Optimal $C^2$ two-dimensional interpolatory ternary subdivision schemes with two-ring stencils

For any interpolatory ternary subdivision scheme with two-ring stencils for a regular triangular or quadrilateral mesh, we show that the critical Hölder smoothness exponent of its basis function cannot exceed $\log_3 11 (\approx 2.18266)$, where the critical Hölder smoothness exponent of a function $f : \mathbb{R}^2\mapsto \mathbb{R}$ is defined to be

$\nu_\infty(f):=\sup\{ \nu\; : \; f\in \hbox{Lip}\,\nu\}.$

On the other hand, for both regular triangular and quadrilateral meshes, we present several examples of interpolatory ternary subdivision schemes with two-ring stencils such that the critical Hölder smoothness exponents of their basis functions do achieve the optimal smoothness upper bound $\log_3 11$. Consequently, we obtain optimal smoothest $C^2$ interpolatory ternary subdivision schemes with two-ring stencils for the regular triangular and quadrilateral meshes. Our computation and analysis of optimal multidimensional subdivision schemes are based on the projection method and the $\ell_p$-norm joint spectral radius.