Weak solutions of the nonlinear Cauchy–Riemann equation are smooth

Read-back: JHolomorphic.contDiffOn_of_isWeakJHolomorphicOn

Data and standing assumptions. Let $E$ be a finite-dimensional real normed vector space (any norm; no inner product or complex structure on $E$ is assumed). Let $p$ be a real number with $p > 2$. Let $J : E \to \mathcal{L}(E,E)$ be any map assigning to each point $x \in E$ a continuous real-linear operator $J(x)$ on $E$. Let $O \subseteq E$ and $U \subseteq \mathbb{C}$ be sets, with $U$ assumed open. Let $u : \mathbb{C} \to E$ be any function. Throughout, $\mathbb{C}$ is regarded as the real plane, a point is written $z = s + it$, and "measure" on $\mathbb{C}$ means two-dimensional Lebesgue measure.

Hypothesis on $J$ (almost complex on $O$). It is assumed that:

• $O$ is open in $E$;
• $J$, viewed as a map $E \to \mathcal{L}(E,E)$, is $C^\infty$ on $O$ (smooth, i.e. $C^k$ for every finite $k$, in the sense of the within-the-set derivatives of $J$ restricted to $O$);
• for every $x \in O$, $J(x) \circ J(x) = -\mathrm{id}_E$.

Nothing is assumed about $J(x)$ for $x \notin O$.

Hypothesis on $u$ (weakly $J$-holomorphic on $U$ with exponent $p$). It is assumed that:

1. $u$ is continuous on $U$ (continuity of the restriction $u|_U$);
2. $u(z) \in O$ for every $z \in U$;
3. there exist two functions $u_s, u_t : \mathbb{C} \to E$ (defined on all of $\mathbb{C}$; their values outside $U$ are unconstrained) such that:
   • (weak partial derivatives) for every $\varphi : \mathbb{C} \to \mathbb{R}$ that is $C^\infty$ on all of $\mathbb{C}$, has compact support, and whose closed support $\operatorname{tsupp}\varphi$ is contained in $U$,

 where $\partial_s\varphi(z) = D\varphi(z)[1]$ and $\partial_t\varphi(z) = D\varphi(z)[i]$ are the real directional derivatives of $\varphi$ in the directions $1$ and $i$. These are $E$-valued Bochner integrals over all of $\mathbb{C}$, with the convention that the integral of a non-integrable function is defined to be $0$; the identities are required as stated under that convention, and the definition itself imposes no separate integrability or measurability requirement;

• (local $L^p$) for every compact set $K \subseteq U$ (including $K = \emptyset$), each of $u$, $u_s$, $u_t$ belongs to $L^p(K)$, i.e. is almost-everywhere strongly measurable with respect to Lebesgue measure restricted to $K$ and satisfies $\int_K \|f\|^p < \infty$ (here the exponent is the extended non-negative real $p$, which is finite since $p > 2$);
• (the equation) for Lebesgue-almost every $z \in U$,

Conclusion. Under these assumptions, $u$ is $C^\infty$ on $U$: it is $C^k$ on $U$ (as a map from the real plane $\mathbb{C} \cong \mathbb{R}^2$ to $E$, with within-$U$ derivatives) for every finite $k$. Since $U$ is open, this is the same as $u$ being infinitely real-differentiable near every point of $U$. The smoothness index is the "$\infty$" of finite-order smoothness, not real-analyticity. Nothing is asserted about $u$ outside $U$, and nothing is asserted about $u_s$ or $u_t$ (for instance, the conclusion does not say that they agree with the classical partial derivatives).

Edge cases and remarks.

• If $U = \emptyset$, every hypothesis on $u$ holds trivially (take $u_s = u_t = 0$) and the conclusion is trivially true.
• $O$ may be empty. Then condition 2 forces $U = \emptyset$, which is the trivial case above.
• If $E = \{0\}$, the condition $J(x)^2 = -\mathrm{id}$ holds automatically, since every operator is $0 = -0$.
• If $E$ is nonzero of odd real dimension, no operator satisfies $J(x)^2 = -\mathrm{id}_E$ (taking determinants gives $\det(J)^2 = (-1)^{\dim E} = -1$). So the hypothesis on $J$ can hold only when $O = \emptyset$, which again forces $U = \emptyset$.
• The exponent $p$ is a real number strictly greater than $2$. The case $p = \infty$ is not covered, and neither is $p \le 2$.
• The equation is required only almost everywhere on $U$, and it uses $J$ only at the points $u(z) \in O$.
• The statement opens the DbarRegularity namespace, but none of its definitions (the maximal function, closed cubes, the operator $\bar\partial$, the Sobolev norm) appear in the hypotheses or the conclusion. Of the JHolomorphic definitions, only "almost complex on $O$" and "weakly $J$-holomorphic on $U$" (together with its weak-partials clause) are used. The pointwise operator $\bar\partial_J$, strong $J$-holomorphicity and $C^\infty_{\mathrm{loc}}$ convergence do not appear.