Hardy spaces of harmonic functions

Summary

The harmonic Hardy space $h^p$ of the unit ball consists of harmonic functions. Their $L^p$ means over the spheres of radius $r$ must stay bounded as $r \to 1$. For $1 < p \le \infty$ these are exactly the Poisson integrals of functions in $L^p$ of the sphere. The space $h^1$ consists of the Poisson integrals of finite signed measures. They are the harmonic counterparts of the holomorphic Hardy spaces $H^p$ and are the natural classes for the Dirichlet problem with $L^p$ boundary data.