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  "body": "\n## Summary\n\nThe harmonic Hardy space $h^p$ of the unit ball consists of [[harmonic functions|0DEB0JMN0W]].\nTheir $L^p$ means over the spheres of radius $r$ must stay bounded as $r \\to 1$.\nFor $1 \u003c p \\le \\infty$ these are exactly the [[Poisson integrals|TCYSWCTYYM]] of functions in $L^p$ of the sphere.\nThe space $h^1$ consists of the Poisson integrals of finite signed measures.\nThey 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.\n\n## Explanations\n\n### L1 human\n\nA harmonic function on a ball is in $h^p$ if it is not too large on average near the boundary sphere.\nSuch a function has boundary values, and it is recovered from them by the Poisson integral.\nFor instance, every bounded harmonic function lies in $h^\\infty$.\nThe Poisson kernel with its pole at one boundary point is in $h^1$ but in no $h^p$ with $p \u003e 1$, because its boundary values form a point mass.\n",
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