15 June 2022, Volume 38 Issue 6

    

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  Robust Error Density Estimation in Ultrahigh Dimensional Sparse Linear Model

  Feng ZOU, Heng Jian CUI

  Acta Mathematica Sinica. 2022, 38(6): 963-984. https://doi.org/10.1007/s10114-022-1134-2

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  This paper focuses on error density estimation in ultrahigh dimensional sparse linear model, where the error term may have a heavy-tailed distribution. First, an improved two-stage refitted crossvalidation method combined with some robust variable screening procedures such as RRCS and variable selection methods such as LAD-SCAD is used to obtain the submodel, and then the residual-based kernel density method is applied to estimate the error density through LAD regression. Under given conditions, the large sample properties of the estimator are also established. Especially, we explicitly give the relationship between the sparsity and the convergence rate of the kernel density estimator. The simulation results show that the proposed error density estimator has a good performance. A real data example is presented to illustrate our methods.
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  A Smallness Condition Ensuring Boundedness in a Two-dimensional Chemotaxis-Navier–Stokes System involving Dirichlet Boundary Conditions for the Signal

  Yu Lan WANG, Michael WINKLER, Zhao Yin XIANG

  Acta Mathematica Sinica. 2022, 38(6): 985-1001. https://doi.org/10.1007/s10114-022-1093-7

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  The chemotaxis-Navier–Stokes system
  \begin{equation*} \begin{cases} n_t + u\cdot\nabla n = \Delta n - \nabla \cdot (n\nabla c), \\ c_t + u\cdot\nabla c =\Delta c - nc, \\ u_t + (u\cdot\nabla) u = \Delta u + \nabla P + n\nabla \phi, \quad \nabla \cdot u =0 \end{cases}\end{equation*}
  is considered in a smoothly bounded planar domain Ω under the boundary conditions
  $$ (\nabla n - n\nabla c)\cdot\nu= 0, \quad c=c_★, \quad u=0, \quad x\in∂Ω, t>0, $$
  with a given nonnegative constant $c_★$. It is shown that if $(n_0,c_0,u_0)$ is sufficiently regular and such that the product $\|n_0\|_{L^1(Ω)} \|c_0\|_{L^\infty(Ω)}^2$ is suitably small, an associated initial value problem possesses a bounded classical solution with $(n,c,u)|_{t=0}=(n_0,c_0,u_0)$.
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  Proper Holomorphic Mappings Between n-generalized Hartogs Triangles

  Feng RONG, Shuo ZHANG

  Acta Mathematica Sinica. 2022, 38(6): 1002-1014. https://doi.org/10.1007/s10114-022-1422-x

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  In this paper, we first introduce the notion of n-generalized Hartogs triangles. Then, we characterize proper holomorphic mappings between some of these domains, and describe their automorphism groups.
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  On Weighted Compactness of Commutators Related with Schrödinger Operators

  Qian Jun HE, Peng Tao LI

  Acta Mathematica Sinica. 2022, 38(6): 1015-1040. https://doi.org/10.1007/s10114-022-1081-y

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  Let $\mathcal{L}=-\Delta+\mathit{V}$ be a Schrödinger operator, where $\Delta$ is the Laplacian operator on $\mathbb{R}^{d}$ $(d\geq 3)$, while the nonnegative potential $\mathit{V}$ belongs to the reverse Hölder class $B_{q}, q>d/2$. In this paper, we study weighted compactness of commutators of some Schrödinger operators, which include Riesz transforms, standard Calderón--Zygmund operators and Littlewood--Paley functions. These results substantially generalize some well-known results.
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  Imaginary Modules over the Affine Nappi–Witten Algebra

  Yi Xin BAO, Yan An CAI

  Acta Mathematica Sinica. 2022, 38(6): 1041-1053. https://doi.org/10.1007/s10114-022-0246-z

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  In this paper, we consider the imaginary highest weight modules and the imaginary Whittaker modules for the affine Nappi-Witten algebra. We show that simple singular imaginary Whittaker modules at level $(k,c)\, (k\in\mathbb{C}^＊)$ are simple imaginary highest weight modules. The necessary and sufficient conditions for these imaginary modules to be simple are given. All simple imaginary modules are classified.
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  Scattering for 3D Cubic Focusing NLS on the Domain Outside a Convex Obstacle Revisited

  Cheng Bin XU, Teng Fei ZHAO, Ji Qiang ZHENG

  Acta Mathematica Sinica. 2022, 38(6): 1054-1068. https://doi.org/10.1007/s10114-022-1058-x

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  In this article, we consider the focusing cubic nonlinear Schrödinger equation(NLS) in the exterior domain outside of a convex obstacle in $\mathbb{R}^3$ with Dirichlet boundary conditions. We revisit the scattering result below ground state in Killip-Visan-Zhang [The focusing cubic NLS on exterior domains in three dimensions. Appl. Math. Res. Express. AMRX, 1, 146-180 (2016)] by utilizing the method of Dodson and Murphy [A new proof of scattering below the ground state for the 3d radial focusing cubic NLS. Proc. Amer. Math. Soc., 145, 4859-4867 (2017)] and the dispersive estimate in Ivanovici and Lebeau [Dispersion for the wave and the Schrödinger equations outside strictly convex obstacles and counterexamples. Comp. Rend. Math., 355, 774-779 (2017)], which avoids using the concentration compactness. We conquer the difficulty of the boundary in the focusing case by establishing a local smoothing effect of the boundary. Based on this effect and the interaction Morawetz estimates, we prove that the solution decays at a large time interval, which meets the scattering criterion.
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  Embedding Derivatives and Integration Operators on Hardy Type Tent Spaces

  Mao Fa WANG, Lv ZHOU

  Acta Mathematica Sinica. 2022, 38(6): 1069-1093. https://doi.org/10.1007/s10114-022-0405-2

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  In this paper, we completely characterize the positive Borel measures $\mu$ on the unit ball $\mathbb{B}_{n}$ such that the differential type operator $\mathcal{R}^{m}$ of order $m\in \mathbb{N}$ is bounded from Hardy type tent space $\mathcal{HT}_{q,\alpha}^{p}(\mathbb{B}_{n})$ into $L^{s}(\mu)$ for full range of $p, q,s, \alpha$. Subsequently, the corresponding compact description of differential type operator $\mathcal{R}^{m}$ is also characterized. As an application, we obtain the boundedness and compactness of integration operator $J_{g}$ from $\mathcal{HT}_{q,\alpha}^{p}(\mathbb{B}_{n})$ to $\mathcal{HT}_{s,\beta}^{t}(\mathbb{B}_{n})$, and the methods used here are adaptable to the Hardy spaces.
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  Pluriclosed Manifolds with Constant Holomorphic Sectional Curvature

  Pei Pei RAO, Fang Yang ZHENG

  Acta Mathematica Sinica. 2022, 38(6): 1094-1104. https://doi.org/10.1007/s10114-022-1046-1

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  A long-standing conjecture in complex geometry says that a compact Hermitian manifold with constant holomorphic sectional curvature must be Kähler when the constant is non-zero and must be Chern flat when the constant is zero. The conjecture is known in complex dimension 2 by the work of Balas-Gauduchon in 1985 (when the constant is zero or negative) and by Apostolov-Davidov-Muskarov in 1996 (when the constant is positive). For higher dimensions, the conjecture is still largely unknown. In this article, we restrict ourselves to pluriclosed manifolds, and confirm the conjecture for the special case of Strominger Kähler-like manifolds, namely, for Hermitian manifolds whose Strominger connection (also known as Bismut connection) obeys all the Kähler symmetries.
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  On the Eventual Shadowing Property and Eventually Shadowable Point of Set-valued Dynamical Systems

  Xue Rong XIE, Jian Dong YIN

  Acta Mathematica Sinica. 2022, 38(6): 1105-1115. https://doi.org/10.1007/s10114-022-1041-6

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  The authors introduce the concepts of the eventual shadowing property and eventually shadowable point for set-valued dynamical systems and prove that a set-valued dynamical system has the eventual shadowing property if and only if every point in the phase space is eventually shadowable; every chain transitive set-valued dynamical system has either the eventual shadowing property or no eventually shadowable points; and a set-valued dynamical system admits an eventually shadowable point if and only if it admits a minimal eventually shadowable point. Moreover, it is proved that a set-valued dynamical system with the eventual shadowing property is chain mixing if and only if it is mixing and if and only if it has the specification property.
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  The Cell Modules of the Green Algebra of Drinfel'd Quantum Double D(H₄)

  Liu Feng CAO, Hui Xiang CHEN, Li Bin LI

  Acta Mathematica Sinica. 2022, 38(6): 1116-1132. https://doi.org/10.1007/s10114-022-9046-8

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  This paper is devoted to studying the structures of the cell modules of the complexified Green algebra R(D(H₄)), where D(H₄) is the Drinfel'd quantum double of Sweedler's 4-dimensional Hopf algebra H₄. We show that R(D(H₄)) has one infinite dimensional cell module, one 4-dimensional cell module generated by all finite dimensional indecomposable projective modules of D(H₄) and infinitely many 2-dimensional cell modules. More precisely, we obtain the decompositions of all finite dimensional cell modules into the direct sum of indecomposable submodules, and show that the infinite dimensional cell module can be written as the direct sum of two infinite dimensional indecomposable submodules