Local higher derivative bounds for almost complex maps with bounded first derivative

We first record the elementary two-dimensional inequalities used in the proof. For a smooth compactly supported vector function $f$ on ${\mathbb{R}}^2$, the one-dimensional fundamental theorem gives $\mathrm{\mid }f(s,t){\mathrm{\mid }}^2\le 2{\int }_{\mathbb{R}}\mathrm{\mid }f{f}_s\mathrm{\mid }(r,t)dr$ and the analogous bound in the other coordinate. Multiply these inequalities and integrate to obtain $\mathrm{\parallel }f{\mathrm{\parallel }}_4^4\le 4\mathrm{\parallel }f{\mathrm{\parallel }}_2^2\mathrm{\parallel }{f}_s{\mathrm{\parallel }}_2\mathrm{\parallel }{f}_t{\mathrm{\parallel }}_2\le 2\mathrm{\parallel }f{\mathrm{\parallel }}_2^2\mathrm{\parallel }\mathrm{\nabla }f{\mathrm{\parallel }}_2^2\mathrm{.}$ Applying the fundamental theorem twice to $\mathrm{\mid }f{\mathrm{\mid }}^2$ also gives $\mathrm{\mid }f(s,t){\mathrm{\mid }}^2\le 2{\int }_{{\mathbb{R}}^2}(\mathrm{\mid }f{f}_{st}\mathrm{\mid }+\mathrm{\mid }{f}_s{f}_t\mathrm{\mid })\le 2\mathrm{\parallel }f{\mathrm{\parallel }}_2\mathrm{\parallel }{f}_{st}{\mathrm{\parallel }}_2+2\mathrm{\parallel }{f}_s{\mathrm{\parallel }}_2\mathrm{\parallel }{f}_t{\mathrm{\parallel }}_2\le C\mathrm{\parallel }f{\mathrm{\parallel }}_{H^2}^2\mathrm{.}$ These yield local $H^1\to L^4$ and $H^2\to L^{\mathrm{\infty }}$ bounds after multiplication by a scalar cutoff. They hold on half-disks too, away from the curved edge. For $H^1$ reflect evenly across the flat edge. For $H^2$ extend a compactly supported upper-half-plane function to negative $t$ by $F(s,-t)=3f(s,t)-2f(s,2t)$, $t>0$. Its value and first normal derivative match at zero, so the extension is $C^1$, piecewise smooth, has weak second derivatives with no line atom, and has controlled $H^2$ norm. The first calculation remains valid for the continuous, piecewise smooth even extension, which is absolutely continuous on coordinate lines; the second remains valid for the order-two extension because its value and first derivatives match.

For $D_0={\mathrm{\partial }}_s+i{\mathrm{\partial }}_t$ and a smooth compactly supported $w$ on the plane one has $\mathrm{\parallel }{D}_{0}w{\mathrm{\parallel }}_2^2=\mathrm{\parallel }{w}_s{\mathrm{\parallel }}_2^2+\mathrm{\parallel }{w}_t{\mathrm{\parallel }}_2^2$. Indeed the integrated cross term is zero by integration by parts in both coordinates. The same identity holds on the upper half-plane for compactly supported smooth $w$ whose flat-edge values are real: the remaining flat-boundary cross term is the imaginary part of the pairing of the real vectors $w_s$ and $w$, hence zero. Let $w$ satisfy $D_{0}w=(i-J)w_t+F$ on a patch, with this boundary condition when needed, and let $\chi$ be a real cutoff compactly supported away from the curved edge. Apply the identity to $\chi w$ and commute the cutoff through the equation. The principal coefficient bound gives $\mathrm{\parallel }\mathrm{\nabla }(\chi w){\mathrm{\parallel }}_2\le \frac{1}{2}\mathrm{\parallel }\mathrm{\nabla }(\chi w){\mathrm{\parallel }}_2+C_{\chi }\mathrm{\parallel }w{\mathrm{\parallel }}_2+\mathrm{\parallel }\chi F{\mathrm{\parallel }}_2,$ so its first term absorbs. This proves the local first-order estimate on any smaller disk or half-disk, with constants depending on the cutoff margin.

We prove uniform local $H^k$ bounds by induction, using slightly nested radii between $\rho$ and one. The $H^1$ bound follows from the given $C^1$ bound and the bounded area. Differentiate the equation once in $s$ and put $w=u_s$. Then $D_{0}w=(i-J)w_t-({\mathrm{\partial }}_{s}J+(d_{x}J)u_s)u_t\mathrm{.}$ Its forcing is bounded in $L^2$ by the given $C^1$ and coefficient bounds. In a half-disk $w$ has real flat-edge values. Step 1.2 therefore bounds $u_{ss}$ and $u_{st}$ on a smaller patch. The equation rewritten as $u_t=J{u}_s$ gives $u_{tt}=({\mathrm{\partial }}_{t}J+(d_{x}J)u_t)u_s+J{u}_{st},$ which bounds the remaining second derivative. Thus $u$ has a uniform local $H^2$ bound. Differentiating twice in $s$ gives the same estimate for $u_{ss}$: its commutator terms contain at most one second derivative of $u$, multiplied by bounded first derivatives, and bounded products of first derivatives. These terms are controlled by $H^2$ and $C^1$. Recovering the other third derivatives from $u_t=J{u}_s$ proves the $H^3$ bound.

For the general induction assume the local $H^k$ bound, $k\ge 3$, on a slightly larger patch. Differentiating $k$ times in $s$ gives $D_0({\mathrm{\partial }}_s^{k}u)=(i-J){\mathrm{\partial }}_t{\mathrm{\partial }}_s^{k}u+F_k\mathrm{.}$ Repeated chain and product rules express $F_k$ as finitely many products of bounded coefficient derivatives and derivatives of $u$ of orders at most $k$, with sum of their orders at most $k+1$; the sole order-$k+1$ term was separated as the principal term. For $k=3$, the only product not directly bounded by the $C^1$ bound times an $H^3$ norm has two second derivatives, which are $L^4$ by step 1.1 applied to the $H^3$ bound. Their product is $L^2$. For $k\ge 4$, derivatives of orders at most $k-2$ are locally bounded by step 1.1. A factor of order $k$ can only be accompanied by total derivative order at most one, and a factor of order $k-1$ by total order at most two; in either case put that high-order factor in $L^2$ and the remaining factors in $L^{\mathrm{\infty }}$. There cannot be two factors of order $k-1$, since $2k-2>k+1$ for $k\ge 4$. If all factors have orders at most $k-2$, they are bounded and their product is $L^2$ on the finite patch. This controls every term of $F_k$. Step 1.2 now bounds the derivatives of ${\mathrm{\partial }}_s^{k}u$, with real boundary values retained. To recover an order-$k+1$ derivative with at least two $t$ derivatives, differentiate $u_t=J{u}_s$ $k$ times. Its principal term has one fewer $t$ derivative than the derivative on the left, and all other terms are controlled by the same product argument using derivatives through order $k$. Induction on the number of $t$ derivatives bounds them all. This completes the Sobolev induction.

For each requested derivative order $j$, obtain the $H^{j+2}$ bound on a slightly larger patch from the preceding Sobolev induction. Multiply each $D^{j}u$ by a cutoff and use the $H^2\to L^{\mathrm{\infty }}$ estimate of step 1.1, including its explicit half-plane extension when needed. This bounds $\sup \mathrm{\mid }{D}^{j}u\mathrm{\mid }$ on the desired smaller patch. For any fixed $k$ only finitely many nested radii and coefficient derivatives are used; their constants depend solely on the data stated. All differentiations are of given smooth solutions, so no weak regularity or hidden choice-dependent elliptic theorem has been invoked.