Sharp weighted norm estimates beyond Calderón–Zygmund theory

By Frédéric Bernicot, D. Frey, S. Petermichl. 2015

155 citations recorded by the source corpus.

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\le p_0<2< q_0 \le \infty $. The norm estimates obtained are powers $\alpha$ of the characteristic used by Auscher and Martell. The critical exponent in this case is $\mathfrak{p}=1+\frac{p_0}{q'_0}$. We prove $\alpha=\frac1{p-p_0}$ when $p_0 < p \le \mathfrak{p}$ and $\alpha= \frac{q_0-1}{q_0-p}$ when $\mathfrak{p} \le p < q_0$. 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 Calder\'on-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.

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/8d6c7c6c9af5d10288666f3f2054e8bc0e50f910

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In Applied Mathematics

Citations in the corpus

Sparse Domination Theorem for Multilinear Singular Integral Operators with $L^{r}$-Hörmander Condition (2016, background): “In [3], Bernicot, Frey and Petermichl showed that a large class of singular non-integral operators can be dominated by sparse operators (actually in the bilinear form sense).”
On pointwise estimates involving sparse operators (2015, background): “Petermichl [1] obtained sharp weighted estimates for a large class of singular nonintegral operators in a rather general setting using similar ideas based on a domination by sparse operators.”