Some regularity theorems for CR mappings

Localized Regularity of Planar Maps of Finite Distortion

In this article we study fine regularity properties for mappings of finite distortion. Our main theorems yield strongly localized regularity results in the borderline case in the class of maps of exponentially integrable distortion. Analogues of such results were known earlier in the case of quasiconformal mappings. Moreover, we study regularity for maps whose distortion has higher exponential integrability.

Regularity of CR-mappings between Fuchsian type hypersurfaces in $\mathbb C^2$

We investigate regularity of CR-mappings between real-analytic infinite type hypersurfaces in $\mathbb C^2$. We show that, under the Fuchsian type condition, all (respectively formal or smooth) CR-diffeomorphisms between them are automatically analytic. The Fuchsian condition appears to be in a certain sense optimal for the regularity problem.

Regularity And Extremality Of Quasiconformal Homeomorphisms On CR 3-Manifolds

This paper first studies the regularity of conformal homeomorphisms on smooth locally embeddable strongly pseudoconvex CR manifolds. Then moduli of curve families are used to estimate the maximal dilatations of quasiconformal homeomorphisms. On certain CR 3-manifolds, namely, CR circle bundles over flat tori, extremal quasiconformal homeomorphisms in some homotopy classes are constructed. These extremal mappings have similar behaviors to Teichmüller mappings on Riemann surfaces.

Optimal regularity for planar mappings of finite distortion

Let $f:Ω\to\IR^2$ be a mapping of finite distortion, where $Ω\subset\IR^2 .$ Assume that the distortion function $K(x,f)$ satisfies $e^{K(\cdot, f)}\in L^p_{loc}(Ω)$ for some $p>0.$ We establish optimal regularity and area distortion estimates for $f$. Especially, we prove that $|Df|^2 \log^{β-1}(e + |Df|) \in L^1_{loc}(Ω) $ for every $β<p.$ This answers positively well known conjectures due to Iwaniec and Martin \cite{IMbook} and to Iwaniec, Koskela and Martin \cite{IKM}.

On holomorphic mappings of strictly pseudoconvex domains

We study the boundary regularity of proper holomorphic mappings between strictly pseudoconvex domains with $C^2$-boundaries.

Extremal Problems in Bergman Spaces and an Extension of Ryabykh's $H^p$ Regularity Theorem For $1<p<\infty$

We study linear extremal problems in the Bergman space $A^p$ of the unit disc, where $1 < p < \infty$. Given a functional on the dual space of $A^p$ with representing kernel $k \in A^q$, where $1/p + 1/q = 1$, we show that if $q \le q_1 < \infty$ and $k \in H^{q_1}$, then $F \in H^{(p-1)q_1}$. This result was previously known only in the case where $p$ is an even integer. We also discuss related results.

Planar mappings of subexponentially integrable distortion -- integrability of distortion of inverses

We establish the optimal regularity for the distortion of inverses of mappings of finite distortion with logarithm-iterated style subexponentially integrable distortion, which generalizes the Theorem 1. of [J. Gill, Ann. Acad. Sci. Fenn. Math. 35 (2010), no. 1, 197--207].

On Riemann's mapping theorem

In this paper we give a new proof of Riemann's well known mapping theorem. The suggested method permits to prove an analog of that theorem for the three dimensional case.

Distortion and covering theorems of pluriharmonic mappings

In this paper, we mainly investigate distortion and covering theorems on some classes of pluriharmonic mappings.

Lipschitz regularity of energy-minimal mappings between doubly connected Riemann surfaces

Let $M$ and $N$ be doubly connected Riemann surfaces with $\mathscr{C}^{1,α}$ boundaries and with nonvanishing conformal metrics $σ$ and $\wp$ respectively, and assume that $\wp$ is a smooth metric with bounded Gauss curvature $\mathcal{K}$ and finite area. Assume that ${\Ho}_ρ(M, N)$ is the class of all $\mathscr{W}^{1,2}$ bomeomorphisms between $M$ and $N$ and assume that $\mathcal{E}^\wp: \overline{\Ho}_ρ(M, N)\to \mathbf{R}$ is the Dirichlet-energy functional, where $\overline{\Ho}_ρ(M, N)$ is the closure of ${\Ho}_ρ(M, N)$ in $\mathscr{W}^{1,2}(M,N)$. By using a result of Iwaniec, Kovalev…

An elliptic regularity theorem for fractional partial differential operators

We present and prove a version of the elliptic regularity theorem for partial differential equations involving fractional Riemann-Liouville derivatives. In this case, regularity is defined in terms of Sobolev spaces $H^s(X)$: if the forcing of a linear elliptic fractional PDE is in one Sobolev space, then the solution is in the Sobolev space of increased order corresponding to the order of the derivatives. We also mention a few applications and potential extensions of this result.

A Counterexample to $C^k$-regularity for the Newlander-Nirenberg Theorem

We give an example of $C^k$-integrable almost complex structure that does not admit a corresponding $C^{k+1}$-complex coordinate system.

On Beltrami equations and Hoelder regularity

We estimate the Hoelder exponent $α$ of solutions to the Beltrami equation $\dbar f=μ\de f$, where the Beltrami coefficient satisfies $\|μ\|_\infty<1$. Our estimate improves the classical estimate $α\ge\|K_μ\|^{-1}$, where $K_μ=(1+|μ|)/(1-|μ|)$, and it is sharp, in the sense that it is actually attained in a class of mappings which generalize the radial stretchings. Some other properties of such mappings are also provided.

拟共形映射在Triebel-Lizorkin尺度下的全局光滑性

Global smoothness of quasiconformal mappings in the Triebel-Lizorkin scale

本文研究平面区域上拟共形映射在Sobolev、Bessel位势和Triebel-Lizorkin尺度下的全局正则性，利用边界几何和Beltrami系数光滑性建立最优条件，并证明在光滑性低于1的Triebel-Lizorkin类中该条件同样保证主解的全局正则性。

The Hopf-Laplace equation: harmonicity and regularity

The central theme in this paper is the Hopf-Laplace equation, which represents stationary solutions with respect to the inner variation of the Dirichlet integral. Among such solutions are harmonic maps. Nevertheless, minimization of the Dirichlet energy among homeomorphisms often leads to mappings which are neither harmonic nor homeomorphisms. We prove that such mappings are harmonic outside of a singular set with small image. On the singular set they are locally Lipschitz, but not necessarily differentiable.

平面非散度型椭圆方程的改进Hölder正则性

Improved Hölder regularity for elliptic equations of non-divergence type in the plane

本文改进了Baernstein和Kovalev研究的准正则梯度映射的Hölder正则性，探讨了平面非散度型椭圆方程的正则性问题。

On Boman's Theorem On Partial Regularity Of Mappings

Let Λ\subsetR^{n}\timesR^{m} and k be a positive integer. Let f:R^{n}\rightarrowR^{m} be a locally bounded map such that for each (ξ,η)\inΛ, the derivatives D_{ξ}^{j}f(x):=|((d^{j})/(dt^{j}))f(x+tξ)|_{t=0}, j=1,2,...k, exist and are continuous. In order to conclude that any such map f is necessarily of class C^{k} it is necessary and sufficient that Λ be not contained in the zero-set of a nonzero homogenous polynomial Φ(ξ,η) which is linear in η=(η_{1},η_{2},...,η_{m}) and homogeneous of degree k in ξ=(ξ_{1},ξ_{2},...,ξ_{n}). This generalizes a result of J. Boman for the case k=1. The statemen…

Variation of quasiconformal mappings on lines

We obtain improved regularity of homeomorphic solutions of the reduced Beltrami equation, as compared to the standard Beltrami equation. Such an improvement is not possible in terms of Holder or Sobolev regularity; instead, our results concern the generalized variation of restrictions to lines. Specifically, we prove that the restriction to any line segment has finite p-variation for all p>1 but not necessarily for p=1.

On Skoda's theorem for Nadel-Lebesgue multiplier ideal sheaves on singular complex spaces and regularity of weak Kähler-Einstein metrics

In this article, we will characterize regular points respectively by the local vanishing, positivity of the Ricci curvature and $L^2$-solvability of the $\overline\partial$-equation together with Skoda's theorem for Nadel-Lebesgue multiplier ideal sheaves associated to plurisubharmonic (psh) functions on any (reduced) complex space of pure dimension. As a by-product, we show that any weak Kähler-Einstein metric on \emph{singular} $\mathbb{Q}$-Fano/Calabi-Yau/general type varieties cannot be smooth, and that in general there exists no \emph{singular} normal Kähler complex space such that the Kä…

多微局化子分析层的强正则性与微支撑估计

Strong Regularity and Microsupport Estimates for Multi-Microlocalizations of Subanalytic Sheaves

本文提出子分析层的强正则性概念，研究多微局化下的支撑和微支撑估计，并推导了温和全纯解的初值定理和除法定理，最终得到Bochner管形定理的多微局化版本。

Wiman and Arima theorems for quasiregular mappings

Generalizations of the theorems of Wiman and of Arima on entire functions are proved for spatial quasiregular mappings.

Sharp Hölder Regularity for Nirenberg's Complex Frobenius Theorem

Nirenberg's famous complex Frobenius theorem gives necessary and sufficient conditions on a locally integrable structure for when the manifold is locally diffeomorphic to $\mathbb R^r\times\mathbb C^m\times \mathbb R^{N-r-2m}$ through a coordinate chart $F$ in such a way that the structure is locally spanned by $F^*\frac\partial{\partial t^1},\dots,F^*\frac\partial{\partial t^r},F^*\frac\partial{\partial z^1},\dots,F^*\frac\partial{\partial z^m}$, where we have given $\mathbb R^r\times\mathbb C^m \times\mathbb R^{N-r-2m}$ coordinates $(t,z,s)$. In this paper, we give the optimal Hölder-Zygmund…