Title: Atomic Hardy space theory for unbounded singular integrals

Authors: Ryan Berndt

Issue: Volume 55 (2006), Issue 4, 1461-1482


We examine singular integrals of the form \[ Tf(x) = \lim_{\epsilon \to 0}\int_{|y| \geq \epsilon} \frac{B(y)}{y} f(x - y) dy \] where the function $B$ is non-negative and even, and is allowed to have singularities at zero and infinity. The operators we consider are not generally bounded on $L^{2}(\mathbb{R})$, yet there is a Hardy space theory for them. For each $T$ there are associated atomic Hardy spaces, called $H^{1}_{B}$ and $H^{1,1}_{B}$. The atoms of both spaces possess a size condition involving $B$.  The operator $T$ maps $H^{1,1}_{B}$ and certain $H^{1}_{B}$ continuously into $H^{1} \subset L^{1}$. The dual of $H^{1}_{B}$ is a space we call $\mathrm{BMO}_{B}$. The Hilbert transform is a special case of an operator $T$ and its $H^{1}_{B}$ and $\mathrm{BMO}_{B}$ spaces are $H^{1}$ and $\mathrm{BMO}$.