The sharp weighted bound for general Calderon-Zygmund operators

By Tuomas Hytönen. 2010

343 citations recorded by the source corpus.

for all Muckenhoupt weights w ∈ A2. This optimal estimate was known as the A2 conjecture. A recent result of Perez–Treil–Volberg reduced the problem to a testing condition on indicator functions, which is verified in this paper. The proof consists of the following elements: (i) a variant of the Nazarov– Treil–Volberg method of random dyadic systems with just one random system and completely without “bad” parts; (ii) a resulting representation of a general Calderón–Zygmund operator as an average of “dyadic shifts”; and (iii) improvements of the Lacey–Petermichl–Reguera estimates for these dyadic shifts, which allow summing up the series in the obtained representation.

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/93a163254a6311b1acfdf363cbb9f58df2e32f96

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In Applied Mathematics

Citations in the corpus

Representation of singular integrals by dyadic operators, and the A_2 theorem (2011, methodology): “DYADIC REPRESENTATION AND A2 THEOREM 29 The second displayed line is the original A2 theorem [6], and it follows from the first line by [w]A∞ .”
SHARP BOUNDS FOR GENERAL COMMUTATORS ON WEIGHTED LEBESGUE SPACES (2010, unknown): “2) for any Calderón-Zygmund operator T has been proved in [16] by T.”
On an estimate of Calderón-Zygmund operators by dyadic positive operators (2012, background): “Hytönen [7] (see also [8, 13]), and its “maximal truncated” corollary proved in [11].”
Sharp weighted estimates for classical operators (2010, unknown): “To do so, we will follow the approach used by Hytönen et al. [23] and actually prove the corresponding result for a family of “maximal” dyadic shift operators.”
Sharp weighted estimates of the dyadic shifts and $A_2$ conjecture (2010, methodology): “Using this result of Pérez–Treil–Volberg and the technique developed in [16], the first author [14] was able to prove the A2 conjecture for general Calderón–Zygmund operators, i.”
Quantitative weighted estimates for rough homogeneous singular integrals (2015, methodology): “so any bounds in terms of the weight characteristics (w)Ap and {w}Ap are stronger than similar bounds using [w] max{1,1/(p−1)} Ap , which is the expression that most frequently appears in sharp estimates for Calderón–Zygmund operators, like [8].”
The Sharp Weighted Bound for Multilinear Maximal Functions and Calderón–Zygmund Operators (2012, unknown): “Later is was proven by Hyt¨onen [8] that if T is a Calder´on-Zygmund operator then (1.2) Again, we refer the reader to [9, 12] for the background material and further references.”
One and two weight norm inequalities for Riesz potentials (2012, methodology): “Pérez and Hytönen [15] showed that [w]A′∞ .”
Sharp weighted bounds for the q‐variation of singular integrals (2012, methodology): “However, their arguments used the random dyadic systems, and were derived from the (non-positive) dyadic representation theorem of [6].”
Sharp weighted bounds without testing or extrapolation (2012, methodology): “[9] T. Hytönen and F. Nazarov, The local Tb theorem with rough test functions, preprint.”
Sparse Domination Theorem for Multilinear Singular Integral Operators with $L^{r}$-Hörmander Condition (2016, background): “11 [12] T. Hytönen, The sharp weighted bound for general Calderón-Zygmund operators.”
Representation of bi-parameter singular integrals by dyadic operators (2011, unknown): “Such a representation theorem exists for ordinary Calderón– Zygmund operators, and this was proven by Hytönen [7] in connection with the proof of the A 2 conjecture for general singular integrals.”
A simple proof of the $A_2$ conjecture (2012, methodology): “1) in full generality (Hytönen [7], 2010); • a simplification of the proof (Hytönen et al.”
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…”
Two Weight Inequality for the Hilbert Transform: A Real Variable Characterization, II (2012, result): “Partial information about the two weight problem for singular integrals [21] was basic to the resolution of the A2 conjecture [3], and several related results [4,6,21,22].”