In the disk, we prove that integrals of boundary functions against the square root of the Poisson kernel converge in regions which we call $L^p$ weakly tangential. If $p>1$ these regions are strictly larger than the weakly tangential regions used by Sjögren. We also investigate how sharp these results are.
In the bidisk, we prove that we have convergence in the product region $A\times B$, where $A$ is a nontangential cone, and $B$ is a weakly tangential region. In this case, the kernel will be a tensor product of powers of Poisson kernels, with the exponent larger than 1/2 in the first variable, and the exponent equal to 1/2 in the second variable.
Reviewer: Jan-Olav Rönning (Skövde)