An elementary proof of the A2 bound

By Michael T. Lacey. 2015

216 citations recorded by the source corpus.

A martingale transform T, applied to an integrable locally supported function f, is pointwise dominated by a positive sparse operator applied to |f|, the choice of sparse operator being a function of T and f. As a corollary, one derives the sharp Ap bounds for martingale transforms, recently proved by Thiele-Treil-Volberg, as well as a number of new sharp weighted inequalities for martingale transforms. The (very easy) method of proof (a) only depends upon the weak-L1 norm of maximal truncations of martingale transforms, (b) applies in the vector valued setting, and (c) has an extension to the continuous case, giving a new elementary proof of the A2 bounds in that setting.

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/142dba80c3b041f9bcf1ea1772bb9152310d6171

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In Applied Mathematics

Citations in the corpus

Quantitative estimates and extrapolation for multilinear weight classes (2018, background): “We also refer the reader to [23,26], where the weaker Dini condition was considered in the linear case.”
Sparse bounds for maximal rough singular integrals via the Fourier transform (2017, methodology): “[10] Amalia Culiuc, Robert Kesler, and Michael T. Lacey, Sparse bounds for the discrete cubic Hilbert transform, preprint arXiv:1612.08881.”
Sparse bounds for spherical maximal functions (2017, background): “These arguments have delivered the most powerful known proof [20] of the A2 conjecture.”
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…”
Uniform sparse domination of singular integrals via dyadic shifts (2016, methodology): “However, these methods fall short of tackling the Dini-continuous case first settled in [14].”
Sparse Domination Theorem for Multilinear Singular Integral Operators with $L^{r}$-Hörmander Condition (2016, methodology): “Finally, Lacey [17] relaxed the log-Dini condition to Dini condition and then Hytönen, Roncal and Tapiola [14] refined the proof by tracking the precise dependence on the constants.”
Borderline weak‐type estimates for singular integrals and square functions (2015, background): “Concerning square functions, a variant of the argument in [11] shows that the intrinsic square function is dominated by a sum of at most 3 sparse square functions defined by Sf(2) = ∑ Q∈S 〈f〉(2)Q1Q.”