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.
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 p...
Let $\mathcal H_g f(z)=\int_0^1 f(t)g'(tz),dt$, and let $2<p<\infty$. Set $t=2p/(p-2)$ and $X_j=2^{-j/p'}\Delta_j g'$, where $\Delta_j$ is the hard dyadic Taylor projection. We prove that $\mathcal H_g:H^p\to H^p$ is bounded if and only if $h\mapsto(X_jh)_{j\geq0}$ is bounded from $H^t$ to $\ell^t(H^2)$. The square of...
Let $Z$ be a compact K\"ahler manifold endowed with a Hamiltonian action of a compact connected Lie group $U$, and let $G\subset U^{\mathbb C}$ be a real compatible subgroup. Using the gradient map $\mu_{\mathfrak p}$ associated with the momentum map, we establish a Calabi-Matsushima type decomposition for the Lie alge...
Let $\omega$ be a radial $\widehat{\mathcal D}$-weight and $u$ be a bounded function on the unit disk $\mathbb D$. We prove that the Toeplitz operator \(T_{\omega,u}\) is compact on \(A_\omega^2\) if and only if its Berezin transform vanishes at the boundary. Our approach is based on a polynomial frame for $A_\omega^2$...
For $n\ge2$, we classify the affine diffeomorphisms $f_{A,v}(x)=Ax+v$ on $\mathbb R^n$ that admit a complete Riemannian metric with respect to which they are Anosov. Such a metric exists if and only if $A$ is hyperbolic or $f_{A,v}$ has no fixed point, equivalently $v\notin\operatorname{Im}(I-A)$. In the latter case, $...
Let $\mathbb{G}$ be a stratified Lie group and $\mathcal L$ be its sub-Laplacian. We prove that the full horizontal Riesz transform $\nabla_{H} \mathcal L^{-1/2}$ is of weak type $(1,1)$ on real-valued functions, with constant at most $2$. In particular, the constant is independent of the horizontal dimension, the homo...
Sheng-Chen Mao, Yaojun Wang, Ye Zhang· 1 citation
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