# The order of regularity improving for elliptic operator with rough coefficients

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 $$\alpha>\frac12$$.