Let $\Omega$ be an open subset of the upper half-plane in the complex plane. I am considering the following problem:

(1) $\overline{\partial}u=f,$ $\textrm{Im} f=0$ on the real line for maps complex-valued maps on $\Omega.$

Here, $\overline{\partial}$ denotes the classical Cauchy-Riemann operator. Usually one considers (1) as posed in the space (of complex-valued maps lying in) $W^{1,p}$ for $p>2$ because by Sobolev embedding elements of $W^{1,p}$ are actually continuous and the boundary condition is then well-defined. One then has elliptic regularity in the sense that if $u\in W^{1,p}(\Omega)$ solves (1) for $f\in W^{k,p}(\Omega),$ then $u\in W^{k+1,p}_{loc}(\Omega)$ with corresponding estimates. (By usually I mean the references on J-holomorphic cures like Salamon-McDuff's books or the book by Abbas-Hofer.)

I am actually looking for the corresponding statement for $p\in(1,2].$ Namely consider the space $W^{1,p}(\Omega)\times W^{1,p}_0(\Omega)$ (now both spaces consist of real-valued maps and correspond to real and imaginary part), where $W^{1,p}_0(\Omega)$ is the closure of compactly-supported smooth $u\in C^{\infty}(\Omega),$ s.t. $\textrm{supp } u$ is disjoint from the real axis. Is it then true that a weak solution $u\in W^{1,p}(\Omega)\times W^{1,p}_0(\Omega)$ of $\overline{\partial}u=f$ with $f\in W^{k,p}$ is in fact in W^{k+1,p}?

As I already mentioned the literature I came across only treats the case $p>2.$ Notice also that it does not suffice to use regularity theory for the Laplace operator, since one only gets local regularity for the real part (i.e. no regularity up to the boundary).