Weak Lp regularity for a small Cauchy–Riemann principal perturbation

With Fourier convention $\widehat{{\mathrm{\partial }}_{j}w}=2\pi i{\xi }_j\widehat{w}$, set $r_s(\xi )=\frac{i{\xi }_s}{i{\xi }_s-{\xi }_t},r_t(\xi )=\frac{i{\xi }_t}{i{\xi }_s-{\xi }_t}(\xi \mathrm{\ne }0)\mathrm{.}$ Both symbols are smooth and homogeneous of degree zero, and their first two derivatives are bounded on the unit circle because the denominator there has modulus one. Homogeneity gives the required Mihlin estimates. Let $R_s,R_t$ be their componentwise multiplier operators. By [F1] their $L^p$ norms on the finite-dimensional vector-valued space are bounded by a finite constant $K=K_{m,p}\ge 1$, while by [F3] their $L^2$ norms are at most one since $\mathrm{\mid }{r}_s\mathrm{\mid },\mathrm{\mid }{r}_t\mathrm{\mid }\le 1$. For smooth compactly supported $w$, Fourier differentiation gives $w_j=R_j{D}_{0}w$. Approximation in $W^{1,2}$ by [F2] proves the same identities for the given weak $w$.

Here is the common approximation needed below. For $g\in L^2\cap L^p$ and integers $n\ge 1$, let $g_n=1_{B_n}g$, and fix smooth cutoffs $0\le {\chi }_n\le 1$ equal to one on $B_n$ and supported in $B_{n+1}$. Set $a_n=\mathrm{\mid }{B}_{n+1}{\mathrm{\mid }}^{1\mathrm{/}2-1\mathrm{/}p}$. By [F2] at $k=0$, choose smooth compactly supported $h_n$ with $\mathrm{\parallel }{h}_n-g_n{\mathrm{\parallel }}_p<1\mathrm{/}(n(1+a_n))$, componentwise; Countable Choice supplies this sequence. Then ${\phi }_n={\chi }_n{h}_n$ is smooth and compactly supported and $\mathrm{\parallel }{\phi }_n-g_n{\mathrm{\parallel }}_p\le 1\mathrm{/}n$, while Holder on $B_{n+1}$ gives $\mathrm{\parallel }{\phi }_n-g_n{\mathrm{\parallel }}_2\le a_n\mathrm{\parallel }{\phi }_n-g_n{\mathrm{\parallel }}_p\le 1\mathrm{/}n$. The tails $g-g_n$ tend to zero in both norms by integrability, so ${\phi }_n\to g$ in both norms. Consequently the bounded $L^2$ and $L^p$ extensions of either multiplier in step 1.1 agree as distributions on the intersection: their values on ${\phi }_n$ coincide, and convergence in either norm implies convergence against compactly supported smooth tests by Holder.

Choose ${\eta }_{m,p}=1\mathrm{/}(2K)$. The equation and step 1.1 give $w_t=R_{t}A{w}_t+R_{t}f$ as an $L^2$ identity. The real linear operator $R_{t}A$ has norm at most $1\mathrm{/}2$ on both $L^2$ and $L^p$. Therefore the geometric series ${\sum }_{k\ge 0}(R_{t}A{)}^k{R}_{t}f$ converges in each of those spaces. Its partial sums belong to their intersection and have identical distributional meanings: the $L^2$ and $L^p$ multipliers agree on that intersection, by the common approximation and distributional identification proved in step 2.1. The two limits thus represent the same distribution. Uniqueness of the $L^2$ solution to $(I-R_{t}A)v=R_{t}f$ identifies that limit with the already existing derivative $w_t$. Consequently $\mathrm{\parallel }{w}_t{\mathrm{\parallel }}_p\le 2K\mathrm{\parallel }f{\mathrm{\parallel }}_p$. The equation then gives $w_s=(A-i)w_t+f\in L^p$ with a uniform bound, proving $\mathrm{\parallel }\mathrm{\nabla }w{\mathrm{\parallel }}_p\le C_{m,p}\mathrm{\parallel }f{\mathrm{\parallel }}_p$.

The function itself lies in $L^p$. Take a square $[-R,R{]}^2$ containing its support. The distributional identity ${\mathrm{\partial }}_{s}w=w_s\in L^p$, together with Fubini, gives the one-dimensional primitive formula on almost every horizontal line: subtract ${\int }_{-R-1}^s{w}_s(r,t)dr$ from $w(s,t)$; its distributional $s$ derivative is zero, so it is a constant. A distribution with derivative zero is constant because it annihilates every integral-zero compact test, each of which is a derivative of a compactly supported test. The vanishing of $w$ before its support makes that constant zero. Thus Holder on each finite horizontal interval gives $\mathrm{\parallel }w{\mathrm{\parallel }}_p\le (2R+2)\mathrm{\parallel }{w}_s{\mathrm{\parallel }}_p$. In particular $w\in W^{1,p}$. This argument requires no general ACL or Morrey theorem.

In the half-plane case let $C$ denote componentwise complex conjugation and extend $w$ by $\widehat{w}(s,-t)=Cw(s,t)$ for $t>0$. This is a compactly supported $W^{1,2}$ map on the plane. To verify the boundary claim directly, apply the primitive argument of step 4.1 on almost every vertical line; the continuous boundary values agree with the resulting finite-endpoint primitive limits. Since these values are real, the reflected values match, and integrating by parts on the two sides leaves no line atom. Define $\widehat{A}(s,-t)=-CA(s,t)C$ and $\widehat{f}(s,-t)=Cf(s,t)$, preserving their original values for $t>0$. Then $D_0\widehat{w}=\widehat{A}{\widehat{w}}_t+\widehat{f}$ weakly, since below the line $D_0\widehat{w}=C(D_{0}w)$ and ${\widehat{w}}_t=-C{w}_t$; continuity again removes any distributional boundary term. The coefficient norm is unchanged. Apply the plane result and restrict back. Both the reflected forcing norm and derivative norm gain the same factor $2^{1\mathrm{/}p}$, so the derivative estimate has the same constants.

For a local equation multiply the solution by a real smooth cutoff $\chi$ supported in a smaller disk or half-disk and extend it by zero. The product obeys $D_0(\chi w)=A{\mathrm{\partial }}_t(\chi w)+\chi f+({\chi }_s+i{\chi }_t-A{\chi }_t)w\mathrm{.}$ The cutoff preserves real boundary values, and its forcing has the claimed integrability whenever the displayed localized terms do. The plane or half-plane result proves the local assertion on the region where $\chi =1$. This completes the weak regularity statement, without assuming differentiability of $A$ or regularity of the weak solution beyond the stated hypotheses.