Sharp weighted norm estimates beyond Calderón–Zygmund theory

Abstract

We dominate non-integral singular operators by adapted sparse operators and derive optimal norm estimates in weighted spaces. Our assumptions on the operators are minimal and our result applies to an array of situations, whose prototype are Riesz transforms / multipliers or paraproducts associated with a second order elliptic operator. It also applies to such operators whose unweighted continuity is restricted to Lebesgue spaces with certain ranges of exponents (p 0 , q 0 ) where 1 ≤ p 0 < 2 < q 0 ≤ ∞. The norm estimates obtained are powers α of the characteristic used by Auscher and Martell. The critical exponent in this case is p = 1 + p0 q ′ 0 . We prove α = 1 p−p0 when p 0 < p ≤ p and α = q0−1 q0−p when p ≤ p < q 0 . In particular, we are able to obtain the sharp A 2 estimates for non-integral singular operators which do not fit into the class of Calderón-Zygmund operators. These results are new even in the Euclidean space and are the first ones for operators whose kernel does not satisfy any regularity estimate.

Domination of multilinear singular integrals by positive sparse forms

“…We also point out the recent improvements by Lacey and Lerner , and the analogue for multilinear Calderón‐‐Zygmund operators obtained independently by Conde‐Alonso and Rey and by Lerner and Nazarov . Most recently, Bernicot, Frey and Petermichl extend this approach to nonintegral singular operators associated with a second‐order elliptic operator, lying outside the scope of classical Calderón–Zygmund theory.…”

“…Summarizing, no such pointwise domination principle can be obtained for T m when inf{p 1 , p 2 } < 2 and, most likely, neither for the case when inf{p 1 , p 2 } 2. Our formulation in terms of positive sparse forms overcomes this obstacle: a similar idea, albeit not explicit, appears in the linear setting in [4]. After the first version of this article was made public, several works based on sparse form domination have appeared within and beyond Calderón--Zygmund theory, see for example [2,6,22,27] and references therein.…”

Some Remarks on the Pointwise Sparse Domination

“…We mention only that sparse bounds for Calderón-Zygmund operators can be found in [7,15,16,18,19,20]. Also, there are several general sparse domination principles [5,6,8,19].In [19], a sparse domination principle was obtained in terms of the grand maximal truncated operatordefined for a given operator T .2010 Mathematics Subject Classification. 42B20, 42B25.…”

Quantitative Weighted Estimates for Rubio de Francia’s Littlewood–Paley Square Function

“…The list of papers in this subject is vast and it would be a challenging task by itself to just write all of them down without a miss. Here we refer the reader to [30,13,23,7,24,32,16] and references therein. We bring attention to the recent work of Lerner [31], where he proved the following.…”