Extrapolation and sharp norm estimates for classical operators on weighted Lebesgue spaces

By Oliver Dragičević, Loukas Grafakos, M. C. Pereyra, S. Petermichl. 2005

164 citations recorded by the source corpus.

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 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.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/d703f4b43c1003521fc0bb030a6da5db976bce32

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Citations in the corpus

The sharp weighted bound for the Riesz transforms (2007, background): “1, one employs the linear bound for p = 2 together with the extrapolation theorem below, which is a special case of the result contained in [4]: Theorem 6.”
Sharp weighted estimates for classical operators (2010, methodology): “8 was proved by Wittwer [57] and by Hukovic, Treil and Volberg [22]; this was extended to p < 2 by extrapolation in [12] and examples were given to show that in this range the exponent is the best possible.”
Representation of singular integrals by dyadic operators, and the A_2 theorem (2011, methodology): “[3] Oliver Dragičević, Loukas Grafakos, María Cristina Pereyra, and Stefanie Petermichl.”
Sharp weighted estimates for approximating dyadic operators (2010, methodology): “[6], it suffices to prove this inequality for p = 2, namely (2) ‖T‖L2(w) ≤ cn [w]A2.”
Sharp weighted norm inequalities for Littlewood-Paley operators and singular integrals (2010, background): “By the extrapolation argument [6], the linear ‖w‖A2 bound implies the bound by ‖w‖max(1,1/(p−1)) Ap for all p > 1.”
The sharp weighted bound for general Calderon-Zygmund operators (2010, background): “A. Pigeonholing à la Lacey et al. The bound (7.1) will be accomplished by carefully partitioning the collection K into appropriate subsets, where the weights w and σ are well under control — a procedure introduced by Lacey, Petermichl and Reguera [13].”
Weighted martingale multipliers in non-homogeneous setting and outer measure spaces (2014, background): “and let Φ ∆X (t) := B(X0 + t∆X), Φ [1,2] ∆X (t) = B[1,2](X(t)).”
The sharp weighted bound for general Calderón-Zygmund operators (2010, background): “By the sharp form of Rubio de Francia’s extrapolation theorem due to Dragičević, Grafakos, Pereyra and Petermichl [6], this implies the corresponding weighted Lp bound, (1.”
The sharp bound for the Hilbert transform on weighted Lebesgue spaces in terms of the classical A p characteristic (2007, methodology): “is discussed in [ 8 ] and also employs the result for p = 2 together with extrapolation.”
The A_2 theorem: Remarks and complements (2012, background): “6) which was originally deduced from the A2 theorem after an additional extrapolation argument from [4].”