Heating of the Ahlfors-Beurling operator: weakly quasiregular maps on the plane are quasiregular

By S. Petermichl, Alexander Volberg. 2002

236 citations recorded by the source corpus.

View Paper: https://www.semanticscholar.org/paper/0e4b40944bf5dc092c07bf32903f60c063ebe1a6

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In Applied Mathematics

Citations in the corpus

Sharp weighted estimates of the dyadic shifts and $A_2$ conjecture (2010, methodology): “This result was used in [34] to answer positively an important question in the theory of quasiconformal maps, see [1], about whether a weakly quasiregular map is quasiregular (or equivalently whether there is a self-improvement of a solution of the Beltrami equation in the case of critical exponent)”
SHARP WEIGHTED BOUNDS FOR FRACTIONAL INTEGRAL OPERATORS (2009, background): “eighted norm for singular integral operators is motivated in part by applications in partial diﬀerential equations. We refer the reader to Astala, Iwaniec, and Saksman [1]; and Petermichl and Volberg [23] for such applications. Petermichl [21], [22] showed that (1.3) kTkLp(w)→Lp(w) ≤ c[w] max{1,1/(p”
A1 bounds for Calderón-Zygmund operators related to a problem of Muckenhoupt and Wheeden (2009, unknown): “Volberg [11, 12, 13] inequality (1.”
The sharp weighted bound for general Calderón-Zygmund operators (2010, result): “solving a conjecture by Astala–Iwaniec–Saksman[1, Eq. (45)] with implications to Beltrami equations, the case of the Beurling–Ahlfors transform B ∈ L(L2(C)) was ﬁrst settled by Petermichl and Volberg [24], and with an alternative proof by Dragičević and Volberg [7]. Petermichl also obtained the shar”
Sharp weighted norm inequalities for Littlewood-Paley operators and singular integrals (2010, methodology): “Recently, Lacey, Petermichl and Reguera [11] have established sharp weighted estimates for Haar shift operators without the use of Bellman functions; their proof uses a two-weight “Tb theorem” for Haar shift operators due to Nazarov, Treil and Volberg [15].”
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.”
Two weight estimate for the Hilbert transform and corona decomposition for non-doubling measures (2010, background): “This condition might even happen to be necessary for the two wight boundedness of the Hilbert transform.”
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].”
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 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.”
Quantitative weighted estimates for rough homogeneous singular integrals (2015, background): “1) and this was positively answered in [19].”
SHARP BOUNDS FOR GENERAL COMMUTATORS ON WEIGHTED LEBESGUE SPACES (2010, unknown): “Volberg [32] and by Petermichl [30, 31] for the Hilbert transform and the Riesz transforms.”
The A_2 theorem: Remarks and complements (2012, background): “After being established for several particular operators first [10, 25, 26, 29], I proved (1.”
Heating of the Ahlfors–Beurling operator, and estimates of its norm (2004, methodology): “We prefer another approach, which follows the approach in [30].”