A pointwise estimate for the local sharp maximal function with applications to singular integrals

By Andrei K. Lerner. 2010

167 citations recorded by the source corpus.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/bd6c851ead7b76704223a904d95278c95cfae5aa

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In Applied Mathematics

In Mathematical Physics

Citations in the corpus

A simple proof of the $A_2$ conjecture (2012, background): “The following theorem was proved in [17] (its very similar version can be found in [16]).”
The A_p-A_infty inequality for general Calderon--Zygmund operators (2011, methodology): “As in [6], Lerner’s local oscillation formula from [10] is used to reduce the estimation of a dyadic shift operator to related positive operators of the form f 7→ ∑”
A Simple Proof of the A2 Conjecture (2013, background): “The following theorem was proved in [18] (its very similar version can be found in [17]).”
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].”
Sharp Weighted Bounds for Multilinear Maximal Functions and Calderón–Zygmund Operators (2012, background): “The following theorem was proved in [16] (its very similar version can be found in [15]).”
On Sharp Aperture-Weighted Estimates for Square Functions (2013, background): “The following theorem was proved in [15] (its very similar version can be found in [12]).”
The sharp weighted bound for general Calderon-Zygmund operators (2010, background): “formula of Lerner [18], which gives very precise and useful information on a function in terms of its local oscillations.”
The L-to-L boundedness of commutators with applications to the Jacobian operator (2018, background): “This is more elementary variant of Lerner’s oscillation formula [17]; we recall the idea of the proof.”
Sharp weighted norm inequalities for Littlewood-Paley operators and singular integrals (2010, background): “Very recently, a new, more elementary proof of this result, avoiding the T b theorem, was given by Cruz-Uribe, Martell and Pérez [5]; a key ingredient in [5] was a decomposition of an arbitrary measurable function in terms of local mean oscillations obtained in [15].”
On an estimate of Calderón-Zygmund operators by dyadic positive operators (2012, methodology): “Lacey [9] where they used the “local mean oscillation decomposition” from [17] to bound S D by the sum ∑κ+1 i=1 Ai, where κ = max(k,m, 1) is the complexity of S m,k D , and the operators Ai are defined by Aif(x) = AD,S,if(x) = ∑”
Representation of singular integrals by dyadic operators, and the A_2 theorem (2011, methodology): “Lerner [13], which provides were precise and useful information of a function in terms of its ‘local oscillations’: the local oscillations of Sf (and even S♮f) can be estimated in terms of f with the help of the weak-type (1, 1) inequality for S (and S♮).”
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…”
Sharp weighted estimates of the dyadic shifts and $A_2$ conjecture (2010, background): “Another proof of the linear bound fordyadic shifts was obtained in Cruz-Uribe– Martell–Pérez [4], [5] in a very beautiful and concise approach based on a remarkable “formula” by Lerner [18].”
An elementary proof of the A2 bound (2015, methodology): “ies for martingale transforms were proved by Bonami-Lépingle [BL, Th. 1] in 1978.1 Their proof follows the good-λapproach, which does not yield sharp constants. The Lerner mean oscillation inequality [L2] gives a pointwise bound on an arbitrary measurable function φby a sum over a sparse dyadic coll”
The sharp weighted bound for general Calderón-Zygmund operators (2010, background): “formula of Lerner [18], which gives very precise and useful information on a function in terms of its local oscillations.”