Two Weight Inequalities for Discrete Positive Operators

By Michael T. Lacey, Eric T. Sawyer, Ignacio Uriarte-Tuero. 2009

96 citations recorded by the source corpus.

We characterize two weight inequalities for general positive dyadic operators. We consider both weak and strong type inequalities, and general (p,q) mapping properties. Special cases include Sawyers Fractional Integral operator results from 1988, and the bilinear embedding inequality of Nazarov-Treil-Volberg from 1999. The method of proof is an extension of Sawyer's argument.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/fc94902a79cdf04e24fa065556bd98abc7981e60

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Citations in the corpus

A pointwise estimate for positive dyadic shifts and some applications (2014, background): “The weighted result for the positive dyadic shifts that this contribution reduced the problem to had been already shown before in [10], see also [3] and [4].”
The A_p-A_infty inequality for general Calderon--Zygmund operators (2011, background): “In [6], these operators were further dominated by simpler positive operators previously studied in [7].”
The A_2 theorem: Remarks and complements (2012, methodology): “The proof below follows the main lines of Treil’s argument [28], with one key difference: rather than splitting the summation over Q ∈ D in the expansion of 〈T (fσ), gω〉 into parts according to an ad hoc criterion such as σ(Q)〈f〉pQ ≥ ω(Q)〈g〉p ′ Q , I simply apply the “parallel corona” decomposition ”
Representation of singular integrals by dyadic operators, and the A_2 theorem (2011, methodology): “[8] Tuomas Hytönen, Michael T. Lacey, Henri Martikainen, Tuomas Orponen, Maria Carmen Reguera, Eric T. Sawyer, and Ignacio Uriarte-Tuero.”
Weighted Norm Inequalities for Rough Singular Integral Operators (2017, methodology): “ to the inequality CT ε X Q∈S h|f| Qh˜ui 1 1+ε −1 Q χQ Lp,∞(w) ≤ C(w)kfkLp(˜u1−p), where ˜u = w− 1+ε p−(1+ε). Then by the characterization of weak type two weight inequality for dyadic operators (see [19]) and using the calculations in [23] we can obtain the following bound kTkLp(w)→Lp,∞(w) ≤ CT [w]”
A remark on two weight estimates for positive dyadic operators (2012, background): “Later in [1] this theorem was proved in full generality; in fact the case of operators Lp(μ) → Lq(ν) was also treated there.”
A Fractional Muckenhoupt–Wheeden Theorem and its Consequences (2013, methodology): “To estimate the norm of I α , we will use a testing condition due to Lacey, Sawyer and Uriarte-Tuero [15].”
The two‐weight inequality for the Hilbert transform with general measures (2018, methodology): “Q− gdw The previous studies [3, 9, 13] considered a version with Q− = Q+ = Q.”
One and two weight norm inequalities for Riesz potentials (2012, background): “To prove bounds for I S α we will use a dyadic version of Theorem 1.6 due to Lacey, Sawyer, and Uriarte-Tuero [19] that gives precise bounds in terms of testing conditions.”
Two weight norm inequalities for fractional integral operators and commutators (2014, background): “The fundamental result we discuss is due to Lacey, Sawyer and Uriarte-Tuero [55], but we will present a beautiful simplification of their proof due to Hytönen [42].”
The L-to-L boundedness of commutators with applications to the Jacobian operator (2018, background): “e p≤ q, has also been recently discovered in the context of two-weight norm inequalities for certain discrete positive operators, where the characterisation for p≤ qby Lacey, Sawyer and Uriarte-Tuero [16] is in terms of local “testing conditions” uniform over all dyadic cubes, while the characterisa”