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Let $\frac12<\alpha<1$ and let $L(x)$ be a first order overdetermined elliptic operator with coefficients in $C^\alpha$. We means $L(x)=\sum_{j=1}^nA_j(x)\partial_j$ where $A_j:\mathbb R^n\to\mathbb R^{m\times p}$ are $C^\alpha$-functions, such that for each $x\in\mathbb R^n$, $\xi\in\mathbb S^{n-1}$, the matrix $\sum_{j=1}^nA_j(x)\xi_j\in\mathbb R^{m\times p}$ is of column full rank.

Suppose $u:\mathbb R^n\times\mathbb R^p$ is a $C^\alpha$-functions satisfies $Lu=0$ in distribution sense. In M. Taylor's ``Pseudodifferential Operators And Nonlinear PDEs'', we know $u\in C^{\alpha+\delta}$ for some $\delta>0$.

My question is, what is the optimal $\delta>0$ (depends on $\alpha$), such that whenever $L$ is a first order overdetermined elliptic operator with $C^\alpha$-coefficients and a $C^\alpha$ map $u$ satisfies $Lu=0$ then $u\in C^{\alpha+\delta}$. Is it true that $\delta<1$ when $\alpha<1$?

Here by paradifferential calculus $L(x)u(x)$ is a well-defined $C^{\alpha-1}$-distribution when $\alpha>\frac12$.

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