---
tag: 6R2ZXY5VZW
kind: concept
name: Hardy spaces of harmonic functions
msc: [31B25, 31A20, 30H10]
sources:
  - url: https://en.wikipedia.org/wiki/Hardy_space
---

## Summary

The harmonic Hardy space $h^p$ of the unit ball consists of [[harmonic functions|0DEB0JMN0W]].
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|TCYSWCTYYM]] 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|PNNNFJVF6W]] $H^p$ and are the natural classes for the Dirichlet problem with $L^p$ boundary data.

## Explanations

### L1 human

A harmonic function on a ball is in $h^p$ if it is not too large on average near the boundary sphere.
Such a function has boundary values, and it is recovered from them by the Poisson integral.
For instance, every bounded harmonic function lies in $h^\infty$.
The Poisson kernel with its pole at one boundary point is in $h^1$ but in no $h^p$ with $p > 1$, because its boundary values form a point mass.