Beurling, and martingale transforms.
Abstract. We show that if a linear operator T is bounded on weighted Lebesgue space L 2 (w) and obeys a linear bound with respect to the A 2 constant of the weight, then its commutator [b, T ] with a function b in BM O will obey a quadratic bound with respect to the A 2 constant of the weight. We also prove that the kth-order] will obey a bound that is a power (k + 1) of the A 2 constant of the weight. Sharp extrapolation provides corresponding L p (w) estimates. In particular these estimates hold for T any Calderón-Zygmund singular integral operator. 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 < p < ∞, all k, and all dimensions, as examples involving the Riesz transforms, power functions and power weights show.
We give an explicit construction of Haar functions associated to a system of dyadic cubes in a geometrically doubling quasi-metric space equipped with a positive Borel measure, and show that these Haar functions form a basis for L p . Next we focus on spaces X of homogeneous type in the sense of Coifman and Weiss, where we use these Haar functions to define a discrete square function, and hence to define dyadic versions of the function spaces H 1 (X) and BMO(X). In the setting of product spaces X = X 1 × · · · × X n of homogeneous type, we show that the space BMO( X) of functions of bounded mean oscillation on X can be written as the intersection of finitely many dyadic BMO spaces on X, and similarly for A p ( X), reverse-Hölder weights on X, and doubling weights on X. We also establish that the Hardy space H 1 ( X) is a sum of finitely many dyadic Hardy spaces on X, and that the strong maximal function on X is pointwise comparable to the sum of finitely many dyadic strong maximal functions. These dyadic structure theorems generalize, to product spaces of homogeneous type, the earlier Euclidean analogues for BMO and H 1 due to Mei and to Li, Pipher and Ward.
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