Uniform sparse domination of singular integrals via dyadic shifts

Abstract

Using the Calderón-Zygmund decomposition, we give a novel and simple proof that $L^2$ bounded dyadic shifts admit a domination by positive sparse forms with linear growth in the complexity of the shift. Our estimate, coupled with Hytönen's dyadic representation theorem, upgrades to a positive sparse domination of the class $\mathcal{U}$ of singular integrals satisfying the assumptions of the classical $T(1)$-theorem of David and Journé, with logarithmic-Dini type smoothness of the integral kernel. Furthermore, our proof extends rather easily to the ${\mathbb{R}}^n$-valued case, yielding as a corollary the operator norm bound on the matrix weighted space $L^2(W;{\mathbb{R}}^n),$ \[ \left\|T\otimes \mathrm{Id}_{\mathbb R^n}\right\|_{L^2(W; \mathbb R^n)\rightarrow L^2(W; \mathbb R^n)} \lesssim [W]_{A_2}^{\frac32} \] uniformly over $T\in \mathcal{U}$, which is the currently best known dependence.

To appear in Math. Res. Lett. Some references have been corrected and updated

References in corpus (1)

• The Sparse T1 Theorem

Cited by in corpus (2)

• Sparse bounds for maximal rough singular integrals via the Fourier transform
• Two weight bump conditions for matrix weights