Quantitative weighted estimates for rough homogeneous singular integrals

By Tuomas Hytönen, Luz Roncal, Olli Tapiola. 2015

124 citations recorded by the source corpus.

We consider homogeneous singular kernels, whose angular part is bounded, but need not have any continuity. For the norm of the corresponding singular integral operators on the weighted space L2(w), we obtain a bound that is quadratic in A2 constant $${\left[ w \right]_{{A_2}}}$$[w]A2. We do not know if this is sharp, but it is the best known quantitative result for this class of operators. The proof relies on a classical decomposition of these operators into smooth pieces, for which we use a quantitative elaboration of Lacey's dyadic decomposition of Dini-continuous operators: the dependence of constants on the Dini norm of the kernels is crucial to control the summability of the series expansion of the rough operator. We conclude with applications and conjectures related to weighted bounds for powers of the Beurling transform.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/04d9ffe4a23b2523c729e4624623be982d8238dc

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In Applied Mathematics

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Citations in the corpus

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.”
On pointwise estimates involving sparse operators (2015, methodology): “In the present note we modify a main idea from Lacey’s work [10] with the aim to give a rather short and elementary proof of the result in [9].”
On pointwise and weighted estimates for commutators of Calder\'on-Zygmund operators (2016, background): “1) provides a simple proof of the sharp L(w) bound for T (see [19,24]): ‖T‖Lp(w) ≤ cn,pCT [w] max ( 1, 1 p−1 ) Ap (1 < p < ∞) (1.”