Paraproducts and the exponential-square class
Let \(b\) be a complex-valued bounded weight on \({\mathbb R}\) such that \(|b_D|>\delta\) for all dyadic intervals \(D\) (here \(b_D\) denotes \(b\)'s average over \(I\)), and let \(h_{(D),b}\) be the associated Haar functions introduced by \textit{R. R. Coifman, P. W. Jones} and \textit{S. Semmes} [J. Am. Math. Soc. 2, 553-564 (1989; Zbl 0713.42010)]. The main theorem in the paper under review shows that, if \(f\in L^1_{\text{loc}}\) has a pointwise bounded dyadic square function \[ S_{b}(f)(x)=\left(\sum_D \frac{|\lambda_{D,b}(f)|^2}{|D|}\chi_{D}(x)\right)^{1/2} \] associated to the Haar coefficients \(\lambda_{D,b}(f)\), then \(f\) is locally exponentially square integrable in the sense that \[ |\{ x; |f(x)-f_D|>\lambda\}|\leq c_1 |D|\exp(-c_2 \lambda^2) \] for all \(D\) and all \(\lambda >0\). This result is also extended to \(d\) dimensions for the natural Haar functions on \({\mathbb R}^d\).
scientific article; zbMATH DE number 1836760