In the simpler context of applying the $p$-Laplacian $\Delta_{p}$ to scalar functions $u \, \colon \, \Omega \subset \mathbb{R}^{N} \to \mathbb{R}$, some calculations reveal connections with complementary averaging operators for different values of $p$. Working formally, for smooth $u$, $1 < p < \infty$, and $p^{-1} + q^{-1} = 1$,
$$\Delta_{p} u ~ := ~ \operatorname{div}{ \left( | \nabla u |^{p-2} \nabla u \right) } ~ = ~ p | \nabla u |^{p-2} \left( p^{-1} \Delta_{1} u + q^{-1} \Delta_{\infty} u \right) \ , $$
where
$$ \Delta_{\infty} u ~ := ~ | \nabla u |^{-2} \sum_{i,j}{ u_{i}u_{j} u_{ij} } \quad \mbox{and} \quad \Delta_{1} u ~ := ~ \Delta u - \Delta_{\infty} u$$
are, respectively, the $\infty$-Laplacian and the $1$-Laplacian. (See, e.g., Peres & Sheffield.)
Now note that $\Delta_\infty u(x)$ measures the difference between $u(x)$ and its midrange on a ball (or sphere) centered at $x$,
$$-\Delta_{\infty} u (x) ~ \propto ~ \lim_{h \to 0}{ \frac{2}{h^2} \left( u(x) - \operatorname{midrange}_{B(x,h)}{ u } \right) } \ , $$
where
$$\operatorname{midrange}_{B(x,h)}{ u } ~ := ~ \frac{1}{2} \, \left( \max_{\overline{B(x,h)}}{ u } + \min_{\overline{B(x,h)}}{ u } \right) \ ,$$
while $\Delta_{1} u(x)$ measures the difference between $u(x)$ and its median over a ball (or sphere) centered at $x$,
$$-\Delta_{1} u(x) ~ \propto ~ \lim_{h \to 0}{ \frac{2}{h^2} \left( u(x) - \operatorname{median}_{\overline{B(x,h)}}{ u } \right) } \ , $$
where the median $m$ of $u$ over $\overline{B(x,h)}$ is the number such that
$$ | \left\{ u < m \right\} | \leq \frac{1}{2} |B(x,h)| \quad \mbox{and} \quad | \left\{ u > m \right\} | \leq \frac{1}{2} |B(x,h)| \ .$$
(See, e.g., Kawohl, Manfredi & Parviainen and Rudd.)
Putting all of this together shows that the $p$-Laplacian of $u$ compares $u$ to a linear combination of its neighboring average and midrange when $p > 2$, while it compares $u$ to a linear combination of its neighboring average and median when $1 < p < 2$. These formal calculations can be made rigorous in the context of viscosity solutions.
