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.

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