Connected to the question,
Does Morrey's inequality contextually relate to Rellic-Kondrachov compactness?
An analysis of the well-known nonlinear Aronsson operator gives $C^{(1, \frac{1}{3})}$ Holder regularity in 'free dimension' $n \geq 2$ with Lipschitz data on the boundary. An outline of this is given below, with a link to a longer paper as well.
My question is, why not bypass Morrey's inequality via the Aronsson operator which profitably uses the Cauchy-Riemann operator?
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In $D_n$ and $\Delta_n$ denote respectively the gradient and Laplacian in dim. $n \geq 2$, then the Algebraic lemma,
$\sum_{j > i=1}^{n} (u_{x_{i}}^2 + u_{x_{j}}^2)(u_{x_{i}x_{i}} + u_{x_{j}x_{j}}) = |D_{n}u|^2 \Delta_{n}u + (n-2)\sum_{i=1}^{n} u_{x_{i}}^2 u_{x_{i}x_{i}} $
holds, where $u$ is the 'viscosity solution' of the Aronsson operator in the open bounded domain U with Lipschitz data on its boundary. Following F. Treves, Basic linear PDE (1975), pp 36,36 we note that $u$ can be regarded as 'reduced' to a disc of radius $R$ for the analysis. We apply the CR operator to both sides of the viscosity equality
$u^2_{x}u_{xx} = -2u_{x}u_{y}u_{xy} - u^2_{y}u_{yy}$
and then 'withdraw' the action with the fundamental solution $\frac{1}{\Pi z}$ (integration by parts), noting that the actual convolution needed to obtain the 'first order expression' (say) $\frac{u_x^{3}}{3}$ is found to be $\frac{1}{\Pi z} *_x H$, $H$ the Heaviside function and convolution in $x$ is locally integrable in $\mathbb{R}^2$. That both $u_x$ and $u_y$ are $\frac{1}{3}$ Holder continuous in $\bar{u}$ follows from an application of solving cubics by Cardan's method.
Our point of view is that the Aronsson operator action actually defines a genuine tempered distribution and the so-called "viscosity methods" are just 'transitory'.
Please see this paper for details: https://www.sciencedirect.com/science/article/pii/S0898122107001927
