The semigroup-fication route to Brennan's conjecture: approximating conformal maps by holomorphic semigroups with derivative-integrability control

Brennan's conjecture (1978) asserts that for every conformal map $f$ of the unit disk the derivative satisfies $\int_{\mathbb D}|f'|^{p}\,dA<\infty$ for every $p\in(-2,2/3)$; it remains one of the central open problems of geometric function theory. Aleman and Kouroupis (2024) proved the conjecture for every element of every continuous one-parameter semigroup of holomorphic self-maps of the unit disk, and posed as the follow-up question whether one can advance toward the full conjecture by 'semigroup-fication' — approximation theorems between general conformal maps and semigroup elements. The problem is the following reduction program: construct, for arbitrary conformal maps (it suffices to treat bounded univalent self-maps of the disk, by results of Carleson and Makarov), approximants that are elements of continuous semigroups and whose $p$-integral means of the derivative stay uniformly bounded for all $p\in(-2,2/3)$; by Fatou's lemma, together with the Aleman–Kouroupis theorem, such an approximation would prove Brennan's conjecture in full generality.

Background

A conformal (univalent, that is, holomorphic and injective) map $f$ of the unit disk $\mathbb D$ can have a derivative that either blows up or degenerates near the boundary. Brennan's conjecture (1978) asserts that nevertheless, for every conformal map $f$ on $\mathbb D$ and every exponent $p$ with $-2<p<2/3$, $\int_{\mathbb D}|f'(w)|^{p}\,dA(w)<\infty,$ where $dA$ is normalized planar Lebesgue measure on $\mathbb D$. The exponent range is sharp. The conjecture is equivalent to a sharp bound on the universal integral means spectrum, the function governing the worst-case boundary growth of the integrals $\int_{0}^{2\pi}|f'(re^{i\theta})|^{p}\,d\theta$ as $r\to1^{-}$, and it is one of the famous open problems of geometric function theory, with connections to complex dynamics and to Sobolev regularity of quasiconformal maps (homeomorphisms of the plane that distort infinitesimal distances by at most a bounded multiplicative factor). It is known for geometrically regular subclasses such as starlike maps (whose image domain is starlike with respect to a point) and close-to-convex maps (univalent maps $f$ such that $\mathrm{Re}(e^{i\theta}f'(z)/g'(z))>0$ for all $z\in\mathbb D$, for some real $\theta$ and some convex univalent map $g$, that is, a univalent map onto a convex domain), and for all conformal maps only on strict subintervals $(-p_{0},2/3)$ with $p_{0}<2$ (Hedenmalm–Shimorin 2005; Sola 2006, whose estimate Aleman–Kouroupis quote as $p_{0}\approx 1.76$). A reduction of Carleson and Makarov shows that it suffices to prove the conjecture for bounded conformal maps, i.e. for univalent self-maps of $\mathbb D$.

A continuous one-parameter semigroup of holomorphic functions is a family $(\phi_{t})_{t\ge 0}$ of conformal self-maps of $\mathbb D$ satisfying $\phi_{0}=\mathrm{id}$, $\phi_{t+s}=\phi_{t}\circ\phi_{s}$, and continuity of $t\mapsto\phi_{t}(z)$ for each $z\in\mathbb D$. Every nontrivial such semigroup shares a single Denjoy–Wolff point $\tau\in\overline{\mathbb D}$, the point toward which the iterates of every element converge, and is linearized by its Koenigs map — a conformal map $h$ with $h\circ\phi_{t}=\psi_{t}\circ h$, where $\psi_{t}(z)=e^{-\lambda t}z$ in the elliptic case (then $h(\mathbb D)$ is invariant under the spiral flow $z\mapsto e^{-\lambda t}z$ for all $t\ge 0$; such maps $h$ are called spirallike, and reduce to the starlike maps just defined when $\lambda$ is real and positive) and $\psi_{t}(z)=z+it$ in the non-elliptic case (then $h$ is starlike at infinity, meaning that, up to normalization, $h(\mathbb D)$ is invariant under the translation flow $z\mapsto z+it$ for all $t\ge 0$). Deciding which univalent self-maps of $\mathbb D$ occur as a member of such a semigroup is the embeddability problem, a long-standing question solved only for linear-fractional maps (fractional-linear self-maps of $\mathbb D$, also called disk automorphisms). Aleman and Kouroupis proved that Brennan's integrability holds, for every $p\in(-2,2/3)$, for every element $\phi_{t}$ of every continuous semigroup and for the Koenigs map of a semigroup; their proof combines classical distortion theorems (standard pointwise two-sided bounds on $|f'|$ valid for all univalent $f$) with the theory of weighted Bergman spaces (spaces of holomorphic functions on $\mathbb D$ that are $p$-integrable against a non-negative weight function), using Békollé–Bonami weights — the disk analogues of the Muckenhoupt $A_{p}$ weights of harmonic analysis, characterized by an $A_{p}$-type condition: a uniformly bounded product of the average of the weight and of a suitable negative power of the weight over Carleson boxes (the boundary squares $\{re^{i\theta}:1-h<r<1,\ |\theta-\theta_{0}|<h/2\}$), a condition that characterizes boundedness of the Bergman projection (the orthogonal projection of square-integrable functions on $\mathbb D$ onto the holomorphic ones) — and spectra of Volterra-type integration operators $T_{g}f(z)=\int_{0}^{z}f(w)g'(w)\,dw$. They observe that their theorem suggests a reduction strategy for the full conjecture, referencing the semigroup-fication technique of Bracci and Roth, which constructs for any univalent self-map $f$ a continuous semigroup whose time-one map $\phi_{1}$ is dynamically close to $f$ (same dynamical type, one-to-one matching of boundary regular fixed points — boundary fixed points at which the map has a finite angular derivative — with their multipliers, identical orbit slopes at the Denjoy–Wolff point). However, no known semigroup-fication or approximation theorem controls integral means of the derivative, which is precisely what a reduction of Brennan's conjecture to the semigroup case would require.

Problem Statement

Let $\mathbb D$ be the unit disk with normalized area measure $dA$. Call a conformal self-map $g$ of $\mathbb D$ a semigroup element if $g=\phi_{t_{0}}$ for some $t_{0}\ge 0$ and some continuous one-parameter semigroup $(\phi_{t})_{t\ge 0}$ of holomorphic self-maps of $\mathbb D$, that is, a family satisfying $\phi_{0}=\mathrm{id}$, $\phi_{t+s}=\phi_{t}\circ\phi_{s}$, and continuity of $t\mapsto\phi_{t}(z)$ for each $z\in\mathbb D$. Aleman–Kouroupis proved that $\int_{\mathbb D}|g'(w)|^{p}\,dA(w)<\infty$ for every semigroup element $g$ and every $p\in(-2,2/3)$, and the same for Koenigs maps of semigroups. The problem they pose, phrased as a reduction program from their theorem to the full conjecture, is:

Establish a semigroup-fication or approximation theorem between general conformal maps and semigroup elements that preserves derivative integrability in Brennan's range. Concretely: for every conformal map $f$ of $\mathbb D$ — where, by the Carleson–Makarov reduction, it suffices to treat bounded conformal maps, equivalently (after affine rescaling) univalent self-maps of $\mathbb D$ — construct semigroup elements $g_{n}$ such that $g_{n}\to f$ locally uniformly and, for every $p\in(-2,2/3)$, $\sup_{n}\int_{\mathbb D}|g_{n}'(w)|^{p}\,dA(w)<\infty.$ Since $g_{n}'\to f'$ locally uniformly, Fatou's lemma would then give $\int_{\mathbb D}|f'(w)|^{p}\,dA(w)<\infty$ for all $p\in(-2,2/3)$, i.e. the full Brennan conjecture. The same conclusion holds for any rigorously proved transfer mechanism that deduces Brennan's conjecture for general (or bounded) conformal maps from the Aleman–Kouroupis semigroup theorem.

References

1. Alexandru Aleman and Athanasios Kouroupis, Brennan's conjecture holds for semigroups of holomorphic functions, preprint (2024, v2 April 2025), arXiv:2409.15074, DOI 10.48550/arXiv.2409.15074, https://arxiv.org/abs/2409.15074
2. James E. Brennan, The integrability of the derivative in conformal mapping, Journal of the London Mathematical Society (2) 18 (1978), no. 2, 261–272, DOI 10.1112/jlms/s2-18.2.261, https://doi.org/10.1112/jlms/s2-18.2.261
3. Lennart Carleson and Nikolai G. Makarov, Some results connected with Brennan's conjecture, Arkiv för Matematik 32 (1994), no. 1, 33–62, DOI 10.1007/bf02559522, https://doi.org/10.1007/bf02559522
4. Håkan Hedenmalm and Serguei Shimorin, Weighted Bergman spaces and the integral means spectrum of conformal mappings, Duke Mathematical Journal 127 (2005), no. 2, 341–393, DOI 10.1215/s0012-7094-04-12725-3, https://doi.org/10.1215/s0012-7094-04-12725-3 (preprint arXiv:math/0406345)
5. Alan Sola, An estimate of the universal means spectrum of conformal mappings, Computational Methods and Function Theory 6 (2006), no. 2, 423–436, DOI 10.1007/bf03321620, https://doi.org/10.1007/bf03321620
6. Filippo Bracci and Oliver Roth, Semigroup-fication of univalent self-maps of the unit disc, Annales de l'Institut Fourier 73 (2023), no. 1, 251–277, DOI 10.5802/aif.3517, https://doi.org/10.5802/aif.3517 (preprint arXiv:2002.08252)
7. Manuel D. Contreras, Santiago Díaz-Madrigal, and Pavel Gumenyuk, Centralizers of non-elliptic univalent self-maps and the embeddability problem in the unit disc, Transactions of the American Mathematical Society (2026), DOI 10.1090/tran/9686, https://doi.org/10.1090/tran/9686 (preprint arXiv:2311.04134)
8. Argyrios Christodoulou and Konstantinos Zarvalis, From discrete iteration in the unit disc to continuous semigroups of holomorphic functions, preprint (2025), arXiv:2511.19022, https://arxiv.org/abs/2511.19022
9. Jianjun Jin, Complex exponential integral means spectra of univalent functions and the Brennan conjecture, preprint (2025, revised through June 2026), arXiv:2512.09330, https://arxiv.org/abs/2512.09330
10. Yigang Zheng, Brennan Conjecture for Basin of Attraction at Infinity, preprint (2026), arXiv:2604.12240, https://arxiv.org/abs/2604.12240