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1. CMB Online first

Gurbuz, 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

2. 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 variationCategories:42B25, 46E35

3. CMB Online first

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

4. CMB Online first

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

5. CMB Online first

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

6. CMB Online first

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

7. 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

8. 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

9. 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

10. 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

11. 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

12. 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

13. 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

14. 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

15. 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

16. 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

17. 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

18. 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 theoryCategories:42C15, 47B15, 46C05

19. 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 principleCategories:42C15, 42C40

20. 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, shearletCategories:43A65, 42C40, 42C15

21. CMB 2013 (vol 56 pp. 745)

Fu, Xiaoye; Gabardo, Jean-Pierre
 Dimension Functions of Self-Affine Scaling Sets In this paper, the dimension function of a self-affine generalized scaling set associated with an $n\times n$ integral expansive dilation $A$ is studied. More specifically, we consider the dimension function of an $A$-dilation generalized scaling set $K$ assuming that $K$ is a self-affine tile satisfying $BK = (K+d_1) \cup (K+d_2)$, where $B=A^t$, $A$ is an $n\times n$ integral expansive matrix with $\lvert \det A\rvert=2$, and $d_1,d_2\in\mathbb{R}^n$. We show that the dimension function of $K$ must be constant if either $n=1$ or $2$ or one of the digits is $0$, and that it is bounded by $2\lvert K\rvert$ for any $n$. Keywords:scaling set, self-affine tile, orthonormal multiwavelet, dimension functionCategory:42C40

22. CMB 2012 (vol 56 pp. 801)

Oberlin, Richard
 Estimates for Compositions of Maximal Operators with Singular Integrals We prove weak-type $(1,1)$ estimates for compositions of maximal operators with singular integrals. Our main object of interest is the operator $\Delta^*\Psi$ where $\Delta^*$ is Bourgain's maximal multiplier operator and $\Psi$ is the sum of several modulated singular integrals; here our method yields a significantly improved bound for the $L^q$ operator norm when $1 \lt q \lt 2.$ We also consider associated variation-norm estimates. Keywords:maximal operator calderon-zygmundCategory:42A45

23. CMB 2011 (vol 56 pp. 326)

Erdoğan, M. Burak; Oberlin, Daniel M.
 Restricting Fourier Transforms of Measures to Curves in $\mathbb R^2$ We establish estimates for restrictions to certain curves in $\mathbb R^2$ of the Fourier transforms of some fractal measures. Keywords:Fourier transforms of fractal measures, Fourier restrictionCategories:42B10, 28A12

24. CMB 2011 (vol 56 pp. 3)

Aïssiou, Tayeb
 Semiclassical Limits of Eigenfunctions on Flat $n$-Dimensional Tori We provide a proof of a conjecture by Jakobson, Nadirashvili, and Toth stating that on an $n$-dimensional flat torus $\mathbb T^{n}$, and the Fourier transform of squares of the eigenfunctions $|\varphi_\lambda|^2$ of the Laplacian have uniform $l^n$ bounds that do not depend on the eigenvalue $\lambda$. The proof is a generalization of an argument by Jakobson, et al. for the lower dimensional cases. These results imply uniform bounds for semiclassical limits on $\mathbb T^{n+2}$. We also prove a geometric lemma that bounds the number of codimension-one simplices satisfying a certain restriction on an $n$-dimensional sphere $S^n(\lambda)$ of radius $\sqrt{\lambda}$, and we use it in the proof. Keywords:semiclassical limits, eigenfunctions of Laplacian on a torus, quantum limitsCategories:58G25, 81Q50, 35P20, 42B05

25. CMB 2011 (vol 55 pp. 646)

Zhou, Jiang; Ma, Bolin
 Marcinkiewicz Commutators with Lipschitz Functions in Non-homogeneous Spaces Under the assumption that $\mu$ is a nondoubling measure, we study certain commutators generated by the Lipschitz function and the Marcinkiewicz integral whose kernel satisfies a HÃ¶rmander-type condition. We establish the boundedness of these commutators on the Lebesgue spaces, Lipschitz spaces, and Hardy spaces. Our results are extensions of known theorems in the doubling case. Keywords:non doubling measure, Marcinkiewicz integral, commutator, ${\rm Lip}_{\beta}(\mu)$, $H^1(\mu)$Categories:42B25, 47B47, 42B20, 47A30
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