Loewner theory in annulus. Part I: Evolution families and differential equations (Q2838074)

Classical Loewner theory deals with the semigroup of all holomorphic self-maps of the unit disk. A similar approach was proposed for holomorphic univalent self-maps of the upper half-plane with the hydrodynamic normalization at infinity. Komatu applied Loewner's ideas for univalent functions in the annulus and multiply connected domains. The authors of the present paper address the Problem: Construct a general Loewner theory in the annulus.NEWLINENEWLINEFor a non-empty interval \(E\subset\mathbb R\) and a connected relatively open subset \(\mathcal D\) of \(\mathbb C\times E\), a function \(G:\mathcal D\to\mathbb C\) is said to be a weak holomorphic vector field of order \(d\in[1,\infty]\) in the domain \(\mathcal D\), if it satisfies the following conditions: for each \(z\in\mathbb C\) the function \(G(z,\cdot)\) is measurable in \(E_z:=\{t: (z,t)\in\mathcal D\}\); for each \(t\in E\) the function \(G(\cdot,t)\) is holomorphic in \(D_t:=\{z: (z,t)\in\mathcal D\}\); for each compact set \(K\subset\mathcal D\) there exists a non-negative function \(k_K\in L^d(\text{pr}_{\mathbb R}(K),\mathbb R)\), where \(\text{pr}_{\mathbb R}(K):=\{t\in E: \exists z\in\mathbb C\;(z,t)\in K\}\) such that \(|G(z,t)|\leq k_K(t)\) for almost every \(t\in\text{pr}_{\mathbb R}(K)\) and \((z,t)\in K\). A weak holomorphic vector field \(G: \mathcal D\to\mathbb C\) is semicomplete if, for any \((z,s)\in\mathcal D\), the initial value problem \(\dot w=G(w(t),t)\), \(w(s)=z\), has a solution defined everywhere in \(E^s:=E\cap[s,\infty)\).NEWLINENEWLINEFor a continuous non-decreasing function \(r:[0,\infty)\to[0,1)\), a family \((D_t)_{t\geq0}\) of annuli \(D_t=\mathbb A_{r(t)}:=\{z: r(t)<|z|<1\}\) is a (doubly connected) canonical domain system of order \(d\in[1,\infty]\) if the function \(t\mapsto\omega(r(t)):=-\pi/\log r(t)\) is non-increasing and locally absolutely continuous on \([0,\infty)\) with its derivative belonging to \(L^d_{\text{loc}}[0,\infty)\). The system \((D_t)\) is non-degenerate if \(r(t)\) does not vanish. A family \((\varphi_{s,t})_{0\leq s\leq t<\infty}\) of holomorphic mappings \(\varphi_{s,t}: D_s\to D_t\) is an \(L^d\)-evolution family of order \(d\) over \((D_t)\) if \(\varphi_{s,s}=\text{id}_{D_s}\), \(\varphi_{s,t}=\varphi_{u,t}\circ\varphi_{s,u}\) whenever \(0\leq s\leq u\leq t<\infty\), and for any closed interval \(I:=[S,T]\subset[0,\infty)\) and any \(z\in D_S\) there exists a non-negative function \(k_{z,I}\in L^d([S,T],\mathbb R)\) such that NEWLINE\[NEWLINE|\varphi_{s,u}(z)-\varphi_{s,t}(z)|\leq\int_u^tk_{z,I}(\xi)d\xi,\;\;\;S\leq s\leq u\leq t\leq T.NEWLINE\]NEWLINE A key result is Theorem 4.6: Let \((D_t)=(\mathbb A_{r(t)})\) be a non-degenerate canonical domain system of order \(d\in[1,\infty]\). Then for any \(L^d\)-evolution family \((\varphi_{s,t})\) over \((D_t)\) there exists a unique \(L^d\)-evolution family \((\Psi_{s,t})\) in the strip \(\mathbb S:=\{z: 0<\Re z<1\}\) such that \(W_t\circ\Psi_{s,t}=\varphi_{s,t}\circ W_s\), \(0\leq s\leq t<\infty\), where \(W_{\tau}(\zeta):=\exp(\zeta\log{r(\tau)})\) for all \(\tau\geq0\), \(\zeta\in\mathbb S\).NEWLINENEWLINESection 5 contains the main results. Theorem 5.1: There is a one-to-one correspondence between the evolution families and semicomplete weak holomorphic fields. More precisely, every evolution family \(((D_t),(\varphi_{s,t}))\) of order \(d\in[1,\infty]\) can be described as the non-autonomous semiflow of some semicomplete weak holomorphic vector field \(G\) of order \(d\) in \(\mathcal D:=\{(z,t): t\geq0,z\in D_t\}\). Conversely, every such semiflow constitutes an evolution family.NEWLINENEWLINEFor \(r\in(0,1)\), the class \(\mathcal V_r\) is the collection of all functions \(p\in\text{Hol}(\mathbb A_r,\mathbb C)\) having the integral representation NEWLINE\[NEWLINEp(z)=\int_{\mathbb T}\mathcal K_r(z/\xi)d\mu_1(\xi)+\int_{\mathbb T}(1-\mathcal K_r(r\xi/z))d\mu_2(\xi),\;\;\mathcal K_r(z)=\lim_{n\to\infty}\sum_{\nu=-n}^n\frac{1+r^{2\nu}z}{1-r^{2\nu}z},\;\;z\in\mathbb A_r,NEWLINE\]NEWLINE where \(\mu_1\) and \(\mu_2\) are positive Borel measures on the unit circle \(\mathbb T\), \(\mu_1(\mathbb T)+\mu_2(\mathbb T)=1\).NEWLINENEWLINETheorem 5.6: Let \((D_t)=(\mathbb A_{r(t)})\) be a non-degenerate canonical domain system of order \(d\in[1,\infty]\). Then a function \(G:\mathcal D\to\mathbb C\), where \(\mathcal D:=\{(z,t): t\geq0,z\in D_t\}\), is a semicomplete weak holomorphic vector field of order \(d\) if and only if there exist functions \(p:\mathcal D\to\mathbb C\) and \(C:[0,\infty)\to\mathbb R\) such thatNEWLINENEWLINE(i) \(G(w,t)=w(iC(t)+r'(t)p(w,t)/r(t))\) for a.e. \(t\geq0\) and all \(w\in D_t\);NEWLINENEWLINE(ii) for each \(w\in D:=\cup_{t\geq0}D_t\) the function \(p(w,\cdot)\) is measurable in \(E_w:=\{t\geq0: (w,t)\in\mathcal D\}\);NEWLINENEWLINE(iii) for each \(t\geq0\) the function \(p(\cdot,t)\) belongs to the class \(\mathcal V_{r(t)}\);NEWLINENEWLINE(iv) \(C\in L^d_{\text{loc}}([0,\infty),\mathbb R)\).