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For $\mathrm{\Omega }\subset {𝐑}^n$; $n\ge 2$, and $N\ge 1$ we consider vector valued minimizers $u\in W_{loc}^{m,p(\cdot )}(\mathrm{\Omega },{𝐑}^N)$ of a uniformly convex integral functional of the type $$ℱ[u,\mathrm{\Omega }]:={\int }_{\mathrm{\Omega }}f(x,D^{m}u)𝑑x,$$ where $f$ is a Carathéorody function satisfying $p(x)$ growth conditions with respect to the second variable. We show that if the dimension $n$ is small enough, dependent on the structure conditions of the functional, there holds $$D^{k}u\in C_{loc}^{0,\beta }(\mathrm{\Omega })\text{for}k\in \{0,\mathrm{\cdots },m-1\},$$ for some $\beta$, also depending on the structural data, provided that the nonlinearity exponent $p$ is uniformly continuous with modulus of continuity $\omega$ satisfying $$\underset{\rho \downarrow 0}{lim\; sup}\omega (\rho )\log \left(\frac{1}{\rho }\right)=0.$$ ...