Aug 2020, Volume 15 Issue 4

    

• Select all

• RESEARCH ARTICLE

  Oscillation and variation for Riesz transform in setting of Bessel operators on H¹ and BMO

  Xiaona CUI, Jing ZHANG

  2020, 15(4): 617-647. https://doi.org/10.1007/s11464-020-0853-x

  Download PDF

  Let \lambda >\mathrm{0} and let the Bessel operator \mathrm{\Delta }\lambda ：=-{\displaystyle \frac{{\mathrm{d}}^{\mathrm{2}}}{{\mathrm{d}x}^{\mathrm{2}}}}-{\displaystyle \frac{\mathrm{2}\lambda }{x}}{\displaystyle \frac{\mathrm{d}}{\mathrm{d}x}} defined on {{\displaystyle ℝ}}_{+}:=(\mathrm{0},\infty ). We show that the oscillation and \rho-variation operators of the Riesz transform {{\displaystyle R}}_{{\mathrm{\Delta }}_{\lambda }} associated with {\mathrm{\Delta }}_{\lambda } are bounded on BMO({{\displaystyle ℝ}}_{+},{{\displaystyle dm}}_{\lambda }), where \rho >\mathrm{2} and {{\displaystyle \mathrm{d}m}}_{\lambda }=x^{\mathrm{2}\lambda }\mathrm{d}x. Moreover, we construct a {{\displaystyle (\mathrm{1},\infty )}}_{{\mathrm{\Delta }}_{\lambda }}-atom as a counterexample to show that the oscillation and \rho-variation operators of {{\displaystyle R}}_{{\mathrm{\Delta }}_{\lambda }} are not bounded from H^{\mathrm{1}}({ℝ}^{+},{{\displaystyle \mathrm{d}m}}_{\lambda }) to L^{\mathrm{1}}({ℝ}^{+},{{\displaystyle \mathrm{d}m}}_{\lambda }). Finally, we prove that the oscillation and the {{\displaystyle (\mathrm{1},\infty )}}_{{\mathrm{\Delta }}_{\lambda }}-variation operators for the smooth truncations associated with Bessel operators {{\displaystyle \tilde{R}}}_{{\mathrm{\Delta }}_{\lambda }} are bounded from H^{\mathrm{1}}({ℝ}^{+},{{\displaystyle \mathrm{d}m}}_{\lambda }) to L^{\mathrm{1}}({ℝ}^{+},{{\displaystyle \mathrm{d}m}}_{\lambda }).

• RESEARCH ARTICLE

  Weighted product Hardy space associated with operators

  Qingquan DENG, Djalal Eddine GUEDJIBA

  2020, 15(4): 649-683. https://doi.org/10.1007/s11464-020-0852-y

  Download PDF

  Assuming that the operators L₁, L₂ are self-adjoint and {\mathrm{e}}^{-{{\displaystyle tL}}_i}(i=\mathrm{1},\mathrm{2}) satisfy the generalized Davies-Gaffney estimates, we shall prove that the weighted Hardy space {{\displaystyle H}}_{{{\displaystyle L}}_{\mathrm{1}},{{\displaystyle L}}_{\mathrm{2}},\omega }^{\mathrm{1}}({ℝ}^{{{\displaystyle n}}_{\mathrm{1}}}\times {ℝ}^{{{\displaystyle n}}_{\mathrm{2}}}) associated to operators L₁, L₂ on product domain, which is defined in terms of area function, has an atomic decomposition for some weight \omega.

• RESEARCH ARTICLE

  Weighted binary relations involving core-EP inverse

  Yuefeng GAO, Jianlong CHEN, Pedro PATR_ICIO

  2020, 15(4): 685-699. https://doi.org/10.1007/s11464-020-0856-7

  Download PDF

  We study a new binary relation defined on the set of rectangular complex matrices involving the weighted core-EP inverse and give its characterizations. This relation becomes a pre-order. Then, one-sided pre-orders associated to the weighted core-EP inverse are given from two perspectives. Finally, we make a comparison for these two sets of one-sided weighted pre-orders.

• RESEARCH ARTICLE

  Remark on Gauss curvature equations on punctured disk

  Yuxiang LI, Hongyan TANG

  2020, 15(4): 701-707. https://doi.org/10.1007/s11464-020-0855-8

  Download PDF

  We give a new argument on the classification of solutions of Gauss curvature equation on R²; which was first proved by W. Chen and C. Li [Duke Math. J., 1991, 63(3): 615–622]. Our argument bases on the decomposition properties of the Gauss curvature equation on the punctured disk.

• RESEARCH ARTICLE

  Weighted weak group inverse for Hilbert space operators

  Dijana MOSIC, Daochang ZHANG

  2020, 15(4): 709-726. https://doi.org/10.1007/s11464-020-0847-8

  Download PDF

  We present the weighted weak group inverse, which is a new generalized inverse of operators between two Hilbert spaces, and we extend the notation of the weighted weak group inverse for rectangular matrices. Some characterizations and representations of the weighted weak group inverse are investigated. We also apply these results to define and study the weak group inverse for a Hilbert space operator. Using the weak group inverse, we define and characterize various binary relations.

• RESEARCH ARTICLE

  Partial approximate boundary synchronization for a coupled system of wave equations with Dirichlet boundary controls

  Chenmu WANG, Yanyan WANG

  2020, 15(4): 727-748. https://doi.org/10.1007/s11464-020-0848-7

  Download PDF

  In this paper, we propose the concept of partial approximate boundary synchronization for a coupled system of wave equations with Dirichlet boundary controls, and make a deep discussion on it. We analyze the relation-ship between the partial approximate boundary synchronization and the partial exact boundary synchronization, and obtain sufficient conditions to realize the partial approximate boundary synchronization and necessary conditions of Kalman's criterion. In addition, with the help of partial synchronization decomposition, a condition that the approximately synchronizable state does not depend on the sequence of boundary controls is also given.

• RESEARCH ARTICLE

  Moments of integral-type downward functionals for single death processes

  Jing WANG, Yuhui ZHANG

  2020, 15(4): 749-768. https://doi.org/10.1007/s11464-020-0850-0

  Download PDF

  We get an explicit and recursive representation for high order moments of integral-type downward functionals for single death processes. Meanwhile, main results are applied to more general integral-type downward functionals.

• RESEARCH ARTICLE

  Intrinsic square function characterizations of Hardy spaces associated with ball quasi-Banach function spaces

  Xianjie YAN, Dachun YANG, Wen YUAN

  2020, 15(4): 769-806. https://doi.org/10.1007/s11464-020-0849-6

  Download PDF

  Let X be a ball quasi-Banach function space satisfying some mild additional assumptions and H_X({ℝ}^n) the associated Hardy-type space. In this article, we first establish the finite atomic characterization of H_X({ℝ}^n). As an application, we prove that the dual space of H_X({ℝ}^n) is the Campanato space associated with X. For any given \alpha \in (\mathrm{0},\mathrm{1}] and s\in {\mathbb{Z}}_{+}, using the atomic and the Littlewood–Paley function characterizations of H_X({ℝ}^n),we also establish its s-order intrinsic square function characterizations, respectively, in terms of the intrinsic Lusin-area function S_{\alpha ,s},the intrinsic g-function g_{\alpha ,s},and the intrinsic g_{\lambda }^{\ast }-function g_{\lambda ,\alpha ,s}^{\ast }, where λ coincides with the best known range.

• RESEARCH ARTICLE

  Dynamics of a family of rational maps concerning renormalization transformation

  Yuhan ZHANG, Junyang GAO, Jianyong QIAO, Qinghua WANG

  2020, 15(4): 807-833. https://doi.org/10.1007/s11464-020-0854-9

  Download PDF

  Considering a family of rational maps T_{n\lambda }concerning renormalization transformation, we give a perfect description about the dynamical properties of T_{n\lambda } and the topological properties of the Fatou components F (T_{n\lambda }). Furthermore, we discuss the continuity of the Hausdorff dimension HD(J (T_{n\lambda })) about real parameter λ.

• RESEARCH ARTICLE

  Hua’s theorem on five squares of primes

  Wenjia ZHAO

  2020, 15(4): 835-850. https://doi.org/10.1007/s11464-020-0851-z

  Download PDF

  We give an alternative proof of Hua’s theorem that each large N≡5 (mod 24) can be written as a sum of five squares of primes. The proof depends on an estimate of exponential sums involving the Möbius function.