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1. CMB Online first

Dang, Pei; Liu, Hua; Qian, Tao
 Hilbert Transformation and Representation of the $ax+b$ Group In this paper we study the Hilbert transformations over $L^2(\mathbb{R})$ and $L^2(\mathbb{T})$ from the viewpoint of symmetry. For a linear operator over $L^2(\mathbb{R})$ commutative with the ax+b group we show that the operator is of the form $\lambda I+\eta H,$ where $I$ and $H$ are the identity operator and Hilbert transformation respectively, and $\lambda,\eta$ are complex numbers. In the related literature this result was proved through first invoking the boundedness result of the operator, proved though a big machinery. In our setting the boundedness is a consequence of the boundedness of the Hilbert transformation. The methodology that we use is Gelfand-Naimark's representation of the ax+b group. Furthermore we prove a similar result on the unit circle. Although there does not exist a group like ax+b on the unit circle, we construct a semigroup to play the same symmetry role for the Hilbert transformations over the circle $L^2(\mathbb{T}).$ Keywords:singular integral, Hilbert transform, the $ax+b$ groupCategories:30E25, 44A15, 42A50

2. CMB Online first

Li, Junfeng; Yu, Haixia
 Oscillatory Hyper-Hilbert Transform Associated with Plane Curves In this paper, the bounded properties of oscillatory hyper-Hilbert transform along certain plane curves $\gamma(t)$ $$T_{\alpha,\beta}f(x,y)=\int_{0}^1f(x-t,y-\gamma(t))e^{ i t^{-\beta}}\frac{\textrm{d}t}{t^{1+\alpha}}$$ were studied. For a general curves, these operators are bounded in ${L^2(\mathbb{R}^{2})}$, if $\beta\geq 3\alpha$. And their boundedness in $L^p(\mathbb{R}^{2})$ were also obtained, whenever $\beta\gt 3\alpha$, $\frac{2\beta}{2\beta-3\alpha}\lt p\lt \frac{2\beta}{3\alpha}$. Keywords:oscillatory hyper-Hilbert transform, oscillatory integralCategories:42B20, 42B35

3. CMB Online first

Rocha, Pablo Alejandro
 A remark on certain integral operators of fractional type For $m, n \in \mathbb{N}$, $1\lt m \leq n$, we write $n = n_1 + \dots + n_m$ where $\{ n_1, \dots, n_m \} \subset \mathbb{N}$. Let $A_1, \dots, A_m$ be $n \times n$ singular real matrices such that $\bigoplus_{i=1}^{m} \bigcap_{1\leq j \neq i \leq m} \mathcal{N}_j = \mathbb{R}^{n},$ where $\mathcal{N}_j = \{ x : A_j x = 0 \}$, $dim(\mathcal{N}_j)=n-n_j$ and $A_1+ \dots+ A_m$ is invertible. In this paper we study integral operators of the form $T_{r}f(x)= \int_{\mathbb{R}^{n}} \, |x-A_1 y|^{-n_1 + \alpha_1} \cdots |x-A_m y|^{-n_m + \alpha_m} f(y) \, dy,$ $n_1 + \dots + n_m = n$, $\frac{\alpha_1}{n_1} = \dots = \frac{\alpha_m}{n_m}=r$, $0 \lt r \lt 1$, and the matrices $A_i$'s are as above. We obtain the $H^{p}(\mathbb{R}^{n})-L^{q}(\mathbb{R}^{n})$ boundedness of $T_r$ for $0\lt p\lt \frac{1}{r}$ and $\frac{1}{q}=\frac{1}{p} - r$. Keywords:integral operator, Hardy spaceCategories:42B20, 42B30

4. CMB Online first

Ding, Yong; Lai, Xudong
 On a singular integral of Christ-JournÃ© type with homogeneous kernel In this paper, we prove that the following singular integral defined by $$T_{\Omega,a}f(x)=\operatorname{p.v.}\int_{\mathbb{R}^{d}}\frac{\Omega(x-y)}{|x-y|^d}\cdot m_{x,y}a\cdot f(y)dy$$ is bounded on $L^p(\mathbb{R}^d)$ for $1\lt p\lt \infty$ and is of weak type (1,1), where $\Omega\in L\log^+L(\mathbb{S}^{d-1})$ and $m_{x,y}a=:\int_0^1a(sx+(1-s)y)ds$ with $a\in L^\infty(\mathbb{R}^d)$ satisfying some restricted conditions. Keywords:CalderÃ³n commutator, rough kernel, weak type (1,1)Category:42B20

5. CMB 2017 (vol 60 pp. 571)

Li, Ji; Wick, Brett D.
 Weak Factorizations of the Hardy space $H^1(\mathbb{R}^n)$ in terms of Multilinear Riesz Transforms This paper provides a constructive proof of the weak factorization of the classical Hardy space $H^1(\mathbb{R}^n)$ in terms of multilinear Riesz transforms. As a direct application, we obtain a new proof of the characterization of ${\rm BMO}(\mathbb{R}^n)$ (the dual of $H^1(\mathbb{R}^n)$) via commutators of the multilinear Riesz transforms. Keywords:Hardy space, BMO space, multilinear Riesz transform, weak factorizationCategories:42B35, 42B20

6. CMB Online first

Wang, Li-an Daniel
 A Multiplier Theorem on Anisotropic Hardy Spaces We present a multiplier theorem on anisotropic Hardy spaces. When $m$ satisfies the anisotropic, pointwise Mihlin condition, we obtain boundedness of the multiplier operator $T_m : H_A^p (\mathbb R^n) \rightarrow H_A^p (\mathbb R^n)$, for the range of $p$ that depends on the eccentricities of the dilation $A$ and the level of regularity of a multiplier symbol $m$. This extends the classical multiplier theorem of Taibleson and Weiss. Keywords:anisotropic Hardy space, multiplier, Fourier transformCategories:42B30, 42B25, 42B35

7. CMB Online first

Saito, Hiroki; Tanaka, Hitoshi
 The Fefferman-Stein type inequalities for strong and directional maximal operators in the plane The Fefferman-Stein type inequalities for strong maximal operator and directional maximal operator are verified with an additional composition of the Hardy-Littlewood maximal operator in the plane. Keywords:directional maximal operator, Fefferman-Stein type inequality, Hardy-Littlewood maximal operator, strong maximal operatorCategories:42B25, 42B35

8. CMB 2016 (vol 60 pp. 655)

Zhuo, Ciqiang; Sickel, Winfried; Yang, Dachun; Yuan, Wen
 Characterizations of Besov-Type and Triebel-Lizorkin-Type Spaces via Averages on Balls Let $\ell\in\mathbb N$ and $\alpha\in (0,2\ell)$. In this article, the authors establish equivalent characterizations of Besov-type spaces, Triebel-Lizorkin-type spaces and Besov-Morrey spaces via the sequence $\{f-B_{\ell,2^{-k}}f\}_{k}$ consisting of the difference between $f$ and the ball average $B_{\ell,2^{-k}}f$. These results give a way to introduce Besov-type spaces, Triebel-Lizorkin-type spaces and Besov-Morrey spaces with any smoothness order on metric measure spaces. As special cases, the authors obtain a new characterization of Morrey-Sobolev spaces and $Q_\alpha$ spaces with $\alpha\in(0,1)$, which are of independent interest. Keywords:Besov space, Triebel-Lizorkin space, ball average, CalderÃ³n reproducing formulaCategories:42B25, 46E35, 42B35

9. CMB 2016 (vol 60 pp. 131)

Gürbüz, Ferit
 Some Estimates for Generalized Commutators of Rough Fractional Maximal and Integral Operators on Generalized Weighted Morrey Spaces In this paper, we establish $BMO$ estimates for generalized commutators of rough fractional maximal and integral operators on generalized weighted Morrey spaces, respectively. Keywords:fractional integral operator, fractional maximal operator, rough kernel, generalized commutator, $A(p,q)$ weight, generalized weighted Morrey spaceCategories:42B20, 42B25

10. CMB 2016 (vol 60 pp. 586)

Liu, Feng; Wu, Huoxiong
 Endpoint Regularity of Multisublinear Fractional Maximal Functions In this paper we investigate the endpoint regularity properties of the multisublinear fractional maximal operators, which include the multisublinear Hardy-Littlewood maximal operator. We obtain some new bounds for the derivative of the one-dimensional multisublinear fractional maximal operators acting on vector-valued function $\vec{f}=(f_1,\dots,f_m)$ with all $f_j$ being $BV$-functions. Keywords:multisublinear fractional maximal operators, Sobolev spaces, bounded variationCategories:42B25, 46E35

11. CMB 2016 (vol 59 pp. 834)

Liao, Fanghui; Liu, Zongguang
 Some Properties of Triebel-Lizorkin and Besov Spaces Associated with Zygmund Dilations In this paper, using CalderÃ³n's reproducing formula and almost orthogonality estimates, we prove the lifting property and the embedding theorem of the Triebel-Lizorkin and Besov spaces associated with Zygmund dilations. Keywords:Triebel-Lizorkin and Besov spaces, Riesz potential, CalderÃ³n's reproducing formula, almost orthogonality estimate, Zygmund dilation, embedding theoremCategories:42B20, 42B35

12. CMB 2016 (vol 59 pp. 528)

Jahan, Qaiser
 Characterization of Low-pass Filters on Local Fields of Positive Characteristic In this article, we give necessary and sufficient conditions on a function to be a low-pass filter on a local field $K$ of positive characteristic associated to the scaling function for multiresolution analysis of $L^2(K)$. We use probability and martingale methods to provide such a characterization. Keywords:multiresolution analysis, local field, low-pass filter, scaling function, probability, conditional probability and martingalesCategories:42C40, 42C15, 43A70, 11S85

13. CMB 2016 (vol 59 pp. 497)

De Carli, Laura; Samad, Gohin Shaikh
 One-parameter Groups of Operators and Discrete Hilbert Transforms We show that the discrete Hilbert transform and the discrete Kak-Hilbert transform are infinitesimal generator of one-parameter groups of operators in $\ell^2$. Keywords:discrete Hilbert transform, groups of operators, isometriesCategories:42A45, 42A50, 41A44

14. CMB 2016 (vol 59 pp. 521)

Hare, Kathryn; Ramsey, L. Thomas
 The Relationship Between $\epsilon$-Kronecker Sets and Sidon Sets A subset $E$ of a discrete abelian group is called $\epsilon$-Kronecker if all $E$-functions of modulus one can be approximated to within $\epsilon$ by characters. $E$ is called a Sidon set if all bounded $E$-functions can be interpolated by the Fourier transform of measures on the dual group. As $% \epsilon$-Kronecker sets with $\epsilon \lt 2$ possess the same arithmetic properties as Sidon sets, it is natural to ask if they are Sidon. We use the Pisier net characterization of Sidonicity to prove this is true. Keywords:Kronecker set, Sidon setCategories:43A46, 42A15, 42A55

15. CMB 2015 (vol 59 pp. 62)

Feng, Han
 Uncertainty Principles on Weighted Spheres, Balls and Simplexes This paper studies the uncertainty principle for spherical $h$-harmonic expansions on the unit sphere of $\mathbb{R}^d$ associated with a weight function invariant under a general finite reflection group, which is in full analogy with the classical Heisenberg inequality. Our proof is motivated by a new decomposition of the Dunkl-Laplace-Beltrami operator on the weighted sphere. Keywords:uncertainty principle, Dunkl theoryCategories:42C10, 42B10

16. CMB 2015 (vol 59 pp. 104)

He, Ziyi; Yang, Dachun; Yuan, Wen
 Littlewood-Paley Characterizations of Second-Order Sobolev Spaces via Averages on Balls In this paper, the authors characterize second-order Sobolev spaces $W^{2,p}({\mathbb R}^n)$, with $p\in [2,\infty)$ and $n\in\mathbb N$ or $p\in (1,2)$ and $n\in\{1,2,3\}$, via the Lusin area function and the Littlewood-Paley $g_\lambda^\ast$-function in terms of ball means. Keywords:Sobolev space, ball means, Lusin-area function, $g_\lambda^*$-functionCategories:46E35, 42B25, 42B20, 42B35

17. CMB 2015 (vol 58 pp. 877)

Zaatra, Mohamed
 Generating Some Symmetric Semi-classical Orthogonal Polynomials We show that if $v$ is a regular semi-classical form (linear functional), then the symmetric form $u$ defined by the relation $x^{2}\sigma u = -\lambda v$, where $(\sigma f)(x)=f(x^{2})$ and the odd moments of $u$ are $0$, is also regular and semi-classical form for every complex $\lambda$ except for a discrete set of numbers depending on $v$. We give explicitly the three-term recurrence relation and the structure relation coefficients of the orthogonal polynomials sequence associated with $u$ and the class of the form $u$ knowing that of $v$. We conclude with an illustrative example. Keywords:orthogonal polynomials, quadratic decomposition, semi-classical forms, structure relationCategories:33C45, 42C05

18. CMB 2015 (vol 59 pp. 211)

Totik, Vilmos
 Universality Under SzegÅ's Condition This paper presents a theorem on universality on orthogonal polynomials/random matrices under a weak local condition on the weight function $w$. With a new inequality for polynomials and with the use of fast decreasing polynomials, it is shown that an approach of D. S. Lubinsky is applicable. The proof works at all points which are Lebesgue-points both for the weight function $w$ and for $\log w$. Keywords:universality, random matrices, Christoffel functions, asymptotics, potential theoryCategories:42C05, 60B20, 30C85, 31A15

19. CMB 2015 (vol 58 pp. 757)

Han, Yanchang
 Embedding Theorem for Inhomogeneous Besov and Triebel-Lizorkin Spaces on RD-spaces In this article we prove the embedding theorem for inhomogeneous Besov and Triebel-Lizorkin spaces on RD-spaces. The crucial idea is to use the geometric density condition on the measure. Keywords:spaces of homogeneous type, test function space, distributions, CalderÃ³n reproducing formula, Besov and Triebel-Lizorkin spaces, embeddingCategories:42B25, 46F05, 46E35

20. CMB 2015 (vol 58 pp. 507)

Hsu, Ming-Hsiu; Lee, Ming-Yi
 VMO Space Associated with Parabolic Sections and its Application In this paper we define $VMO_\mathcal{P}$ space associated with a family $\mathcal{P}$ of parabolic sections and show that the dual of $VMO_\mathcal{P}$ is the Hardy space $H^1_\mathcal{P}$. As an application, we prove that almost everywhere convergence of a bounded sequence in $H^1_\mathcal{P}$ implies weak* convergence. Keywords:Monge-Ampere equation, parabolic section, Hardy space, BMO, VMOCategory:42B30

21. CMB 2015 (vol 58 pp. 808)

Liu, Feng; Wu, Huoxiong
 On the Regularity of the Multisublinear Maximal Functions This paper is concerned with the study of the regularity for the multisublinear maximal operator. It is proved that the multisublinear maximal operator is bounded on first-order Sobolev spaces. Moreover, two key point-wise inequalities for the partial derivatives of the multisublinear maximal functions are established. As an application, the quasi-continuity on the multisublinear maximal function is also obtained. Keywords:regularity, multisublinear maximal operator, Sobolev spaces, partial deviative, quasicontinuityCategories:42B25, 46E35

22. CMB 2014 (vol 58 pp. 432)

Yang, Dachun; Yang, Sibei
 Second-order Riesz Transforms and Maximal Inequalities Associated with Magnetic SchrÃ¶dinger Operators Let $A:=-(\nabla-i\vec{a})\cdot(\nabla-i\vec{a})+V$ be a magnetic SchrÃ¶dinger operator on $\mathbb{R}^n$, where $\vec{a}:=(a_1,\dots, a_n)\in L^2_{\mathrm{loc}}(\mathbb{R}^n,\mathbb{R}^n)$ and $0\le V\in L^1_{\mathrm{loc}}(\mathbb{R}^n)$ satisfy some reverse HÃ¶lder conditions. Let $\varphi\colon \mathbb{R}^n\times[0,\infty)\to[0,\infty)$ be such that $\varphi(x,\cdot)$ for any given $x\in\mathbb{R}^n$ is an Orlicz function, $\varphi(\cdot,t)\in {\mathbb A}_{\infty}(\mathbb{R}^n)$ for all $t\in (0,\infty)$ (the class of uniformly Muckenhoupt weights) and its uniformly critical upper type index $I(\varphi)\in(0,1]$. In this article, the authors prove that second-order Riesz transforms $VA^{-1}$ and $(\nabla-i\vec{a})^2A^{-1}$ are bounded from the Musielak-Orlicz-Hardy space $H_{\varphi,\,A}(\mathbb{R}^n)$, associated with $A$, to the Musielak-Orlicz space $L^{\varphi}(\mathbb{R}^n)$. Moreover, the authors establish the boundedness of $VA^{-1}$ on $H_{\varphi, A}(\mathbb{R}^n)$. As applications, some maximal inequalities associated with $A$ in the scale of $H_{\varphi, A}(\mathbb{R}^n)$ are obtained. Keywords:Musielak-Orlicz-Hardy space, magnetic SchrÃ¶dinger operator, atom, second-order Riesz transform, maximal inequalityCategories:42B30, 42B35, 42B25, 35J10, 42B37, 46E30

23. CMB 2014 (vol 58 pp. 19)

Chen, Jiecheng; Hu, Guoen
 Compact Commutators of Rough Singular Integral Operators Let $b\in \mathrm{BMO}(\mathbb{R}^n)$ and $T_{\Omega}$ be the singular integral operator with kernel $\frac{\Omega(x)}{|x|^n}$, where $\Omega$ is homogeneous of degree zero, integrable and has mean value zero on the unit sphere $S^{n-1}$. In this paper, by Fourier transform estimates and approximation to the operator $T_{\Omega}$ by integral operators with smooth kernels, it is proved that if $b\in \mathrm{CMO}(\mathbb{R}^n)$ and $\Omega$ satisfies a certain minimal size condition, then the commutator generated by $b$ and $T_{\Omega}$ is a compact operator on $L^p(\mathbb{R}^n)$ for appropriate index $p$. The associated maximal operator is also considered. Keywords:commutator,singular integral operator, compact operator, maximal operatorCategory:42B20

24. CMB 2014 (vol 58 pp. 144)

Olevskii, Victor
 Localization and Completeness in $L_2({\mathbb R})$ We give a necessary and sufficient condition for a sequence to be a localization set for a determining average sampler. Keywords:localization, completeness, average samplingCategories:42C30, 94A20

25. CMB 2014 (vol 57 pp. 834)

Koh, Doowon
 Restriction Operators Acting on Radial Functions on Vector Spaces Over Finite Fields We study $L^p-L^r$ restriction estimates for algebraic varieties $V$ in the case when restriction operators act on radial functions in the finite field setting. We show that if the varieties $V$ lie in odd dimensional vector spaces over finite fields, then the conjectured restriction estimates are possible for all radial test functions. In addition, assuming that the varieties $V$ are defined in even dimensional spaces and have few intersection points with the sphere of zero radius, we also obtain the conjectured exponents for all radial test functions. Keywords:finite fields, radial functions, restriction operatorsCategories:42B05, 43A32, 43A15
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