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Hi!

Let $L$ be a linear elliptic operator of order $4$ with smooth and bounded coefficients on the ball

$B_1$ of $R^{2n}$ and let $N\in C_{loc}^{0,\alpha}(R^{3})$.

Let $f\in C^{0,\alpha}(B_1)$ and suppose i have a function $u\in C^{4,\gamma}(B_{1})$ with $\gamma<\alpha$ s.t.

$$Lu+ N(\nabla^{2} u,\nabla^{3}u,\nabla^{4}u)=f$$

Moreover i know that

$$ \left\|N(\nabla^{2} u,\nabla^{3}u,\nabla^{4}u)\right\|_{C^{0,\gamma}} <\left\|Lu\right\|_{C^{0,\gamma}}$$

does it follow that $u\in C^{4,\alpha}(B_{1/2})$?

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