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1. CJM Online first

Cao, Jun; Jiang, Renjin; Yang, Dachun; Zhang, Junqiang
Non-tangential Maximal Function Characterizations of Hardy Spaces Associated with Degenerate Elliptic Operators
Let $w$ be either in the Muckenhoupt class of $A_2(\mathbb{R}^n)$ weights or in the class of $QC(\mathbb{R}^n)$ weights, and $L_w:=-w^{-1}\mathop{\mathrm{div}}(A\nabla)$ the degenerate elliptic operator on the Euclidean space $\mathbb{R}^n$, $n\ge 2$. In this article, the authors establish the non-tangential maximal function characterization of the Hardy space $H_{L_w}^p(\mathbb{R}^n)$ associated with $L_w$ for $p\in (0,1]$ and, when $p\in (\frac{n}{n+1},1]$ and $w\in A_{q_0}(\mathbb{R}^n)$ with $q_0\in[1,\frac{p(n+1)}n)$, the authors prove that the associated Riesz transform $\nabla L_w^{-1/2}$ is bounded from $H_{L_w}^p(\mathbb{R}^n)$ to the weighted classical Hardy space $H_w^p(\mathbb{R}^n)$.

Keywords:degenerate elliptic operator, Hardy space, square function, maximal function, molecule, Riesz transform
Categories:42B30, 42B35, 35J70

2. CJM 2014 (vol 66 pp. 1358)

Osėkowski, Adam
Sharp Localized Inequalities for Fourier Multipliers
In the paper we study sharp localized $L^q\colon L^p$ estimates for Fourier multipliers resulting from modulation of the jumps of Lévy processes. The proofs of these estimates rest on probabilistic methods and exploit related sharp bounds for differentially subordinated martingales, which are of independent interest. The lower bounds for the constants involve the analysis of laminates, a family of certain special probability measures on $2\times 2$ matrices. As an application, we obtain new sharp bounds for the real and imaginary parts of the Beurling-Ahlfors operator .

Keywords:Fourier multiplier, martingale, laminate
Categories:42B15, 60G44, 42B20

3. CJM 2013 (vol 66 pp. 284)

Eikrem, Kjersti Solberg
Random Harmonic Functions in Growth Spaces and Bloch-type Spaces
Let $h^\infty_v(\mathbf D)$ and $h^\infty_v(\mathbf B)$ be the spaces of harmonic functions in the unit disk and multi-dimensional unit ball which admit a two-sided radial majorant $v(r)$. We consider functions $v $ that fulfill a doubling condition. In the two-dimensional case let $u (re^{i\theta},\xi) = \sum_{j=0}^\infty (a_{j0} \xi_{j0} r^j \cos j\theta +a_{j1} \xi_{j1} r^j \sin j\theta)$ where $\xi =\{\xi_{ji}\}$ is a sequence of random subnormal variables and $a_{ji}$ are real; in higher dimensions we consider series of spherical harmonics. We will obtain conditions on the coefficients $a_{ji} $ which imply that $u$ is in $h^\infty_v(\mathbf B)$ almost surely. Our estimate improves previous results by Bennett, Stegenga and Timoney, and we prove that the estimate is sharp. The results for growth spaces can easily be applied to Bloch-type spaces, and we obtain a similar characterization for these spaces, which generalizes results by Anderson, Clunie and Pommerenke and by Guo and Liu.

Keywords:harmonic functions, random series, growth space, Bloch-type space
Categories:30B20, 31B05, 30H30, 42B05

4. CJM 2013 (vol 66 pp. 1382)

Wu, Xinfeng
Weighted Carleson Measure Spaces Associated with Different Homogeneities
In this paper, we introduce weighted Carleson measure spaces associated with different homogeneities and prove that these spaces are the dual spaces of weighted Hardy spaces studied in a forthcoming paper. As an application, we establish the boundedness of composition of two Calderón-Zygmund operators with different homogeneities on the weighted Carleson measure spaces; this, in particular, provides the weighted endpoint estimates for the operators studied by Phong-Stein.

Keywords:composition of operators, weighted Carleson measure spaces, duality
Categories:42B20, 42B35

5. CJM 2013 (vol 65 pp. 1217)

Cruz, Victor; Mateu, Joan; Orobitg, Joan
Beltrami Equation with Coefficient in Sobolev and Besov Spaces
Our goal in this work is to present some function spaces on the complex plane $\mathbb C$, $X(\mathbb C)$, for which the quasiregular solutions of the Beltrami equation, $\overline\partial f (z) = \mu(z) \partial f (z)$, have first derivatives locally in $X(\mathbb C)$, provided that the Beltrami coefficient $\mu$ belongs to $X(\mathbb C)$.

Keywords:quasiregular mappings, Beltrami equation, Sobolev spaces, Calderón-Zygmund operators
Categories:30C62, 35J99, 42B20

6. CJM 2012 (vol 66 pp. 102)

Birth, Lidia; Glöckner, Helge
Continuity of convolution of test functions on Lie groups
For a Lie group $G$, we show that the map $C^\infty_c(G)\times C^\infty_c(G)\to C^\infty_c(G)$, $(\gamma,\eta)\mapsto \gamma*\eta$ taking a pair of test functions to their convolution is continuous if and only if $G$ is $\sigma$-compact. More generally, consider $r,s,t \in \mathbb{N}_0\cup\{\infty\}$ with $t\leq r+s$, locally convex spaces $E_1$, $E_2$ and a continuous bilinear map $b\colon E_1\times E_2\to F$ to a complete locally convex space $F$. Let $\beta\colon C^r_c(G,E_1)\times C^s_c(G,E_2)\to C^t_c(G,F)$, $(\gamma,\eta)\mapsto \gamma *_b\eta$ be the associated convolution map. The main result is a characterization of those $(G,r,s,t,b)$ for which $\beta$ is continuous. Convolution of compactly supported continuous functions on a locally compact group is also discussed, as well as convolution of compactly supported $L^1$-functions and convolution of compactly supported Radon measures.

Keywords:Lie group, locally compact group, smooth function, compact support, test function, second countability, countable basis, sigma-compactness, convolution, continuity, seminorm, product estimates
Categories:22E30, 46F05, 22D15, 42A85, 43A10, 43A15, 46A03, 46A13, 46E25

7. CJM 2012 (vol 65 pp. 510)

Blasco de la Cruz, Oscar; Villarroya Alvarez, Paco
Transference of vector-valued multipliers on weighted $L^p$-spaces
We prove restriction and extension of multipliers between weighted Lebesgue spaces with two different weights, which belong to a class more general than periodic weights, and two different exponents of integrability which can be below one. We also develop some ad-hoc methods which apply to weights defined by the product of periodic weights with functions of power type. Our vector-valued approach allow us to extend results to transference of maximal multipliers and provide transference of Littlewood-Paley inequalities.

Keywords:Fourier multipliers, restriction theorems, weighted spaces
Categories:42B15, 42B35

8. CJM 2012 (vol 65 pp. 299)

Grafakos, Loukas; Miyachi, Akihiko; Tomita, Naohito
On Multilinear Fourier Multipliers of Limited Smoothness
In this paper, we prove certain $L^2$-estimate for multilinear Fourier multiplier operators with multipliers of limited smoothness. As a result, we extend the result of Calderón and Torchinsky in the linear theory to the multilinear case. The sharpness of our results and some related estimates in Hardy spaces are also discussed.

Keywords:multilinear Fourier multipliers, Hörmander multiplier theorem, Hardy spaces
Categories:42B15, 42B20

9. CJM 2012 (vol 65 pp. 600)

Kroó, A.; Lubinsky, D. S.
Christoffel Functions and Universality in the Bulk for Multivariate Orthogonal Polynomials
We establish asymptotics for Christoffel functions associated with multivariate orthogonal polynomials. The underlying measures are assumed to be regular on a suitable domain - in particular this is true if they are positive a.e. on a compact set that admits analytic parametrization. As a consequence, we obtain asymptotics for Christoffel functions for measures on the ball and simplex, under far more general conditions than previously known. As another consequence, we establish universality type limits in the bulk in a variety of settings.

Keywords:orthogonal polynomials, random matrices, unitary ensembles, correlation functions, Christoffel functions
Categories:42C05, 42C99, 42B05, 60B20

10. CJM 2011 (vol 64 pp. 1036)

Koh, Doowon; Shen, Chun-Yen
Harmonic Analysis Related to Homogeneous Varieties in Three Dimensional Vector Spaces over Finite Fields
In this paper we study the extension problem, the averaging problem, and the generalized Erdős-Falconer distance problem associated with arbitrary homogeneous varieties in three dimensional vector spaces over finite fields. In the case when the varieties do not contain any plane passing through the origin, we obtain the best possible results on the aforementioned three problems. In particular, our result on the extension problem modestly generalizes the result by Mockenhaupt and Tao who studied the particular conical extension problem. In addition, investigating the Fourier decay on homogeneous varieties enables us to give complete mapping properties of averaging operators. Moreover, we improve the size condition on a set such that the cardinality of its distance set is nontrivial.

Keywords:extension problems, averaging operator, finite fields, Erdős-Falconer distance problems, homogeneous polynomial
Categories:42B05, 11T24, 52C17

11. CJM 2011 (vol 64 pp. 1201)

Aistleitner, Christoph; Elsholtz, Christian
The Central Limit Theorem for Subsequences in Probabilistic Number Theory
Let $(n_k)_{k \geq 1}$ be an increasing sequence of positive integers, and let $f(x)$ be a real function satisfying \begin{equation} \tag{1} f(x+1)=f(x), \qquad \int_0^1 f(x) ~dx=0,\qquad \operatorname{Var_{[0,1]}} f \lt \infty. \end{equation} If $\lim_{k \to \infty} \frac{n_{k+1}}{n_k} = \infty$ the distribution of \begin{equation} \tag{2} \frac{\sum_{k=1}^N f(n_k x)}{\sqrt{N}} \end{equation} converges to a Gaussian distribution. In the case $$ 1 \lt \liminf_{k \to \infty} \frac{n_{k+1}}{n_k}, \qquad \limsup_{k \to \infty} \frac{n_{k+1}}{n_k} \lt \infty $$ there is a complex interplay between the analytic properties of the function $f$, the number-theoretic properties of $(n_k)_{k \geq 1}$, and the limit distribution of (2). In this paper we prove that any sequence $(n_k)_{k \geq 1}$ satisfying $\limsup_{k \to \infty} \frac{n_{k+1}}{n_k} = 1$ contains a nontrivial subsequence $(m_k)_{k \geq 1}$ such that for any function satisfying (1) the distribution of $$ \frac{\sum_{k=1}^N f(m_k x)}{\sqrt{N}} $$ converges to a Gaussian distribution. This result is best possible: for any $\varepsilon\gt 0$ there exists a sequence $(n_k)_{k \geq 1}$ satisfying $\limsup_{k \to \infty} \frac{n_{k+1}}{n_k} \leq 1 + \varepsilon$ such that for every nontrivial subsequence $(m_k)_{k \geq 1}$ of $(n_k)_{k \geq 1}$ the distribution of (2) does not converge to a Gaussian distribution for some $f$. Our result can be viewed as a Ramsey type result: a sufficiently dense increasing integer sequence contains a subsequence having a certain requested number-theoretic property.

Keywords:central limit theorem, lacunary sequences, linear Diophantine equations, Ramsey type theorem
Categories:60F05, 42A55, 11D04, 05C55, 11K06

12. CJM 2011 (vol 64 pp. 892)

Hytönen, Tuomas; Liu, Suile; Yang, Dachun; Yang, Dongyong
Boundedness of Calderón-Zygmund Operators on Non-homogeneous Metric Measure Spaces
Let $({\mathcal X}, d, \mu)$ be a separable metric measure space satisfying the known upper doubling condition, the geometrical doubling condition, and the non-atomic condition that $\mu(\{x\})=0$ for all $x\in{\mathcal X}$. In this paper, we show that the boundedness of a Calderón-Zygmund operator $T$ on $L^2(\mu)$ is equivalent to that of $T$ on $L^p(\mu)$ for some $p\in (1, \infty)$, and that of $T$ from $L^1(\mu)$ to $L^{1,\,\infty}(\mu).$ As an application, we prove that if $T$ is a Calderón-Zygmund operator bounded on $L^2(\mu)$, then its maximal operator is bounded on $L^p(\mu)$ for all $p\in (1, \infty)$ and from the space of all complex-valued Borel measures on ${\mathcal X}$ to $L^{1,\,\infty}(\mu)$. All these results generalize the corresponding results of Nazarov et al. on metric spaces with measures satisfying the so-called polynomial growth condition.

Keywords:upper doubling, geometrical doubling, dominating function, weak type $(1,1)$ estimate, Calderón-Zygmund operator, maximal operator
Categories:42B20, 42B25, 30L99

13. CJM 2011 (vol 64 pp. 257)

Chen, Yanping; Ding, Yong; Wang, Xinxia
Compactness of Commutators for Singular Integrals on Morrey Spaces
In this paper we characterize the compactness of the commutator $[b,T]$ for the singular integral operator on the Morrey spaces $L^{p,\lambda}(\mathbb R^n)$. More precisely, we prove that if $b\in \operatorname{VMO}(\mathbb R^n)$, the $\operatorname {BMO} (\mathbb R^n)$-closure of $C_c^\infty(\mathbb R^n)$, then $[b,T]$ is a compact operator on the Morrey spaces $L^{p,\lambda}(\mathbb R^n)$ for $1\lt p\lt \infty$ and $0\lt \lambda\lt n$. Conversely, if $b\in \operatorname{BMO}(\mathbb R^n)$ and $[b,T]$ is a compact operator on the $L^{p,\,\lambda}(\mathbb R^n)$ for some $p\ (1\lt p\lt \infty)$, then $b\in \operatorname {VMO}(\mathbb R^n)$. Moreover, the boundedness of a rough singular integral operator $T$ and its commutator $[b,T]$ on $L^{p,\,\lambda}(\mathbb R^n)$ are also given. We obtain a sufficient condition for a subset in Morrey space to be a strongly pre-compact set, which has interest in its own right.

Keywords:singular integral, commutators, compactness, VMO, Morrey space
Categories:42B20, 42B99

14. CJM 2011 (vol 63 pp. 798)

Daws, Matthew
Representing Multipliers of the Fourier Algebra on Non-Commutative $L^p$ Spaces
We show that the multiplier algebra of the Fourier algebra on a locally compact group $G$ can be isometrically represented on a direct sum on non-commutative $L^p$ spaces associated with the right von Neumann algebra of $G$. The resulting image is the idealiser of the image of the Fourier algebra. If these spaces are given their canonical operator space structure, then we get a completely isometric representation of the completely bounded multiplier algebra. We make a careful study of the non-commutative $L^p$ spaces we construct and show that they are completely isometric to those considered recently by Forrest, Lee, and Samei. We improve a result of theirs about module homomorphisms. We suggest a definition of a Figa-Talamanca-Herz algebra built out of these non-commutative $L^p$ spaces, say $A_p(\widehat G)$. It is shown that $A_2(\widehat G)$ is isometric to $L^1(G)$, generalising the abelian situation.

Keywords:multiplier, Fourier algebra, non-commutative $L^p$ space, complex interpolation
Categories:43A22, 43A30, 46L51, 22D25, 42B15, 46L07, 46L52

15. CJM 2011 (vol 63 pp. 689)

Olphert, Sean; Power, Stephen C.
Higher Rank Wavelets
A theory of higher rank multiresolution analysis is given in the setting of abelian multiscalings. This theory enables the construction, from a higher rank MRA, of finite wavelet sets whose multidilations have translates forming an orthonormal basis in $L^2(\mathbb R^d)$. While tensor products of uniscaled MRAs provide simple examples we construct many nonseparable higher rank wavelets. In particular we construct \emph{Latin square wavelets} as rank~$2$ variants of Haar wavelets. Also we construct nonseparable scaling functions for rank $2$ variants of Meyer wavelet scaling functions, and we construct the associated nonseparable wavelets with compactly supported Fourier transforms. On the other hand we show that compactly supported scaling functions for biscaled MRAs are necessarily separable.

Keywords: wavelet, multi-scaling, higher rank, multiresolution, Latin squares
Categories:42C40, 42A65, 42A16, 43A65

16. CJM 2010 (vol 63 pp. 181)

Ismail, Mourad E. H.; Obermaier, Josef
Characterizations of Continuous and Discrete $q$-Ultraspherical Polynomials
We characterize the continuous $q$-ultraspherical polynomials in terms of the special form of the coefficients in the expansion $\mathcal{D}_q P_n(x)$ in the basis $\{P_n(x)\}$, $\mathcal{D}_q$ being the Askey--Wilson divided difference operator. The polynomials are assumed to be symmetric, and the connection coefficients are multiples of the reciprocal of the square of the $L^2$ norm of the polynomials. A similar characterization is given for the discrete $q$-ultraspherical polynomials. A new proof of the evaluation of the connection coefficients for big $q$-Jacobi polynomials is given.

Keywords:continuous $q$-ultraspherical polynomials, big $q$-Jacobi polynomials, discrete $q$-ultra\-spherical polynomials, Askey--Wilson operator, $q$-difference operator, recursion coefficients
Categories:33D45, 42C05

17. CJM 2010 (vol 62 pp. 1419)

Yang, Dachun; Yang, Dongyong
BMO-Estimates for Maximal Operators via Approximations of the Identity with Non-Doubling Measures
Let $\mu$ be a nonnegative Radon measure on $\mathbb{R}^d$ that satisfies the growth condition that there exist constants $C_0>0$ and $n\in(0,d]$ such that for all $x\in\mathbb{R}^d$ and $r>0$, ${\mu(B(x,\,r))\le C_0r^n}$, where $B(x,r)$ is the open ball centered at $x$ and having radius $r$. In this paper, the authors prove that if $f$ belongs to the $\textrm {BMO}$-type space $\textrm{RBMO}(\mu)$ of Tolsa, then the homogeneous maximal function $\dot{\mathcal{M}}_S(f)$ (when $\mathbb{R}^d$ is not an initial cube) and the inhomogeneous maximal function $\mathcal{M}_S(f)$ (when $\mathbb{R}^d$ is an initial cube) associated with a given approximation of the identity $S$ of Tolsa are either infinite everywhere or finite almost everywhere, and in the latter case, $\dot{\mathcal{M}}_S$ and $\mathcal{M}_S$ are bounded from $\textrm{RBMO}(\mu)$ to the $\textrm {BLO}$-type space $\textrm{RBLO}(\mu)$. The authors also prove that the inhomogeneous maximal operator $\mathcal{M}_S$ is bounded from the local $\textrm {BMO}$-type space $\textrm{rbmo}(\mu)$ to the local $\textrm {BLO}$-type space $\textrm{rblo}(\mu)$.

Keywords:Non-doubling measure, maximal operator, approximation of the identity, RBMO(mu), RBLO(mu), rbmo(mu), rblo(mu)
Categories:42B25, 42B30, 47A30, 43A99

18. CJM 2010 (vol 62 pp. 1182)

Yue, Hong
A Fractal Function Related to the John-Nirenberg Inequality for $Q_{\alpha}({\mathbb R^n})$
A borderline case function $f$ for $ Q_{\alpha}({\mathbb R^n})$ spaces is defined as a Haar wavelet decomposition, with the coefficients depending on a fixed parameter $\beta>0$. On its support $I_0=[0, 1]^n$, $f(x)$ can be expressed by the binary expansions of the coordinates of $x$. In particular, $f=f_{\beta}\in Q_{\alpha}({\mathbb R^n})$ if and only if $\alpha<\beta<\frac{n}{2}$, while for $\beta=\alpha$, it was shown by Yue and Dafni that $f$ satisfies a John--Nirenberg inequality for $ Q_{\alpha}({\mathbb R^n})$. When $\beta\neq 1$, $f$ is a self-affine function. It is continuous almost everywhere and discontinuous at all dyadic points inside $I_0$. In addition, it is not monotone along any coordinate direction in any small cube. When the parameter $\beta\in (0, 1)$, $f$ is onto from $I_0$ to $[-\frac{1}{1-2^{-\beta}}, \frac{1}{1-2^{-\beta}}]$, and the graph of $f$ has a non-integer fractal dimension $n+1-\beta$.

Keywords:Haar wavelets, Q spaces, John-Nirenberg inequality, Greedy expansion, self-affine, fractal, Box dimension
Categories:42B35, 42C10, 30D50, 28A80

19. CJM 2010 (vol 62 pp. 827)

Ouyang, Caiheng; Xu, Quanhua
BMO Functions and Carleson Measures with Values in Uniformly Convex Spaces
This paper studies the relationship between vector-valued BMO functions and the Carleson measures defined by their gradients. Let $dA$ and $dm$ denote Lebesgue measures on the unit disc $D$ and the unit circle $\mathbf{T}$, respectively. For $1< q<\infty$ and a Banach space $B$, we prove that there exists a positive constant $c$ such that $$\sup_{z_0\in D}\int_{D}(1-|z|)^{q-1}\|\nabla f(z)\|^q P_{z_0}(z) dA(z) \le c^q\sup_{z_0\in D}\int_{\mathbf{T}}\|f(z)-f(z_0)\|^qP_{z_0}(z) dm(z)$$ holds for all trigonometric polynomials $f$ with coefficients in $B$ if and only if $B$ admits an equivalent norm which is $q$-uniformly convex, where $$P_{z_0}(z)=\frac{1-|z_0|^2}{|1-\bar{z_0}z|^2} .$$ The validity of the converse inequality is equivalent to the existence of an equivalent $q$-uniformly smooth norm.

Keywords:BMO, Carleson measures, Lusin type, Lusin cotype, uniformly convex spaces, uniformly smooth spaces
Categories:46E40, 42B25, 46B20

20. CJM 2009 (vol 61 pp. 807)

21. CJM 2009 (vol 61 pp. 141)

Green, Ben; Konyagin, Sergei
On the Littlewood Problem Modulo a Prime
Let $p$ be a prime, and let $f \from \mathbb{Z}/p\mathbb{Z} \rightarrow \mathbb{R}$ be a function with $\E f = 0$ and $\Vert \widehat{f} \Vert_1 \leq 1$. Then $\min_{x \in \Zp} |f(x)| = O(\log p)^{-1/3 + \epsilon}$. One should think of $f$ as being ``approximately continuous''; our result is then an ``approximate intermediate value theorem''. As an immediate consequence we show that if $A \subseteq \Zp$ is a set of cardinality $\lfloor p/2\rfloor$, then $\sum_r |\widehat{1_A}(r)| \gg (\log p)^{1/3 - \epsilon}$. This gives a result on a ``mod $p$'' analogue of Littlewood's well-known problem concerning the smallest possible $L^1$-norm of the Fourier transform of a set of $n$ integers. Another application is to answer a question of Gowers. If $A \subseteq \Zp$ is a set of size $\lfloor p/2 \rfloor$, then there is some $x \in \Zp$ such that \[ | |A \cap (A + x)| - p/4 | = o(p).\]

Categories:42A99, 11B99

22. CJM 2008 (vol 60 pp. 1283)

Ho, Kwok-Pun
Remarks on Littlewood--Paley Analysis
Littlewood--Paley analysis is generalized in this article. We show that the compactness of the Fourier support imposed on the analyzing function can be removed. We also prove that the Littlewood--Paley decomposition of tempered distributions converges under a topology stronger than the weak-star topology, namely, the inductive limit topology. Finally, we construct a multiparameter Littlewood--Paley analysis and obtain the corresponding ``renormalization'' for the convergence of this multiparameter Littlewood--Paley analysis.

Keywords:Littlewood--Paley analysis, distributions
Category:42B25

23. CJM 2008 (vol 60 pp. 685)

Savu, Anamaria
Closed and Exact Functions in the Context of Ginzburg--Landau Models
For a general vector field we exhibit two Hilbert spaces, namely the space of so called \emph{closed functions} and the space of \emph{exact functions} and we calculate the codimension of the space of exact functions inside the larger space of closed functions. In particular we provide a new approach for the known cases: the Glauber field and the second-order Ginzburg--Landau field and for the case of the fourth-order Ginzburg--Landau field.

Keywords:Hermite polynomials, Fock space, Fourier coefficients, Fourier transform, group of symmetries
Categories:42B05, 81Q50, 42A16

24. CJM 2008 (vol 60 pp. 334)

Curry, Eva
Low-Pass Filters and Scaling Functions for Multivariable Wavelets
We show that a characterization of scaling functions for multiresolution analyses given by Hern\'{a}ndez and Weiss and that a characterization of low-pass filters given by Gundy both hold for multivariable multiresolution analyses.

Keywords:multivariable multiresolution analysis, low-pass filter, scaling function
Categories:42C40, 60G35

25. CJM 2007 (vol 59 pp. 1223)

Buraczewski, Dariusz; Martinez, Teresa; Torrea, José L.
Calderón--Zygmund Operators Associated to Ultraspherical Expansions
We define the higher order Riesz transforms and the Littlewood--Paley $g$-function associated to the differential operator $L_\l f(\theta)=-f''(\theta)-2\l\cot\theta f'(\theta)+\l^2f(\theta)$. We prove that these operators are Calder\'{o}n--Zygmund operators in the homogeneous type space $((0,\pi),(\sin t)^{2\l}\,dt)$. Consequently, $L^p$ weighted, $H^1-L^1$ and $L^\infty-BMO$ inequalities are obtained.

Keywords:ultraspherical polynomials, Calderón--Zygmund operators
Categories:42C05, 42C15frcs
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