Sharp weighted estimates for classical operators

By D. Cruz-Uribe, J. M. Martell, C. P'erez. 2010

193 citations recorded by the source corpus.

Abstract We give a general method based on dyadic Calderon–Zygmund theory to prove sharp one- and two-weight norm inequalities for some of the classical operators of harmonic analysis: the Hilbert and Riesz transforms, the Beurling–Ahlfors operator, the maximal singular integrals associated to these operators, the dyadic square function and the vector-valued maximal operator. In the one-weight case we prove the sharp dependence on the A p constant by finding the best value for the exponent α ( p ) such that ‖ T f ‖ L p ( w ) ⩽ C n , T [ w ] A p α ( p ) ‖ f ‖ L p ( w ) . For the Hilbert transform, the Riesz transforms and the Beurling–Ahlfors operator the sharp value of α ( p ) was found by Petermichl and Volberg (2007, 2008, 2002) [47] , [48] , [49] ; their proofs used approximations by the dyadic Haar shift operators, Bellman function techniques, and two-weight norm inequalities. Our proofs again depend on dyadic approximation, but avoid Bellman functions and two-weight norm inequalities. We instead use a recent result due to A. Lerner (2010) [34] to estimate the oscillation of dyadic operators. By applying this we get a straightforward proof of the sharp dependence on the A p constant for any operator that can be approximated by Haar shift operators. In particular, we provide a unified approach for the Hilbert and Riesz transforms, the Beurling–Ahlfors operator (and their corresponding maximal singular integrals), dyadic paraproducts and Haar multipliers. Furthermore, we completely solve the open problem of sharp dependence for the dyadic square functions and vector-valued Hardy–Littlewood maximal function. In the two-weight case we use the very same techniques to prove sharp results in the scale of A p bump conditions. For the singular integrals considered above, we show they map L p ( v ) into L p ( u ) , 1 p ∞ , if the pair ( u , v ) satisfies sup Q ‖ u 1 / p ‖ A , Q ‖ v − 1 / p ‖ B , Q ∞ , where A ¯ ∈ B p ′ and B ¯ ∈ B p are Orlicz functions. This condition is sharp. Furthermore, this condition characterizes (in the scale of these A p bump conditions) the corresponding two-weight norm inequality for the Hardy–Littlewood maximal operator M and its dual: i.e., M : L p ( v ) → L p ( u ) and M : L p ′ ( u 1 − p ′ ) → L p ( v 1 − p ′ ) . Muckenhoupt and Wheeden conjectured that these two inequalities for M are sufficient for the Hilbert transform to be bounded from L p ( v ) into L p ( u ) . Thus, in the scale of A p bump conditions, we prove their conjecture. We prove similar, sharp two-weight results for the dyadic square function and the vector-valued maximal operator.

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/fe26a07f4668a51d7f39795d06f89e7d51dca0d6

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

Citations in the corpus

SHARP BOUNDS FOR GENERAL COMMUTATORS ON WEIGHTED LEBESGUE SPACES (2010, unknown): “…[8], no Bellman function techniques nor any two weight results are used, and the methods can be extended to other important operators in Harmonic Analysis such as dyadic square functions and paraproducts, maximal singular integrals and the vector-valued maximal function as can be found in [9].”
Extrapolation for multilinear Muckenhoupt classes and applications (2018, background): “To proceed we follow an argument in [15], see also [29], which in turn is a bilinear extension of the linear case proof in [13].”
On Sharp Aperture-Weighted Estimates for Square Functions (2013, result): “In the case when m = 0, the following result was proved in [4].”
Sharp weighted bounds without testing or extrapolation (2012, methodology): “Most known proofs of (1) follow this paradigm, beginning with p = 2 and then extrapolating [2, 5, 10, 13, 16, 19, 20].”
Sharp Bekolle estimates for the Bergman projection (2012, methodology): “Here, we use the easier approach from Cruz-Uribe, Martell and Pérez in [7, 8].”
On an estimate of Calderón-Zygmund operators by dyadic positive operators (2012, methodology): “After that, using the same decomposition and the operator A, the conjecture was proved for the Hilbert, Riesz and Beurling transforms in [4].”
One and two weight norm inequalities for Riesz potentials (2012, background): “For the complete history of the problem we also refer the reader to [2, 18, 22, 24, 32] and the references they contain.”
A simple proof of the $A_2$ conjecture (2012, background): “This was first observed in [5, 6] (see also [17]).”
A Simple Proof of the A2 Conjecture (2013, background): “[5, 6], 2010); • the L2(w) bound for general T by ‖w‖A2 log(1+ ‖w‖A2) (Pérez et al.”
Sharp weighted estimates for approximating dyadic operators (2010, methodology): “All these results can be found in [5].”
The Sharp Weighted Bound for Multilinear Maximal Functions and Calderón–Zygmund Operators (2012, background): “2) Again, we refer the reader to [5,10,15,17] for the background material and further references.”
The A_p-A_infty inequality for general Calderon--Zygmund operators (2011, methodology): “The application of Lerner’s formula [10] to Haar shift operators is well-known, see [1], [2], [9], [6].”
Two weight norm inequalities for fractional integral operators and commutators (2014, background): “For a more detailed overview of these conjectures, see [20, 21].”