No, you need some information on what $\psi$ does on the real line. A counterexample for your estimate is given by the bounded harmonic function $$ \psi(x,y)=\arctan\frac xy, $$ which does not even belong to $W^{1,p}(B_{1/2})$ for $p\ge 2$.

No, you need some information on what $\psi$ does on the real line. A counterexample for your estimate is given by the bounded harmonic function $$ \psi(x,y)=\arctan\frac xy, $$ which does not even belong to $W^{1,p}(B_{1/2})$ for $p\ge 2$.

To see that Schauder estimates fail, you can similarly use the counterexample $$\psi(z)=\mathrm{Re} \, (z^{\alpha})$$ for $0<\alpha<1$. Then $\psi$ is harmonic and $\|\psi\|_{C^\alpha(B_1)}<\infty$, but as $\psi$ is not even differentiable at $0$, $\psi\notin C^{2,\alpha}(B_{1/2})$.