The sharp bound for the Hilbert transform on weighted Lebesgue spaces in terms of the classical A p characteristic

By S. Petermichl. 2007

257 citations recorded by the source corpus.

Subfields: Applied Mathematics, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/9ca23e014a0f269f312d133a0ec9a1ec0e5f6413

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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, methodology): “The result in [32] is a considerable strengthening of Fefferman–Pipher’s theorem.”
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.”
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.”
An elementary proof of the A2 bound (2015, background): “nhoupt A2condition [M2]. One could have asked, even then, what the sharp dependence on the norm of His, in terms of the A2characteristic of the weight. This was not resolved until 2007, by Petermichl [P], showing that norm of the Hilbert transform is linear in the A2 characteristic. Five years later”
Sharp weighted norm estimates beyond Calderón–Zygmund theory (2015, unknown): “As done in previous works (see for example [Petermichl 2007; Hytönen 2012; Lerner 2010; 2013a; 2013b; Lacey 2015]), the analysis will involve a discrete stopping-time argument that relies on nice properties associated with a dyadic structure, which is by now well-known in the context of doubling…”
The A_2 theorem: Remarks and complements (2012, background): “After being established for several particular operators first [10, 25, 26, 29], I proved (1.”
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].”
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.”
The sharp weighted bound for the Riesz transforms (2007, methodology): “Without loss of generality, we fix out shift operator here so that L0 is the north west corner child of Ll. Proof. As in [ 12 ] we consider the function”
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.”
On $A_2$ conjecture and corona decomposition of weights (2010, background): “tor, and its sharp weighted estimate allowed the authors to solve a problem of Iwaniec on a borderline regularity 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.”