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26. 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

27. CMB 2011 (vol 56 pp. 194)

Stefánsson, Úlfar F.
 On the Smallest and Largest Zeros of MÃ¼ntz-Legendre Polynomials MÃ¼ntz-Legendre polynomials $L_n(\Lambda;x)$ associated with a sequence $\Lambda=\{\lambda_k\}$ are obtained by orthogonalizing the system $(x^{\lambda_0}, x^{\lambda_1}, x^{\lambda_2}, \dots)$ in $L_2[0,1]$ with respect to the Legendre weight. If the $\lambda_k$'s are distinct, it is well known that $L_n(\Lambda;x)$ has exactly $n$ zeros $l_{n,n}\lt l_{n-1,n}\lt \cdots \lt l_{2,n}\lt l_{1,n}$ on $(0,1)$. First we prove the following global bound for the smallest zero, $$\exp\biggl(-4\sum_{j=0}^n \frac{1}{2\lambda_j+1}\biggr) \lt l_{n,n}.$$ An important consequence is that if the associated MÃ¼ntz space is non-dense in $L_2[0,1]$, then $$\inf_{n}x_{n,n}\geq \exp\biggl({-4\sum_{j=0}^{\infty} \frac{1}{2\lambda_j+1}}\biggr)\gt 0,$$ so the elements $L_n(\Lambda;x)$ have no zeros close to 0. Furthermore, we determine the asymptotic behavior of the largest zeros; for $k$ fixed, $$\lim_{n\rightarrow\infty} \vert \log l_{k,n}\vert \sum_{j=0}^n (2\lambda_j+1)= \Bigl(\frac{j_k}{2}\Bigr)^2,$$ where $j_k$ denotes the $k$-th zero of the Bessel function $J_0$. Keywords:MÃ¼ntz polynomials, MÃ¼ntz-Legendre polynomialsCategories:42C05, 42C99, 41A60, 30B50

28. CMB 2011 (vol 55 pp. 555)

Michalowski, Nicholas; Rule, David J.; Staubach, Wolfgang
 Weighted $L^p$ Boundedness of Pseudodifferential Operators and Applications In this paper we prove weighted norm inequalities with weights in the $A_p$ classes, for pseudodifferential operators with symbols in the class ${S^{n(\rho -1)}_{\rho, \delta}}$ that fall outside the scope of CalderÃ³n-Zygmund theory. This is accomplished by controlling the sharp function of the pseudodifferential operator by Hardy-Littlewood type maximal functions. Our weighted norm inequalities also yield $L^{p}$ boundedness of commutators of functions of bounded mean oscillation with a wide class of operators in $\mathrm{OP}S^{m}_{\rho, \delta}$. Keywords:weighted norm inequality, pseudodifferential operator, commutator estimatesCategories:42B20, 42B25, 35S05, 47G30

29. CMB 2011 (vol 55 pp. 708)

Demeter, Ciprian
 Improved Range in the Return Times Theorem We prove that the Return Times Theorem holds true for pairs of $L^p-L^q$ functions, whenever $\frac{1}{p}+\frac{1}{q}<\frac{3}{2}$. Keywords:Return Times Theorem, maximal multiplier, maximal inequalityCategories:42B25, 37A45

30. CMB 2011 (vol 55 pp. 424)

Yang, Jianbin; Li, Song
 Convergence Rates of Cascade Algorithms with Infinitely Supported Masks We investigate the solutions of refinement equations of the form $$\phi(x)=\sum_{\alpha\in\mathbb Z^s}a(\alpha)\:\phi(Mx-\alpha),$$ where the function $\phi$ is in $L_p(\mathbb R^s)$$(1\le p\le\infty), a is an infinitely supported sequence on \mathbb Z^s called a refinement mask, and M is an s\times s integer matrix such that \lim_{n\to\infty}M^{-n}=0. Associated with the mask a and M is a linear operator Q_{a,M} defined on L_p(\mathbb R^s) by Q_{a,M} \phi_0:=\sum_{\alpha\in\mathbb Z^s}a(\alpha)\phi_0(M\cdot-\alpha). Main results of this paper are related to the convergence rates of (Q_{a,M}^n \phi_0)_{n=1,2,\dots} in L_p(\mathbb R^s) with mask a being infinitely supported. It is proved that under some appropriate conditions on the initial function \phi_0, Q_{a,M}^n \phi_0 converges in L_p(\mathbb R^s) with an exponential rate. Keywords:refinement equations, infinitely supported mask, cascade algorithms, rates of convergenceCategories:39B12, 41A25, 42C40 31. CMB 2011 (vol 55 pp. 303) Han, Yongsheng; Lee, Ming-Yi; Lin, Chin-Cheng  Atomic Decomposition and Boundedness of Operators on Weighted Hardy Spaces In this article, we establish a new atomic decomposition for f\in L^2_w\cap H^p_w, where the decomposition converges in L^2_w-norm rather than in the distribution sense. As applications of this decomposition, assuming that T is a linear operator bounded on L^2_w and 0 Keywords:A_p weights, atomic decomposition, CalderÃ³n reproducing formula, weighted Hardy spacesCategories:42B25, 42B30 32. CMB 2011 (vol 55 pp. 689) Berndt, Ryan  A Pointwise Estimate for the Fourier Transform and Maxima of a Function We show a pointwise estimate for the Fourier transform on the line involving the number of times the function changes monotonicity. The contrapositive of the theorem may be used to find a lower bound to the number of local maxima of a function. We also show two applications of the theorem. The first is the two weight problem for the Fourier transform, and the second is estimating the number of roots of the derivative of a function. Keywords:Fourier transform, maxima, two weight problem, roots, norm estimates, Dirichlet-Jordan theoremCategories:42A38, 65T99 33. CMB 2010 (vol 54 pp. 113) Hytönen, Tuomas P.  On the Norm of the Beurling-Ahlfors Operator in Several Dimensions The generalized Beurling-Ahlfors operator S on L^p(\mathbb{R}^n;\Lambda), where \Lambda:=\Lambda(\mathbb{R}^n) is the exterior algebra with its natural Hilbert space norm, satisfies the estimate$$\|S\|_{\mathcal{L}(L^p(\mathbb{R}^n;\Lambda))}\leq(n/2+1)(p^*-1),\quad p^*:=\max\{p,p'\}$$This improves on earlier results in all dimensions n\geq 3. The proof is based on the heat extension and relies at the bottom on Burkholder's sharp inequality for martingale transforms. Categories:42B20, 60G46 34. CMB 2010 (vol 54 pp. 159) Sababheh, Mohammad  Hardy Inequalities on the Real Line We prove that some inequalities, which are considered to be generalizations of Hardy's inequality on the circle, can be modified and proved to be true for functions integrable on the real line. In fact we would like to show that some constructions that were used to prove the Littlewood conjecture can be used similarly to produce real Hardy-type inequalities. This discussion will lead to many questions concerning the relationship between Hardy-type inequalities on the circle and those on the real line. Keywords:Hardy's inequality, inequalities including the Fourier transform and Hardy spacesCategories:42A05, 42A99 35. CMB 2010 (vol 54 pp. 100) Fan, Dashan; Wu, Huoxiong  On the Generalized Marcinkiewicz Integral Operators with Rough Kernels A class of generalized Marcinkiewicz integral operators is introduced, and, under rather weak conditions on the integral kernels, the boundedness of such operators on L^p and Triebel--Lizorkin spaces is established. Keywords: Marcinkiewicz integral, Littlewood--Paley theory, Triebel--Lizorkin space, rough kernel, product domainCategories:42B20, , , , , 42B25, 42B30, 42B99 36. CMB 2010 (vol 54 pp. 172) Shayya, Bassam  Measures with Fourier Transforms in L^2 of a Half-space We prove that if the Fourier transform of a compactly supported measure is in L^2 of a half-space, then the measure is absolutely continuous to Lebesgue measure. We then show how this result can be used to translate information about the dimensionality of a measure and the decay of its Fourier transform into geometric information about its support. Categories:42B10, 28A75 37. CMB 2010 (vol 53 pp. 491) Jizheng, Huang; Liu, Heping  The Weak Type (1,1) Estimates of Maximal Functions on the Laguerre Hypergroup In this paper, we discuss various maximal functions on the Laguerre hypergroup \mathbf{K} including the heat maximal function, the Poisson maximal function, and the Hardy--Littlewood maximal function which is consistent with the structure of hypergroup of \mathbf{K}. We shall establish the weak type (1,1) estimates for these maximal functions. The L^p estimates for p>1 follow from the interpolation. Some applications are included. Keywords:Laguerre hypergroup, maximal function, heat kernel, Poisson kernelCategories:42B25, 43A62 38. CMB 2009 (vol 53 pp. 133) Moritoh, Shinya; Tomoeda, Kyoko  A Further Decay Estimate for the DziubaÅski-HernÃ¡ndez Wavelets We give a further decay estimate for the DziubaÅski-HernÃ¡ndez wavelets that are band-limited and have subexponential decay. This is done by constructing an appropriate bell function and using the Paley-Wiener theorem for ultradifferentiable functions. Keywords:wavelets, ultradifferentiable functionsCategories:42C40, 46E10 39. CMB 2009 (vol 53 pp. 263) Feuto, Justin; Fofana, Ibrahim; Koua, Konin  Weighted Norm Inequalities for a Maximal Operator in Some Subspace of Amalgams We give weighted norm inequalities for the maximal fractional operator \mathcal M_{q,\beta } of HardyÂLittlewood and the fractional integral I_{\gamma}. These inequalities are established between (L^{q},L^{p}) ^{\alpha }(X,d,\mu ) spaces (which are superspaces of Lebesgue spaces L^{\alpha}(X,d,\mu) and subspaces of amalgams (L^{q},L^{p})(X,d,\mu)) and in the setting of space of homogeneous type (X,d,\mu). The conditions on the weights are stated in terms of Orlicz norm. Keywords:fractional maximal operator, fractional integral, space of homogeneous typeCategories:42B35, 42B20, 42B25 40. CMB 2009 (vol 52 pp. 627) Yu, Dan Sheng; Zhou, Ping; Zhou, Song Ping  On L^{1}-Convergence of Fourier Series under the MVBV Condition Let f\in L_{2\pi } be a real-valued even function with its Fourier series % \frac{a_{0}}{2}+\sum_{n=1}^{\infty }a_{n}\cos nx, and let S_{n}(f,x) ,\;n\geq 1, be the n-th partial sum of the Fourier series. It is well known that if the nonnegative sequence \{a_{n}\} is decreasing and \lim_{n\rightarrow \infty }a_{n}=0, then% \begin{equation*} \lim_{n\rightarrow \infty }\Vert f-S_{n}(f)\Vert _{L}=0 \text{ if and only if }\lim_{n\rightarrow \infty }a_{n}\log n=0. \end{equation*}% We weaken the monotone condition in this classical result to the so-called mean value bounded variation (MVBV) condition. The generalization of the above classical result in real-valued function space is presented as a special case of the main result in this paper, which gives the L^{1}% -convergence of a function f\in L_{2\pi } in complex space. We also give results on L^{1}-approximation of a function f\in L_{2\pi } under the MVBV condition. Keywords:complex trigonometric series, L^{1} convergence, monotonicity, mean value bounded variationCategories:42A25, 41A50 41. CMB 2009 (vol 52 pp. 521) Chen, Yanping; Ding, Yong  The Parabolic Littlewood--Paley Operator with Hardy Space Kernels In this paper, we give the L^p boundedness for a class of parabolic Littlewood--Paley g-function with its kernel function \Omega is in the Hardy space H^1(S^{n-1}). Keywords:parabolic Littlewood-Paley operator, Hardy space, rough kernelCategories:42B20, 42B25 42. CMB 2009 (vol 52 pp. 95) Miranian, L.  Matrix Valued Orthogonal Polynomials on the Unit Circle: Some Extensions of the Classical Theory In the work presented below the classical subject of orthogonal polynomials on the unit circle is discussed in the matrix setting. An explicit matrix representation of the matrix valued orthogonal polynomials in terms of the moments of the measure is presented. Classical recurrence relations are revisited using the matrix representation of the polynomials. The matrix expressions for the kernel polynomials and the Christoffel--Darboux formulas are presented for the first time. Keywords:Matrix valued orthogonal polynomials, unit circle, Schur complements, recurrence relations, kernel polynomials, Christoffel-DarbouxCategory:42C99 43. CMB 2008 (vol 51 pp. 487) Betancor, Jorge J.; Mart\'{\i}nez, Teresa; Rodr\'{\i}guez-Mesa, Lourdes  Laplace Transform Type Multipliers for Hankel Transforms In this paper we establish that Hankel multipliers of Laplace transform type are bounded from L^p(w) into itself when 1 Keywords:Hankel transform, Laplace transform, multiplier, CalderÃ³n--ZygmundCategory:42 44. CMB 2008 (vol 51 pp. 348) Casazza, Peter G.; Christensen, Ole  The Reconstruction Property in Banach Spaces and a Perturbation Theorem Perturbation theory is a fundamental tool in Banach space theory. However, the applications of the classical results are limited by the fact that they force the perturbed sequence to be equivalent to the given sequence. We will develop a more general perturbation theory that does not force equivalence of the sequences. Category:42C15 45. CMB 2007 (vol 50 pp. 85) Han, Deguang  Classification of Finite Group-Frames and Super-Frames Given a finite group G, we examine the classification of all frame representations of G and the classification of all G-frames, \emph{i.e.,} frames induced by group representations of G. We show that the exact number of equivalence classes of G-frames and the exact number of frame representations can be explicitly calculated. We also discuss how to calculate the largest number L such that there exists an L-tuple of strongly disjoint G-frames. Keywords:frames, group-frames, frame representations, disjoint framesCategories:42C15, 46C05, 47B10 46. CMB 2006 (vol 49 pp. 414) Jiang, Liya; Jia, Houyu; Xu, Han  Commutators Estimates on Triebel--Lizorkin Spaces In this paper, we consider the behavior of the commutators of convolution operators on the Triebel--Lizorkin spaces \dot{F}^{s, q} _p. Keywords:commutators, Triebel--Lizorkin spaces, paraproductCategories:42B, 46F 47. CMB 2006 (vol 49 pp. 438) Mercer, Idris David  Unimodular Roots of\\ Special Littlewood Polynomials We call \alpha(z) = a_0 + a_1 z + \dots + a_{n-1} z^{n-1} a Littlewood polynomial if a_j = \pm 1 for all j. We call \alpha(z) self-reciprocal if \alpha(z) = z^{n-1}\alpha(1/z), and call \alpha(z) skewsymmetric if n = 2m+1 and a_{m+j} = (-1)^j a_{m-j} for all j. It has been observed that Littlewood polynomials with particularly high minimum modulus on the unit circle in \bC tend to be skewsymmetric. In this paper, we prove that a skewsymmetric Littlewood polynomial cannot have any zeros on the unit circle, as well as providing a new proof of the known result that a self-reciprocal Littlewood polynomial must have a zero on the unit circle. Categories:26C10, 30C15, 42A05 48. CMB 2006 (vol 49 pp. 3) Al-Salman, Ahmad  On a Class of Singular Integral Operators With Rough Kernels In this paper, we study the L^p mapping properties of a class of singular integral operators with rough kernels belonging to certain block spaces. We prove that our operators are bounded on L^p provided that their kernels satisfy a size condition much weaker than that for the classical Calder\'{o}n--Zygmund singular integral operators. Moreover, we present an example showing that our size condition is optimal. As a consequence of our results, we substantially improve a previously known result on certain maximal functions. Keywords:Singular integrals, Rough kernels, Square functions,, Maximal functions, Block spacesCategories:42B20, 42B15, 42B25 49. CMB 2005 (vol 48 pp. 382) De Carli, Laura  Uniform Estimates of Ultraspherical Polynomials of Large Order In this paper we prove the sharp inequality$$ |P_n^{(s)}(x)|\leq P_n^{(s)}(1)\bigl(|x|^n +\frac{n-1}{2 s+1}(1-|x|^n)\bigr),$$where$P_n^{(s)}(x)$is the classical ultraspherical polynomial of degree$n$and order$s\ge n\frac{1+\sqrt 5}{4}$. This inequality can be refined in$[0,z_n^s]$and$[z_n^s,1]$, where$z_n^s$denotes the largest zero of$P_n^{(s)}(x)$. Categories:42C05, 33C47 50. CMB 2005 (vol 48 pp. 370) Daly, J. E.; Fridli, S.  Trigonometric Multipliers on$H_{2\pi}$In this paper we consider multipliers on the real Hardy space$H_{2\pi}$. It is known that the Marcinkiewicz and the H\"ormander--Mihlin conditions are sufficient for the corresponding trigonometric multiplier to be bounded on$L_{2\pi}^p$,$1 Keywords:Multipliers, Hardy spaceCategories:42A45, 42A50, 42A85
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