arXiv subjects

Liu Tailiang

Publications and source records attributed to Liu Tailiang.

5 recordsLinked to original sources

arXiv2026

Real-analytic realization of universal Teichm\"uller space via complex-structures on $H^{1/2}$

Let $H$ be the Hilbert transform, let $h$ be a quasisymmetric homeomorphism of the unit circle $S^1$, and set $V_hu=u\circ h$ and $J_h=V_hHV_h^{-1}$, defined on the Sobolev space $H^{1/2}(S^1)$. We prove that $h\mapsto V_h$ is nowhere continuous in operator norm, although it is continuous in the strong operator topology. By contrast, the induced map $h\mapsto J_h$ is a real-analytic diffeomorphism onto its image in the operator-norm topology. Based on this, we further compute the differential at the identity and show that it is precisely the Calder\'on commutator. In graph coordinates, the tangent map admits a weighted Hankel matrix representation, whose Hilbert-Schmidt norm recovers the Weil-Petersson tangent quadratic form.

math.CV↗

arXiv2026

A Compactness Characterization of Strongly Symmetric Homeomorphisms

A self-homeomorphism $h$ of the unit circle $\mathbb{T}$ is strongly symmetric if it is absolutely continuous and $\log h'\in\text{VMO}(\mathbb{T})$. Let $P_h^-$ be the anti-analytic component of the pullback operator $P_h: F\mapsto F\circ h$ on $\text{BMOA}(\mathbb{D})$, where $\mathbb{D}$ is the unit disk. P. Jones proved $P_h$ is bounded on BMO if and only if $h$ is strongly quasisymmetric. Fan, Hu, and Shen showed that the strong symmetry of $h$ yields the compactness of $P_h^-$, and raised the question of whether the converse is true. We answer this question affirmatively, establishing that $P_h^-$ is compact if and only if $\log h'\in\text{VMO}(\mathbb{T}))$, which completes the VMO theory of the universal Teichm\"uller space.

math.CV↗

arXiv2026

Conformal welding of chord-arc curves

We study the relation between the geometric properties of a chord-arc curve and its conformal welding. Let $h$ be the conformal welding of a closed quasicircle $\Gamma$. By Jones's theorem, the pull-back operator $C_h u=u\circ h$ is bounded on BMO if and only if $h$ corresponds to the welding of a Bishop-Jones quasicircle, equivalently, $h$ is strongly quasisymmetric. Let $A_h$ denote the analytic projection of $C_h$. We prove that $A_h$ is a bounded isomorphism on BMOA if and only if $\Gamma$ is a chord-arc curve. More strongly, the same characterization holds if invertibility is replaced by Fredholmness. This gives an intrinsic conformal-welding characterization of chord-arc curves and a complete geometric answer to the invertibility problem posed by Semmes in the 1980s. Furthermore, we establish an exact correspondence between the inverse of $A_h$ and the classical Faber integral operator, showing that for a rectifiable quasicircle, the Faber operator is a bounded isomorphism on BMOA if and only if the curve satisfies the chord-arc condition.

math.CV↗

arXiv2026

Conformally Invariant Besov Spaces on Chord-Arc Domains

Let $\Gamma$ be a rectifiable Jordan curve. We introduce the associated Besov $p$-spaces on $\Gamma$ and in its complementary domains, and characterize chord-arc curves by the natural trace isomorphisms among these spaces. We prove the characterization for the previously unresolved range $1<p<2$, thereby extending the known results $p\geq2$ to $1<p<\infty$. We further give an alternative proof for $p\geq2$.

math.CV↗

arXiv2024

The boundary correspondence under quasiconformal mappings and VMO-Teichmuller space

In this paper, we introduce a class of vanishing Carleson measures with conformal invariance and corresponding strongly vanishing symmetric homeomorphisms on the real line and prove that they can be mutually generated under quasiconformal mappings. This is motivated by constructing a nice VMO-Teichmuller space on the real line, which completely removes the obstacle of the usual VMO-Teichmuller space that lacks conformal invariance and is repeatedly encountered in the papers [17,22-24].

math.CV↗