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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).

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1 Answer 1

up vote 2 down vote accepted

I'm almost certain that Abbas-Hofer does this in the appendix in which they prove the $L^p$ estimates. If I recall correctly, the proof shows the estimate in weak $L^1$ and for $L^2$, and then uses interpolation to get $1 < p \le 2$. You then obtain the $2 < p < \infty$ by duality. If you don't have access to it, I will try to find some references to the published literature.

The $p>2$ condition comes in to play once we want to show that a sequence holomorphic curves with gradient bounds have $C^\infty$ bounds. The induction uses that $W^{1,p}$ is a Banach algebra (notably to deal with the nonlinear almost complex structure $J$).

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@Sam: Thanks for the answer! In the version of Abbas/Hofer that I have (possibly not the latest one) they prove interior $L^p$-estimates ($p>1$) for the CR-operator. They only discuss regularity in $W^{1,p}$ for $p>2.$ Nevertheless I will accept your answer since I realized that I have stupidly missed that the corresponding statement is proved in Appendix B of the newer McDuff-Salamon treatise. They prove interior $L^p$-regularity ($p>1$) and reduce the boundary regularity (with vanishing imaginary part a.e.) to the interior regularity via Schwarz reflection. –  Orbicular Oct 21 '11 at 17:50

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