The sharp weighted bound for the Riesz transforms

By S. Petermichl. 2007

148 citations recorded by the source corpus.

We establish the best possible bound on the norm of the Riesz transforms as operators in the weighted space L p Rn (ω) for 1 < p < ∞ in terms of the classical Ap characteristic of the weight.

Subfields: Applied Mathematics, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/59eba98eeb12ee68da13d8524cfa5f0249367866

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In Applied Mathematics

In Numerical Analysis

Citations in the corpus

Sharp weighted estimates of the dyadic shifts and $A_2$ conjecture (2010, background): “Further cases of the A2 conjecture were settled in [2, 33].”
Representation of singular integrals by dyadic operators, and the A_2 theorem (2011, background): “[3] Oliver Dragičević, Loukas Grafakos, María Cristina Pereyra, and Stefanie Petermichl.”
Sharp A2 inequality for Haar shift operators (2009, methodology): “‖w‖A2‖f‖L2(w) , where T is either the Hilbert transform [12], a Riesz transform [13], or the Beurling operator [14].”
Sharp A1 Bounds for Calderón-Zygmund Operators and the Relationship with a Problem of Muckenhoupt and Wheeden (2008, result): “We mention some recent results by Petermichl and Volberg [15] for the Ahlfors– Beurling transform and by Petermichl [13, 14] for the Hilbert transform and the Riesz transforms.”
SHARP BOUNDS FOR GENERAL COMMUTATORS ON WEIGHTED LEBESGUE SPACES (2010, unknown): “This linear growth was proved by S. Petermichl and A. Volberg [32] and by Petermichl [30, 31] for the Hilbert transform and the Riesz Transforms.”
Sharp weighted estimates for approximating dyadic operators (2010, methodology): “In each of the known cases, the proof used a technique developed by Petermichl [17] to reduce the problem to proving the analogous inequality for a corresponding Haar shift operator.”
A simple proof of the $A_2$ conjecture (2012, background): “• Beurling transform (Petermichl and Volberg [23], 2002); • Hilbert transform (Petermichl [21], 2007); • Riesz transform (Petermichl [22], 2008); • Dyadic paraproduct (Beznosova [2], 2008); • Haar shift (Lacey et al.”
On $A_2$ conjecture and corona decomposition of weights (2010, background): “rity of Beltrami PDE. The Hilbert transform turned out to be more diﬃcult to treat, but in [44] Petermichl proved the linear estimate in terms of norm [w]A2 for the Hilbert transform as well. Then in [45] she did this for the Riesz transforms. The Ahlfors–Beurling operator is the averaging of Martin”
Sharp weighted estimates for classical operators (2010, methodology): “When p = 2, Theorem 1.8 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.”
Commutators in the two-weight setting (2015, background): “, n and all w ∈ Ar with 1 < r < ∞ [36], we have that [b,K] is bounded L(μ) → L(λ) for all K that are polynomials in the Riesz transforms and the norm of this operator is at most n ∑”
The sharp weighted bound for general Calderón-Zygmund operators (2010, background): “f by Dragičević and Volberg [7]. Petermichl also obtained the sharp bounds for the Hilbert transform H ∈ L(L2(R)) [22], and then for the Riesz transforms Ri ∈ L(L2(RN)) in arbitrary dimension N ∈ Z + [23]. All these results relied on ad hoc representations based on speciﬁc symmetries of the operator”
The sharp weighted bound for general Calderon-Zygmund operators (2010, background): “Petermichl also obtained the sharp bounds for the Hilbert transform H ∈ L (L2(R)) [27] and then for the Riesz transforms Ri ∈ L (L2(RN )) in arbitrary dimension N ∈ Z+ [28].”
Two Weight Inequalities for Discrete Positive Operators (2009, result): “And it has been subsequently used in the proofs of several results, such as [2, 14,15,24,25] among other papers.”
A1 bounds for Calderón-Zygmund operators related to a problem of Muckenhoupt and Wheeden (2009, unknown): “Volberg [11, 12, 13] inequality (1.”
A Simple Proof of the A2 Conjecture (2013, background): “• Beurling transform (Petermichl and Volberg [23], 2002); • Hilbert transform (Petermichl [21], 2007); • Riesz transform (Petermichl [22], 2008); • Dyadic paraproduct (Beznosova [2], 2008); • Haar shift (Lacey et al.”
Sharp weighted norm inequalities for Littlewood-Paley operators and singular integrals (2010, methodology): “1) is proved for: the Hilbert transform and Riesz transforms (Petermichl [17,18]), the Ahlfors–Beurling operator (Petermichl and Volberg [19]), any onedimensional Calderón–Zygmund convolution operator with sufficiently smooth kernel (Vagharshakyan [21]).”
Sharp weighted norm estimates beyond Calderón–Zygmund theory (2015, unknown): “This conjecture was solved by Petermichl and Volberg [2002] for the Beurling–Ahlfors transform, by Petermichl [2007; 2008] for the Hilbert and Riesz transforms (see also the alternative proof by Lacey, Petermichl and Reguera [Lacey et al. 2010]) and by Hytönen [2012] for arbitrary Calderón–Zygmund…”