On an estimate of Calderón-Zygmund operators by dyadic positive operators

By Andrei K. Lerner. 2012

205 citations recorded by the source corpus.

Given a general dyadic grid D and a sparse family of cubes S = {Qjk ∈ D, define a dyadic positive operator AD,S by $${A_{D,S}}f(x) = \sum\limits_{j,k} {{f_{Q_j^k}}{\chi _{Q_j^k}}} (x)$$. Given a Banach function space X(ℝn) and the maximal Calderón-Zygmund operator $${T_\natural }$$, we show that $${\left\| {{T_\natural}f} \right\|_X} \leqslant c(T,n)\mathop {\sup }\limits_{D,S} {\left\| {{A_{D,S}}|f|} \right\|_X}$$ This result is applied to weighted inequalities. In particular, it implies (i) the “twoweight conjecture” by D. Cruz-Uribe and C. Pérez in full generality; (ii) a simplification of the proof of the “A2 conjecture”; (iii) an extension of certain mixed Ap−Ar estimates to general Calderón-Zygmund operators; (iv) an extension of sharp A1 estimates (known for T ) to the maximal Calderón-Zygmund operator $$\natural $$.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/dbf498897b13ac8a2285824b205cb2ba6125f088

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Citations in the corpus

Quantitative weighted estimates for rough homogeneous singular integrals (2015, background): “Lerner [16, 17], and keeps developing with new extensions, among them the recent approach of M.”
Representation of singular integrals by dyadic operators, and the A_2 theorem (2011, methodology): “[13] Andrei K. Lerner.”
One and two weight norm inequalities for Riesz potentials (2012, background): “The importance of these grids is shown by the following proposition; a proof can be found in [22].”
Sharp Weighted Bounds for Multilinear Maximal Functions and Calderón–Zygmund Operators (2012, background): “Next we get a bound for multilinear Calderón-Zygmund operators in terms of dyadic positive multilinear operators in the spirit of the recent work [16].”
A simple proof of the $A_2$ conjecture (2012, background): “2) in [17] still depended on the representation of T in terms of the Haar shift operators.”
A pointwise estimate for positive dyadic shifts and some applications (2014, methodology): “Lerner gave a simplification of the A2 theorem in [13] which avoided the use of most of the complicated machinery in [8]; it mainly relied on a general pointwise estimate for functions in terms of positive dyadic operators which had already been proven in [12].”
A Simple Proof of the A2 Conjecture (2013, background): “2) in [17] still depended on the representation of T in terms of the Haar shift operators.”
Sharp weighted bounds for the q‐variation of singular integrals (2012, methodology): “For these special shifts, the ideas of Lerner [12, 13] simplify the proof of Proposition 4.”
Sparse bilinear forms for Bochner Riesz multipliers and applications (2016, unknown): “Firstly, Lerner [24, 25] proved that the norm of a C-Z operator in a Banach function space is dominated by sparse positive operators.”
Lp theory for outer measures and two themes of Lennart Carleson united (2013, methodology): “We conclude this section by pointing at an alternative approach to Calderón– Zygmund operators used in Lerner’s work [13], who essentially controls Calderón– Zygmund operators by a superposition of “sparse” operators.”
Two weight norm inequalities for fractional integral operators and commutators (2014, background): “Our understanding of this approach continues to evolve: see for instance, the very recent lecture notes on dyadic approximation by Lerner and Nazarov [60].”
The singular integral operator and its commutator on weighted Morrey spaces (2016, background): “There exists a close relation between these two types of the sharp maximal operators provided by Jawerth and Torchinsky in [18]: (13) M M ♯,d λ f(x) ∼η,λ f (x), for sufficiently small λ, where M (η) denotes the powered Hardy-Littlewood maximal operator defined by M f(x) := sup Q∈Q ( 1 |Q| ∫”
Sharp weighted norm estimates beyond Calderón–Zygmund theory (2015, unknown): “…this extra property, we are going to detail an “elementary” proof of the sharp weighted estimates (simpler than all the existing proofs, such as [Lerner 2013b; Lacey 2015]), which has also the very important property that is extends to nonintegral operators with no regularity property on the…”