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Planar transport equations

Written by Claude Opus 5.5 (Anthropic), October 2026. Self-checked by the writing AI. Public domain (CC0).

For a complex vector \(\theta=\alpha+i\beta\) with \(\theta\cdot\theta=0\), the derivative \(D_\theta=\theta\cdot\nabla\) acts only in the plane spanned by \(\alpha\) and \(\beta\), where it is a multiple of the Cauchy–Riemann operator \(\partial_{\bar z}\). The leading terms of the solutions built in Exact solutions with prescribed leading terms satisfy first-order systems \(D_\theta V=MV\), and their comparison in Matching the transports needs two facts about such planar systems: invertible solutions exist over a whole disk, and a solution with compact support in a disk is determined by its equation [OpenAI-EL, Lemmas 4.1 and 5.2]. This lesson proves both, the first from a named hypothesis on holomorphic frames.

We use from the core course Complex Analysis (Lebl, Guide to Cultivating Complex Analysis, Theorems 2.4.7 and 4.6.3) the identity theorem and the Cauchy–Pompeiu formula; Plancherel's identity and the Sobolev norms \(\|F\|_{H^k}^2=(2\pi)^{-2}\int(1+|\xi|^2)^k|\widehat F|^2d\xi\) on \(\mathbb R^2\), which for integers \(k\ge0\) are equivalent to the sums of the \(L^2\) norms of the derivatives of order at most \(k\), from Sections 2 and 7 of Fourier transforms, finite spectra and convex separation; and from Section 1 of Symbols, finite defects, and the index on a closed manifold the compactness (I2), in its local form for functions supported in one fixed compact set, and the embedding of \(\bigcap_kH^k\) into smooth functions (I1).

1. The Cauchy transform and regularity for ∂/∂z̄

Write points of \(\mathbb R^2=\mathbb C\) as \(y=(y_1,y_2)\) or \(z=y_1+iy_2\), let \(dA\) be the area measure, \(\partial_{\bar z}=\frac12(\partial_{y_1}+i\partial_{y_2})\), and \(D^2_R=\{|y|<R\}\). For a bounded measurable function \(f\) with bounded support, matrix-valued or vector-valued, the Cauchy transform is \[ (\mathcal Tf)(z)=\frac1\pi\int_{\mathbb C}\frac{f(w)}{z-w}\,dA(w). \]

Lemma 1.1. Let \(f\) be bounded and measurable with support in a disk of radius \(\delta\).

  1. \(|\mathcal Tf(z)|\le2\delta\sup|f|\) for every \(z\).
  2. \(\mathcal Tf\) is continuous.
  3. \(\partial_{\bar z}\mathcal Tf=f\) in the sense of distributions.

Proof. (1) Let \(D\) be the support disk and \(B=B(z,\delta)\), of the same area. As \(|z-w|^{-1}\ge\delta^{-1}\) on \(B\) and \(\le\delta^{-1}\) off \(B\), \[ \begin{aligned}&\int_{D\setminus B}|z-w|^{-1}\le\operatorname{area}(D\setminus B)/\delta\\&\quad=\operatorname{area}(B\setminus D)/\delta\le\int_{B\setminus D}|z-w|^{-1}.\end{aligned} \] So \(\int_D|z-w|^{-1}dA\le\int_B|z-w|^{-1}dA\), and the latter equals \(2\pi\delta\).

(2) After the substitution \(w\mapsto z-w\), \[ \begin{aligned}&\mathcal Tf(z)-\mathcal Tf(z')\\&\quad=\frac1\pi\int\bigl(f(z-w)-f(z'-w)\bigr)\\&\qquad\qquad\times w^{-1}\,dA(w).\end{aligned} \] The part with \(|w|<\eta\) is at most \(4\eta\sup|f|\), and the rest at most \(\eta^{-1}\pi^{-1}\int|f(z-w)-f(z'-w)|\,dA(w)\), which tends to zero as \(z'\to z\) by continuity of translation in \(L^1\).

(3) For a test function \(\varphi\), the Cauchy–Pompeiu formula on a large circle gives \(\varphi(w)=-\frac1\pi\int\frac{\partial_{\bar z}\varphi(z)}{z-w}\,dA(z)\). By Fubini's theorem, which applies because \(|z-w|^{-1}\) is locally integrable, \[ \begin{aligned}&-\int\mathcal Tf\,\partial_{\bar z}\varphi\,dA\\&\quad=-\frac1\pi\int f(w)\\&\qquad\times\int\frac{\partial_{\bar z}\varphi(z)}{z-w}\,dA(z)\,dA(w)\\&\quad=\int f\varphi\,dA .\qquad\square\end{aligned} \]

Lemma 1.2 (regularity). Let \(U\subset\mathbb C\) be open, \(b\in C^\infty(U;\mathbb C^{n\times n})\), \(g\in C^\infty(U;\mathbb C^n)\), and let \(W\in L^2_{\mathrm{loc}}(U;\mathbb C^n)\) satisfy \(\partial_{\bar z}W=bW+g\) in the sense of distributions. Then \(W\in C^\infty(U)\).

Proof. For \(F\in L^2(\mathbb R^2)\) with compact support, \(\widehat{\partial_{\bar z}F}(\xi)=\frac i2(\xi_1+i\xi_2)\widehat F(\xi)\), of modulus \(\frac12|\xi||\widehat F|\). Hence \[ \begin{aligned}(1+|\xi|^2)^{k+1}|\widehat F|^2&=(1+|\xi|^2)^k|\widehat F|^2\\&\quad+4(1+|\xi|^2)^k\bigl|\widehat{\partial_{\bar z}F}\bigr|^2,\end{aligned} \] so \(F\in H^k\) and \(\partial_{\bar z}F\in H^k\) give \(F\in H^{k+1}\). Suppose \(W\in H^k_{\mathrm{loc}}(U)\) for some integer \(k\ge0\), and let \(\chi\in C^\infty_c(U)\). Then \(F=\chi W\in H^k\), and \(\partial_{\bar z}F=\chi(bW+g)+(\partial_{\bar z}\chi)W\in H^k\) by the product rule for weak derivatives. So \(\chi W\in H^{k+1}\), and \(W\in H^{k+1}_{\mathrm{loc}}(U)\). By induction \(W\in H^k_{\mathrm{loc}}\) for every \(k\), and by (I1) \(W\) is smooth. \(\square\)

Corollary 1.3. If \(w\in L^2_{\mathrm{loc}}(U)\) and \(\partial_{\bar z}w=0\) in the sense of distributions, then \(w\) is holomorphic. \(\square\)

2. Frames and inhomogeneous equations

A frame of the equation \(\partial_{\bar z}V=AV\), for a smooth \(A\) with values in \(\mathbb C^{n\times n}\), is a smooth solution \(V\) with values in invertible \(n\times n\) matrices. Its columns are \(n\) solutions that are linearly independent at every point.

Lemma 2.1 (local frames). Let \(A\in C^\infty(U;\mathbb C^{n\times n})\) on an open set \(U\subset\mathbb C\). Every \(z_0\in U\) has a disk \(D(z_0,\delta)\subset U\) on which \(\partial_{\bar z}V=AV\) has a frame \(V\) with \(|V-\mathrm{Id}|\le\frac12\) in operator norm.

Proof. Choose \(\delta_0\) with \(\overline{D(z_0,2\delta_0)}\subset U\), let \(M=\max|A|\) on that closed disk, and take \(\delta\le\delta_0\) with \(4\delta M\le\frac13\). Let \(\chi\in C^\infty_c(D(z_0,2\delta))\) with \(0\le\chi\le1\) and \(\chi=1\) on \(D(z_0,\delta)\). On bounded measurable matrix functions with the supremum norm, \(KV=\mathcal T(\chi AV)\) has norm at most \(\frac13\) by Lemma 1.1(1). The Neumann series \(V=\sum_{j\ge0}K^j\mathrm{Id}\) solves \(V=\mathrm{Id}+KV\) and satisfies \(\sup|V-\mathrm{Id}|\le\frac{1/3}{1-1/3}=\frac12\), so \(V(z)\) is invertible for every \(z\). By Lemma 1.1(3), \(\partial_{\bar z}V=\chi AV\), which is \(AV\) on \(D(z_0,\delta)\). As \(V\) is bounded, Lemma 1.2 makes it smooth there. \(\square\)

If \(V_1,V_2\) are frames on overlapping disks, then on the overlap \[ \begin{aligned}\partial_{\bar z}(V_1^{-1}V_2)&=-V_1^{-1}(\partial_{\bar z}V_1)V_1^{-1}V_2\\&\quad+V_1^{-1}AV_2=0 ,\end{aligned} \] so the transition matrices are holomorphic. Frames over a whole disk are given by the following hypothesis.

Hypothesis (HF) (frames over a disk; [Forstnerič 2009, Theorem 1.1]). Let \(R>0\) and \(A\in C^\infty(D^2_R;\mathbb C^{n\times n})\). Then \(\partial_{\bar z}V=AV\) has a frame on all of \(D^2_R\).

The local frames of Lemma 2.1 and their holomorphic transition matrices define a holomorphic vector bundle over the disk, and the frames identify its underlying smooth bundle with the trivial bundle; so its holomorphic frame bundle, a holomorphic fiber bundle with fiber the complex Lie group \(\mathrm{GL}_n(\mathbb C)\), has a continuous section. The disk is a Stein manifold and \(\mathrm{GL}_n(\mathbb C)\) is a complex homogeneous manifold, and the Oka principle for sections of such bundles [Forstnerič 2009, Theorem 1.1], which in the homogeneous case goes back to Grauert, gives a holomorphic section: a holomorphic frame of the bundle, which in the smooth trivialization is a frame as stated.

Lemma 2.2 (frames depending on a parameter; [OpenAI-EL, Lemma 4.1]). Let \(I\subset\mathbb R\) be an open interval and \(A\in C^\infty(D^2_R\times I;\mathbb C^{n\times n})\). For every \(R'<R\) and \(t_0\in I\) there are an open interval \(I_0\ni t_0\) and a smooth function \(V\) on \(D^2_{R'}\times I_0\), with invertible values, such that \(\partial_{\bar z}V=AV\) for each \(t\in I_0\).

Proof. Let \(V_0\) be a frame of \(\partial_{\bar z}V=A(\cdot,t_0)V\) on \(D^2_R\), by Hypothesis (HF). Writing \(V=V_0W\), the equation becomes \[ \begin{aligned}\partial_{\bar z}W&=B(z,t)W,\qquad\text{where}\\ B(z,t)&=V_0(z)^{-1}\bigl(A(z,t)\\&\qquad-A(z,t_0)\bigr)V_0(z).\end{aligned} \] Choose \(\chi\in C^\infty_c(D^2_R)\) with \(0\le\chi\le1\), equal to one near \(\overline{D^2_{R'}}\). On the compact set \(\operatorname{supp}\chi\), \(B(\cdot,t)\to0\) uniformly as \(t\to t_0\); choose a compact interval \(I_1\ni t_0\) in \(I\) with \(2R\sup_{\operatorname{supp}\chi\times I_1}|B|\le\frac13\), and let \(I_0\) be its interior. The operators \(K_t:W\mapsto\mathcal T(\chi B(\cdot,t)W)\) on bounded measurable matrix functions then have norm at most \(\frac13\), and \(W_t=(I-K_t)^{-1}\mathrm{Id}\) satisfies \(\sup|W_t-\mathrm{Id}|\le\frac12\) and \(\partial_{\bar z}W_t=B(\cdot,t)W_t\) on \(D^2_{R'}\).

Smoothness. By Lemma 1.1(1), \(\|K_t-K_s\|\) is at most \(2R\sup|\chi(B(\cdot,t)-B(\cdot,s))|\), and the same bound for the difference quotients shows that \(t\mapsto K_t\) is \(C^\infty\) in operator norm, with derivatives \(W\mapsto\mathcal T(\chi\,\partial_t^kB\,W)\); hence so is \(t\mapsto(I-K_t)^{-1}\), and \(t\mapsto W_t\) is \(C^\infty\) as a map into bounded functions, with derivatives \(\partial_t^kW_t\) satisfying \(\partial_{\bar z}(\partial_t^kW_t)=\sum_{j\le k}\binom kj(\partial_t^jB)(\partial_t^{k-j}W_t)\) on \(D^2_{R'}\). The proof of Lemma 1.2 is quantitative: for \(F=\chi'W\) it gives \(\|F\|_{H^{q+1}}^2=\|F\|_{H^q}^2+4\|\partial_{\bar z}F\|_{H^q}^2\), so on a smaller disk the \(H^{q+1}\) norm of a solution of \(\partial_{\bar z}W=bW+g\) is bounded by its \(H^q\) norm and that of \(g\) on a larger disk, linearly. Applied to \(\partial_t^kW_t-\partial_t^kW_s\), whose equation has the right side \(B(\cdot,t)(\partial_t^kW_t-\partial_t^kW_s)\) plus terms that tend to zero in every \(H^q\) as \(s\to t\), and to the difference quotients of \(\partial_t^kW_t\), this shows that \(t\mapsto\partial_t^kW_t\) is continuous into \(H^m\) of every smaller disk, for every \(k\) and \(m\), and is the \(k\)-th derivative there. By (I1) the same holds with \(C^m\) in place of \(H^m\). Hence all mixed derivatives of \(W\) in \((z,t)\) exist and are continuous on \(D^2_{R''}\times I_0\) for every \(R''<R'\). Replacing \(R'\) by a number between \(R'\) and \(R\) at the start gives the statement on \(D^2_{R'}\times I_0\), and \(V=V_0W\). \(\square\)

Lemma 2.3 (inhomogeneous equations; [OpenAI-EL, Lemma 4.1]). Let \(V\) be as in Lemma 2.2 on \(D^2_{R'}\times I_0\), let \(R''<R'\), and let \(F\in C^\infty(D^2_{R'}\times I_0;\mathbb C^n)\). Then \(\partial_{\bar z}v=Av+F\) has a smooth solution on \(D^2_{R''}\times I_0\). If \(F(\cdot,t)=0\) for \(t\) outside a compact set \(K\subset I_0\), the solution can be chosen with \(v(\cdot,t)=0\) for \(t\notin K\).

Proof. Choose \(\chi\in C^\infty_c(D^2_{R'})\) equal to one near \(\overline{D^2_{R''}}\), and put \(v=V\,\mathcal T(\chi V^{-1}F)\), the transform acting in \(z\) for each \(t\). Writing \(\mathcal Tg(z)=\frac1\pi\int g(z-w)\,w^{-1}dA(w)\), derivatives of a smooth \(g\) with support in a fixed compact set can be taken under the integral, so \(v\) is smooth. By Lemma 1.1(3), \(\partial_{\bar z}v=AV\mathcal T(\chi V^{-1}F)+V\chi V^{-1}F\), which is \(Av+F\) on \(D^2_{R''}\). If \(F(\cdot,t)=0\), then \(v(\cdot,t)=0\). \(\square\)

The same statements hold for \(c\,\partial_{\bar z}\) with a nonzero constant \(c\), and for systems \(Q\,\partial_{\bar z}v=Av+F\) with a smooth invertible matrix \(Q\), after dividing by \(c\) or multiplying by \(Q^{-1}\).

3. Supported solutions

Lemma 3.1 (supported planar transport; [OpenAI-EL, Lemma 5.2]). Let \(r>0\), let \(\mathcal U\subset\mathbb R^d\) be open, \(n\ge1\), and \[ \begin{gathered}\mathcal P_\rho=\gamma(\rho)(\partial_{y_1}+i\partial_{y_2})+A(y,\rho),\\ \rho\in\mathcal U,\end{gathered} \] with \(\gamma\) smooth and nonzero and \(A\) a smooth complex \(n\times n\) matrix on a neighbourhood of \(\overline{D^2_{2r}}\times\mathcal U\).

  1. If \(W\in L^2(D^2_{2r};\mathbb C^n)\) has compact support in \(D^2_{2r}\) and \(\mathcal P_\rho W=0\) in the sense of distributions, then \(W=0\).
  2. For every integer \(j\ge0\) there is \(C_j\), locally uniform in \(\rho\), such that every \(W\in H^{j+1}(\mathbb R^2;\mathbb C^n)\) with \(\operatorname{supp}W\subset\overline{D^2_r}\) satisfies \[ \begin{aligned}&\|W\|_{H^{j+1}}\\&\quad\le C_j\|\mathcal P_\rho W\|_{H^j}.\end{aligned}\tag{3.1} \]
  3. Let \(f\) be smooth on a neighbourhood of \(\overline{D^2_{2r}}\times\mathcal U\). If \(\mathcal P_\rho W=f(\cdot,\rho)\) has a solution \(W\in L^2\) with support in \(\overline{D^2_r}\) for a dense set of \(\rho\), then it has one for every \(\rho\). These solutions are unique and form a smooth function of \((y,\rho)\).

The same holds for matrix-valued unknowns, whose entries are regarded as a vector.

Proof. Note that \(\mathcal P_\rho W=0\) means \(\partial_{\bar z}W=-(2\gamma)^{-1}AW\), and \(\mathcal P_\rho W=f\) means \(\partial_{\bar z}W=-(2\gamma)^{-1}AW+(2\gamma)^{-1}f\), at fixed \(\rho\).

(1) By Lemma 1.2, \(W\) is smooth. Let \(O\) be the set of points of \(D^2_{2r}\) near which \(W\) vanishes; it is open, and nonempty because the support of \(W\) is compact. Let \(z\in D^2_{2r}\) be a limit of points of \(O\), and let \(F\) be a frame of \(\partial_{\bar z}F=-(2\gamma)^{-1}AF\) on a disk \(D\ni z\) (Lemma 2.1). Then \(\partial_{\bar z}(F^{-1}W)=0\) on \(D\), so \(F^{-1}W\) is holomorphic, and it vanishes on the open set \(O\cap D\), which is nonempty. By the identity theorem \(F^{-1}W=0\) on \(D\), so \(z\in O\). Thus \(O\) is open and closed in the connected disk, and \(W=0\).

(2) Extend \(W\) by zero to \(\mathbb R^2\), and cut \(A\) off outside a neighbourhood of \(\overline{D^2_r}\); this changes nothing on the support of \(W\). The Fourier transform of \(\gamma(\partial_{y_1}+i\partial_{y_2})W\) is \(i\gamma(\xi_1+i\xi_2)\widehat W\), of modulus \(|\gamma||\xi||\widehat W|\), and \((1+|\xi|^2)^{j+1}\) is \((1+|\xi|^2)^j+(1+|\xi|^2)^j|\xi|^2\); so \[ \begin{aligned}&\|W\|_{H^{j+1}}\\&\quad\le\|W\|_{H^j}\\&\qquad+|\gamma|^{-1}\|\gamma(\partial_{y_1}+i\partial_{y_2})W\|_{H^j}\\&\quad\le\|W\|_{H^j}\\&\qquad+|\gamma|^{-1}\bigl(\|\mathcal P_\rho W\|_{H^j}+\|AW\|_{H^j}\bigr).\end{aligned} \] By the product rule \(\|AW\|_{H^j}\le C\|W\|_{H^j}\), and \(\|W\|_{H^j}\le\varepsilon\|W\|_{H^{j+1}}+C_\varepsilon\|W\|_{L^2}\), because \((1+|\xi|^2)^j\le\varepsilon^2(1+|\xi|^2)^{j+1}+C_\varepsilon^2\). Absorbing, \[ \begin{aligned}&\|W\|_{H^{j+1}}\le C\bigl(\|\mathcal P_\rho W\|_{H^j}\\&\qquad+\|W\|_{L^2}\bigr).\end{aligned}\tag{3.2} \] If (3.1) failed at a fixed \(\rho\), there would be \(W_\nu\) supported in \(\overline{D^2_r}\) with \(\|W_\nu\|_{H^{j+1}}=1\) and \(\|\mathcal P_\rho W_\nu\|_{H^j}\to0\). By (I2) a subsequence converges in \(L^2\) to some \(W\), supported in \(\overline{D^2_r}\), with \(\mathcal P_\rho W=0\) in the sense of distributions. By (1), \(W=0\), and then (3.2) contradicts \(\|W_\nu\|_{H^{j+1}}=1\). So (3.1) holds at each \(\rho_0\). For \(\rho\) near \(\rho_0\), \(\|(\mathcal P_\rho-\mathcal P_{\rho_0})W\|_{H^j}\le\eta(\rho)\|W\|_{H^{j+1}}\) with \(\eta(\rho)\to0\), since the coefficients depend smoothly on \(\rho\); absorbing this term gives (3.1) near \(\rho_0\) with twice the constant.

(3) Let \(X_j\) be the closed subspace of \(H^{j+1}(\mathbb R^2;\mathbb C^n)\) of the functions with support in \(\overline{D^2_r}\), and \(Z_j=H^j(\mathbb R^2;\mathbb C^n)\), with their Hermitian inner products. The map \(\rho\mapsto\mathcal P_\rho\in\mathcal L(X_j,Z_j)\) is \(C^\infty\) in operator norm, because its \(\rho\)-derivatives are differential operators of order at most one with coefficients smooth in \(\rho\). By (3.1), \((\mathcal P_\rho^*\mathcal P_\rho W,W)=\|\mathcal P_\rho W\|^2\ge C_j^{-2}\|W\|^2\), where \(\mathcal P_\rho^*\) is the Hilbert space adjoint; so the self-adjoint operator \(\mathcal P_\rho^*\mathcal P_\rho\) on \(X_j\) is injective with closed range, its range is dense because its orthogonal complement is its kernel, and it is invertible. Thus \[ T_{\rho,j}=(\mathcal P_\rho^*\mathcal P_\rho)^{-1}\mathcal P_\rho^*:Z_j\to X_j \] is a bounded left inverse of \(\mathcal P_\rho\), and \(\rho\mapsto T_{\rho,j}\) is \(C^\infty\) in operator norm, as adjoints and inverses of bounded invertible operators depend smoothly on the operator. An \(L^2\) solution of \(\mathcal P_\rho W=f(\cdot,\rho)\) supported in \(\overline{D^2_r}\) is smooth by Lemma 1.2, hence lies in every \(X_j\), and equals \(T_{\rho,j}f(\cdot,\rho)\). Put \(W_j(\rho)=T_{\rho,j}f(\cdot,\rho)\), a \(C^\infty\) map into \(X_j\). The residual \(\mathcal P_\rho W_j(\rho)-f(\cdot,\rho)\) is continuous in \(\rho\) with values in \(Z_j\) and vanishes on a dense set, hence everywhere: a supported solution exists for every \(\rho\). It is unique by (1), so it does not depend on \(j\). As \(\rho\mapsto W(\rho)\) is \(C^\infty\) into every \(H^{j+1}\), (I1) shows that all mixed derivatives in \((y,\rho)\) exist and are continuous. \(\square\)

4. Exercises

4.1. Show that \(\mathcal T\) applied to the indicator function of the unit disk is \(\bar z\) for \(|z|\le1\) and \(1/z\) for \(|z|\ge1\), and check Lemma 1.1(1) for it.

4.2. Let \(\theta=\alpha+i\beta\) with \(|\alpha|=|\beta|=\rho_0>0\) and \(\alpha\perp\beta\), and coordinates \(y_1,y_2\) along \(\alpha/\rho_0\) and \(\beta/\rho_0\). Show that \(D_\theta=\rho_0(\partial_{y_1}+i\partial_{y_2})=2\rho_0\partial_{\bar z}\).

4.3. For \(n=1\) and a smooth scalar \(a\) on a neighbourhood of \(\overline{D^2_R}\), show that \(V=\exp(\mathcal T(\chi a))\), with \(\chi\in C^\infty_c\) equal to one near \(\overline{D^2_R}\), is a frame of \(\partial_{\bar z}V=aV\) on \(D^2_R\). Why does this argument not work for matrices?

4.4. Show that Lemma 3.1(1) fails without the support condition: find a nonzero \(W\) on \(D^2_{2r}\) with \((\partial_{y_1}+i\partial_{y_2})W=0\).

4.5. In the proof of Lemma 3.1(3), why is the left inverse \(T_{\rho,j}\) used instead of an inverse of \(\mathcal P_\rho\)?

5. Solutions

4.1. For \(|z|>1\), \(w\mapsto(z-w)^{-1}\) is holomorphic near the closed unit disk, and by the mean value property its average over the disk is \(z^{-1}\); so \(\mathcal T\mathbf 1(z)=1/z\), also for \(|z|=1\) by continuity (Lemma 1.1(2)). Let \(u=\bar z\) for \(|z|\le1\) and \(u=1/z\) for \(|z|\ge1\). It is continuous, as \(\bar z=1/z\) on the circle, and the divergence theorem on the disk and on its complement, whose boundary terms cancel by continuity, gives \(\partial_{\bar z}u=\mathbf 1\) in the sense of distributions. So \(\mathcal T\mathbf 1-u\) is continuous with \(\partial_{\bar z}(\mathcal T\mathbf 1-u)=0\), hence entire by Corollary 1.3, and it tends to zero at infinity; by Liouville's theorem it is zero. Thus \(\mathcal T\mathbf 1(z)=\bar z\) for \(|z|\le1\), and \(|\mathcal T\mathbf 1|\le1\le2\), as Lemma 1.1(1) with \(\delta=1\) requires.

4.2. \(\theta\cdot\nabla=\alpha\cdot\nabla+i\beta\cdot\nabla=\rho_0\partial_{y_1}+i\rho_0\partial_{y_2}\), and \(\partial_{y_1}+i\partial_{y_2}=2\partial_{\bar z}\).

4.3. \(\mathcal T(\chi a)\) is smooth, as in the proof of Lemma 2.3, and \(\partial_{\bar z}\exp(\mathcal T(\chi a))=\exp(\mathcal T(\chi a))\,\partial_{\bar z}\mathcal T(\chi a)\), which is \(aV\) on \(D^2_R\) by Lemma 1.1(3); \(V\) never vanishes. For matrices, \(\partial_{\bar z}e^{M(z)}\ne(\partial_{\bar z}M)e^{M(z)}\) unless \(M\) commutes with its derivative, so the exponential does not solve the equation.

4.4. Any holomorphic function, for example \(W=1\) or \(W=z\).

4.5. \(\mathcal P_\rho\) is injective on \(X_j\) but need not be surjective onto \(Z_j\): for most right sides there is no solution supported in \(\overline{D^2_r}\). The left inverse exists without surjectivity, and its smoothness in \(\rho\) is what transfers smoothness to the solutions on the set of parameters where they exist.

References