Let \(0< \rho< r< R\) and \(d\ge 2\). There is a constant \(\alpha>0\), depending on \(\frac{\rho}{R}\), \(\frac{r}{R}\) and \(d\), such that for all harmonic functions \(u\colon B_R \to \mathbb{C}\),

\[ \lVert u\rVert_{L^\infty (B_r)}\le \lVert u\rVert_{L^\infty (B_\rho)}^\alpha \lVert u\rVert_{L^\infty (B_R)}^{1-\alpha}, \]

where \(B_R\) denotes a ball in \(\mathbb{R}^d\) [1, Theorem 1.2].

Remarks
  • In [1, Theorem 1.1] the authors deduce a three region inequality by incorporating the above theorem.
  • This property of harmonic functions is also known as propagation of smallness.

References Link to heading

  1. J. Korevaar and J. Meyers, Logarithmic Convexity for Supremum Norms of Harmonic Functions, Bulletin of the London Mathematical Society, vol. 26, no. 4, pp. 353–362, 1994. doi:10.1112/blms/26.4.353