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  "body": "\n## Statement\n\nLet $n \\ge 2$ and let $\\Omega \\subseteq \\R^n$ be a bounded [[Lipschitz domain|1GNKQ7NBAG]] that is connected.\nLet $\\sigma$ be the surface measure on $\\partial\\Omega$, that is, the $(n-1)$-dimensional [[Hausdorff measure|Q1TGZK8RN6]] restricted to $\\partial\\Omega$.\nLet $x \\in \\Omega$ and let $\\omega^x$ be the [[harmonic measure|CFD7BPM51W]] of $\\Omega$ at $x$.\nThen $\\omega^x$ and $\\sigma$ are mutually [[absolutely continuous|P0P21KDV6E]], and the density $k^x = d\\omega^x / d\\sigma$ belongs to [[Lp space|SASP14MA1Z]] $L^2(\\partial\\Omega, \\sigma)$.\n\n## Remarks\n\nMore precisely, $k^x$ satisfies a reverse Hölder inequality with exponent $2$ on surface balls, and by Gehring's lemma it lies in $L^{2 + \\varepsilon}(\\sigma)$ for some $\\varepsilon \u003e 0$ depending on $\\Omega$.\nAs a consequence, the Dirichlet problem on $\\Omega$ is solvable for boundary data in $L^p(\\sigma)$ with $p \\ge 2$, with nontangential convergence to the data and a nontangential maximal function estimate.\nIdea of the proof: the [[boundary Harnack principle|PYJ7T889KC]] compares $\\omega^x$ of a surface ball with the Green function at a nearby interior point.\nOne proof then uses a Rellich-type integral identity for harmonic functions on Lipschitz domains to get the $L^2$ bound.\nExample: for a ball, $k^x$ is the Poisson kernel, which is bounded, so $k^x \\in L^p$ for every $p$.\nThe exponent $2$ cannot be replaced by a fixed larger one for all Lipschitz domains.\nIn the plane, at the vertex of a sector of opening $\\theta \\in (\\pi, 2\\pi)$, the density behaves like $t^{\\pi/\\theta - 1}$ at distance $t$ from the vertex.\nIt lies in $L^p$ near the vertex only for $p \u003c 1 / (1 - \\pi/\\theta)$, and this bound tends to $2$ as $\\theta \\to 2\\pi$.\n\n## Explanations\n\n### L1 human\n\nHarmonic measure tells where a random path started at $x$ first hits the boundary.\nOn a domain with corners but no cusps, this hitting distribution has a density with respect to surface area, and the density is square-integrable.\nSo a set of zero area is never hit, and a set that is hit with probability zero has zero area.\n\n### L2 human\n\nDahlberg's theorem made Lipschitz domains the natural setting for harmonic analysis on rough domains.\nIt is the basis of the $L^p$ theory of the Dirichlet and Neumann problems and of layer potentials on Lipschitz boundaries.\nIt shows that harmonic measure is an $A_\\infty$ weight with respect to surface measure, a property later studied on much more general domains.\nIn the plane, for a Jordan domain with rectifiable boundary, the [[F. and M. Riesz theorem|X9V1E3V145]] already gives mutual absolute continuity with arc length.\n",
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