Harmonic Replacement

Let \(u \in C^0(\Omega)\) be subharmonic , and \(B(x, r) \subseteq \Omega\).

The harmonic replacement of \(u\) on \(B(x, r)\) is defined by:

\[P_{B(x)} u := v(y) = \begin{cases} \int_{\partial B(x)} K(y, z) u(z) \, d\sigma(z), & \text{if } y \in B(x, r), \\ u(y), & \text{if } y \in \Omega \setminus B(x, r), \end{cases}\]

where \(K\) denotes the Poisson’s kernel on the ball \(B(x,r)\).