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1. CMB Online first

Ding, Yong; Lai, Xudong
On a singular integral of Christ-Journé 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:Calderón commutator, rough kernel, weak type (1,1)

2. CMB Online first

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 factorization
Categories:42B35, 42B20

3. 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 transform
Categories:42B30, 42B25, 42B35

4. 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 operator
Categories:42B25, 42B35

5. CMB Online first

Sickel, Winfried; Yang, Dachun; Yuan, Wen; Zhuo, Ciqiang
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, Calderón reproducing formula
Categories:42B25, 46E35, 42B35

6. 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 space
Categories:42B20, 42B25

7. CMB Online first

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 variation
Categories:42B25, 46E35

8. 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 Calderón'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, Calderón's reproducing formula, almost orthogonality estimate, Zygmund dilation, embedding theorem
Categories:42B20, 42B35

9. 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 martingales
Categories:42C40, 42C15, 43A70, 11S85

10. 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, isometries
Categories:42A45, 42A50, 41A44

11. 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 set
Categories:43A46, 42A15, 42A55

12. 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 theory
Categories:42C10, 42B10

13. 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, $g_\lambda^*$-function
Categories:46E35, 42B25, 42B20, 42B35

14. 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 relation
Categories:33C45, 42C05

15. CMB 2015 (vol 59 pp. 211)

Totik, Vilmos
Universality Under Szegő'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 theory
Categories:42C05, 60B20, 30C85, 31A15

16. 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, Calderón reproducing formula, Besov and Triebel-Lizorkin spaces, embedding
Categories:42B25, 46F05, 46E35

17. 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, VMO

18. 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, quasicontinuity
Categories:42B25, 46E35

19. CMB 2014 (vol 58 pp. 432)

Yang, Dachun; Yang, Sibei
Second-order Riesz Transforms and Maximal Inequalities Associated with Magnetic Schrödinger Operators
Let $A:=-(\nabla-i\vec{a})\cdot(\nabla-i\vec{a})+V$ be a magnetic Schrödinger 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 Hölder 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 Schrödinger operator, atom, second-order Riesz transform, maximal inequality
Categories:42B30, 42B35, 42B25, 35J10, 42B37, 46E30

20. 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 operator

21. 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 sampling
Categories:42C30, 94A20

22. 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 operators
Categories:42B05, 43A32, 43A15

23. CMB 2013 (vol 57 pp. 463)

Bownik, Marcin; Jasper, John
Constructive Proof of Carpenter's Theorem
We give a constructive proof of Carpenter's Theorem due to Kadison. Unlike the original proof our approach also yields the real case of this theorem.

Keywords:diagonals of projections, the Schur-Horn theorem, the Pythagorean theorem, the Carpenter theorem, spectral theory
Categories:42C15, 47B15, 46C05

24. CMB 2013 (vol 57 pp. 254)

Christensen, Ole; Kim, Hong Oh; Kim, Rae Young
On Parseval Wavelet Frames with Two or Three Generators via the Unitary Extension Principle
The unitary extension principle (UEP) by Ron and Shen yields a sufficient condition for the construction of Parseval wavelet frames with multiple generators. In this paper we characterize the UEP-type wavelet systems that can be extended to a Parseval wavelet frame by adding just one UEP-type wavelet system. We derive a condition that is necessary for the extension of a UEP-type wavelet system to any Parseval wavelet frame with any number of generators, and prove that this condition is also sufficient to ensure that an extension with just two generators is possible.

Keywords:Bessel sequences, frames, extension of wavelet Bessel system to tight frame, wavelet systems, unitary extension principle
Categories:42C15, 42C40

25. CMB 2013 (vol 56 pp. 729)

Currey, B.; Mayeli, A.
The Orthonormal Dilation Property for Abstract Parseval Wavelet Frames
In this work we introduce a class of discrete groups containing subgroups of abstract translations and dilations, respectively. A variety of wavelet systems can appear as $\pi(\Gamma)\psi$, where $\pi$ is a unitary representation of a wavelet group and $\Gamma$ is the abstract pseudo-lattice $\Gamma$. We prove a condition in order that a Parseval frame $\pi(\Gamma)\psi$ can be dilated to an orthonormal basis of the form $\tau(\Gamma)\Psi$ where $\tau$ is a super-representation of $\pi$. For a subclass of groups that includes the case where the translation subgroup is Heisenberg, we show that this condition always holds, and we cite familiar examples as applications.

Keywords:frame, dilation, wavelet, Baumslag-Solitar group, shearlet
Categories:43A65, 42C40, 42C15
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