# Frontiers of Mathematics in China

ISSN 1673-3452 (Print)
ISSN 1673-3576 (Online)
CN 11-5739/O1
Postal Subscription Code 80-964
2019 Impact Factor: 1.03

#### , Volume 16 Issue 1 dateToAbbreviationMonthYear('2021-02-15 00:00:00.0','CurrentIssueDate');

SURVEY ARTICLE
 Select Function characterizations via commutators of Hardy operator Shanzhen LU Front. Math. China. 2021, 16 (1): 1-12.  https://doi.org/10.1007/s11464-021-0894-9 Abstract   PDF (300KB) This paper is a summary of the research on the characterizations of central function spaces by the author and his collaborators in the past ten years. More precisely, the author gives some characterizations of central Campanato spaces via the boundedness and compactness of commutators of Hardy operator.
RESEARCH ARTICLE
 Select Singular integral operators on product domains along twisted surfaces Ahmad AL-SALMAN Front. Math. China. 2021, 16 (1): 13-28.  https://doi.org/10.1007/s11464-021-0911-z Abstract   PDF (299KB) We introduce a class of singular integral operators on product domains along twisted surfaces. We prove that the operators are bounded on Lp provided that the kernels satisfy weak conditions.
 Select Grothendieck rings of a class of Hopf algebras of Kac-Paljutkin type Jialei CHEN, Shilin YANG, Dingguo WANG Front. Math. China. 2021, 16 (1): 29-47.  https://doi.org/10.1007/s11464-021-0893-x Abstract   PDF (337KB) We construct the Grothendieck rings of a class of $2n2$ dimensional semisimple Hopf Algebras $H2n2$,which can be viewed as a generalization of the 8 dimensional Kac-Paljutkin Hopf algebra $H8$.All irreducible $H2n2$-modules are classified. Furthermore, we describe the Grothendieck rings $r(H2n2)$ by generators and relations explicitly.
 Select Exceptional sets in Waring-Goldbach problem for fifth powers Zhenzhen FENG, Zhixin LIU Front. Math. China. 2021, 16 (1): 49-58.  https://doi.org/10.1007/s11464-021-0899-4 Abstract   PDF (278KB) We consider exceptional sets in the Waring-Goldbach problem for fifth powers. For example, we prove that all but O(N131/132) integers satisfying the necessary local conditions can be represented as the sum of 11 fifth powers of primes, which improves the previous results due to A. V. Kumchev [Canad. J. Math., 2005, 57: 298–327] and Z. X. Liu [Int. J. Number Theory, 2012, 8: 1247–1256].
 Select Dynamical behaviors of non-autonomous fractional FitzHugh-Nagumo system driven by additive noise in unbounded domains Chunxiao GUO, Yiju CHEN, Ji SHU, Xinguang YANG Front. Math. China. 2021, 16 (1): 59-93.  https://doi.org/10.1007/s11464-021-0896-7 Abstract   PDF (365KB) The regularity of random attractors is considered for the nonautonomous fractional stochastic FitzHugh-Nagumo system. We prove that the system has a pullback random attractor that is compact in $Hs(ℝn)×L2(ℝn)$ and attracts all tempered random sets of $Ls(ℝn)×L2(ℝn)$ in the topology of $Hs(ℝn)×L2(ℝn)$ with $s∈(0,1)$. By the idea of positive and negative truncations, spectral decomposition in bounded domains, and tail estimates, we achieved the desired results.
 Select Proper resolutions and Gorensteinness in extriangulated categories Jiangsheng HU, Dondong ZHANG, Panyue ZHOU Front. Math. China. 2021, 16 (1): 95-117.  https://doi.org/10.1007/s11464-021-0887-8 Abstract   PDF (317KB) Let $(ℓ,E,s)$ be an extriangulated category with a proper class $ξ$ of $E$-triangles, and $W$ an additive full subcategory of $(ℓ,E,s)$. We provide a method for constructing a proper $Wξ$-resolution (resp., coproper $Wξ$- coresolution) of one term in an $E$-triangle in $ξ$ from that of the other two terms. By using this way, we establish the stability of the Gorenstein category $GWξ$ in extriangulated categories. These results generalize the work of Z. Y. Huang [J. Algebra, 2013, 393: 142{169] and X. Y. Yang and Z. C. Wang [Rocky Mountain J. Math., 2017, 47: 1013{1053], but the proof is not too far from their case. Finally, we give some applications about our main results.
 Select Fourier transform of anisotropic mixed-norm Hardy spaces Long HUANG, Der-Chen CHANG, Dachun YANG Front. Math. China. 2021, 16 (1): 119-139.  https://doi.org/10.1007/s11464-021-0906-9 Abstract   PDF (349KB) Let $a→：=(a1,...,an)∈[1,∞)n,p→:=(p1,...pn)∈(0,1]n,Ha→p→(ℝn)$ be the anisotropic mixed-norm Hardy space associated with $a→$ defined via the radial maximal function, and let f belong to the Hardy space $Ha→p→(ℝn)$. In this article, we show that the Fourier transform $f^$ coincides with a continuous function g on $ℝn$ in the sense of tempered distributions and, moreover, this continuous function g; multiplied by a step function associated with $a→$; can be pointwisely controlled by a constant multiple of the Hardy space norm of f: These proofs are achieved via the known atomic characterization of $Ha→p→(ℝn)$ and the establishment of two uniform estimates on anisotropic mixed-norm atoms. As applications, we also conclude a higher order convergence of the continuous function g at the origin. Finally, a variant of the Hardy{Littlewood inequality in the anisotropic mixed-norm Hardy space setting is also obtained. All these results are a natural generalization of the well-known corresponding conclusions of the classical Hardy spaces $Hp(ℝn)$ with $p∈(0,1]$, and are even new for isotropic mixed-norm Hardy spaces on $ℝn$.
 Select Bi-block positive semidefiniteness of bi-block symmetric tensors Zheng-Hai HUANG, Xia LI, Yong WANG Front. Math. China. 2021, 16 (1): 141-169.  https://doi.org/10.1007/s11464-021-0874-0 Abstract   PDF (367KB) The positive definiteness of elasticity tensors plays an important role in the elasticity theory. In this paper, we consider the bi-block symmetric tensors, which contain elasticity tensors as a subclass. First, we define the bi-block M-eigenvalue of a bi-block symmetric tensor, and show that a bi-block symmetric tensor is bi-block positive (semi)definite if and only if its smallest bi-block M-eigenvalue is (nonnegative) positive. Then, we discuss the distribution of bi-block M-eigenvalues, by which we get a sufficient condition for judging bi-block positive (semi)definiteness of the bi-block symmetric tensor involved. Particularly, we show that several classes of bi-block symmetric tensors are bi-block positive definite or bi-block positive semidefinite, including bi-block (strictly) diagonally dominant symmetric tensors and bi-block symmetric (B)B0-tensors. These give easily checkable sufficient conditions for judging bi-block positive (semi)definiteness of a bi-block symmetric tensor. As a byproduct, we also obtain two easily checkable suffcient conditions for the strong ellipticity of elasticity tensors.
 Select Convergence of complex martingale for a branching random walk in an independent and identically distributed environment Xin WANG, Xingang LIANG, Chunmao HUANG Front. Math. China. 2021, 16 (1): 187-209.  https://doi.org/10.1007/s11464-021-0882-0 Abstract   PDF (341KB) We consider an $ℝd$-valued discrete time branching random walk in an independent and identically distributed environment indexed by time $n∈ℕ$. Let $Wn(z)(z∈ℂd)$ be the natural complex martingale of the process. We show necessary and sufficient conditions for the $Lα$-convergence of $Wn(z)$ for $α$>1, as well as its uniform convergence region.
 Select Boundedness of some integral operators and commutators on homogeneous Herz spaces with three variable exponents Xia YU, Zongguang LIU Front. Math. China. 2021, 16 (1): 211-237.  https://doi.org/10.1007/s11464-021-0897-6 Abstract   PDF (335KB) We obtain the boundedness of some integral operators and commutators on homogeneous Herz spaces with three variable exponents $K ˙p(⋅),q(⋅)α(⋅)$, such as some sublinear operators, the fractional integral and its commutator.