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26. CJM 2011 (vol 64 pp. 183)

Nowak, Adam; Stempak, Krzysztof
Negative Powers of Laguerre Operators
We study negative powers of Laguerre differential operators in $\mathbb{R}^d$, $d\ge1$. For these operators we prove two-weight $L^p-L^q$ estimates with ranges of $q$ depending on $p$. The case of the harmonic oscillator (Hermite operator) has recently been treated by Bongioanni and Torrea by using a straightforward approach of kernel estimates. Here these results are applied in certain Laguerre settings. The procedure is fairly direct for Laguerre function expansions of Hermite type, due to some monotonicity properties of the kernels involved. The case of Laguerre function expansions of convolution type is less straightforward. For half-integer type indices $\alpha$ we transfer the desired results from the Hermite setting and then apply an interpolation argument based on a device we call the convexity principle to cover the continuous range of $\alpha\in[-1/2,\infty)^d$. Finally, we investigate negative powers of the Dunkl harmonic oscillator in the context of a finite reflection group acting on $\mathbb{R}^d$ and isomorphic to $\mathbb Z^d_2$. The two weight $L^p-L^q$ estimates we obtain in this setting are essentially consequences of those for Laguerre function expansions of convolution type.

Keywords:potential operator, fractional integral, Riesz potential, negative power, harmonic oscillator, Laguerre operator, Dunkl harmonic oscillator
Categories:47G40, 31C15, 26A33

27. CJM 2011 (vol 63 pp. 862)

Hosokawa, Takuya; Nieminen, Pekka J.; Ohno, Shûichi
Linear Combinations of Composition Operators on the Bloch Spaces
We characterize the compactness of linear combinations of analytic composition operators on the Bloch space. We also study their boundedness and compactness on the little Bloch space.

Keywords: composition operator, compactness, Bloch space
Categories:47B33, 30D45, 47B07

28. 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

29. 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

30. CJM 2010 (vol 62 pp. 1037)

Calviño-Louzao, E.; García-Río, E.; Vázquez-Lorenzo, R.
Riemann Extensions of Torsion-Free Connections with Degenerate Ricci Tensor
{Correspondence} between torsion-free connections with {nilpotent skew-symmetric curvature operator} and IP Riemann extensions is shown. Some consequences are derived in the study of four-dimensional IP metrics and locally homogeneous affine surfaces.

Keywords:Walker metric, Riemann extension, curvature operator, projectively flat and recurrent affine connection
Categories:53B30, 53C50

31. CJM 2009 (vol 62 pp. 305)

Hua, He; Yunbai, Dong; Xianzhou, Guo
Approximation and Similarity Classification of Stably Finitely Strongly Irreducible Decomposable Operators
Let $\mathcal H$ be a complex separable Hilbert space and ${\mathcal L}({\mathcal H})$ denote the collection of bounded linear operators on ${\mathcal H}$. In this paper, we show that for any operator $A\in{\mathcal L}({\mathcal H})$, there exists a stably finitely (SI) decomposable operator $A_\epsilon$, such that $\|A-A_{\epsilon}\|<\epsilon$ and ${\mathcal{\mathcal A}'(A_{\epsilon})}/\operatorname{rad} {{\mathcal A}'(A_{\epsilon})}$ is commutative, where $\operatorname{rad}{{\mathcal A}'(A_{\epsilon})}$ is the Jacobson radical of ${{\mathcal A}'(A_{\epsilon})}$. Moreover, we give a similarity classification of the stably finitely decomposable operators that generalizes the result on similarity classification of Cowen-Douglas operators given by C. L. Jiang.

Keywords:$K_{0}$-group, strongly irreducible decomposition, Cowen—Douglas operators, commutant algebra, similarity classification
Categories:47A05, 47A55, 46H20

32. CJM 2009 (vol 62 pp. 218)

Xing, Yang
The General Definition of the Complex Monge--Ampère Operator on Compact Kähler Manifolds
We introduce a wide subclass ${\mathcal F}(X,\omega)$ of quasi-plurisubharmonic functions in a compact Kähler manifold, on which the complex Monge-Ampère operator is well defined and the convergence theorem is valid. We also prove that ${\mathcal F}(X,\omega)$ is a convex cone and includes all quasi-plurisubharmonic functions that are in the Cegrell class.

Keywords:complex Monge--Ampère operator, compact Kähler manifold
Categories:32W20, 32Q15

33. CJM 2009 (vol 62 pp. 202)

Tang, Lin
Interior $h^1$ Estimates for Parabolic Equations with $\operatorname{LMO}$ Coefficients
In this paper we establish \emph{a priori} $h^1$-estimates in a bounded domain for parabolic equations with vanishing $\operatorname{LMO}$ coefficients.

Keywords:parabolic operator, Hardy space, parabolic, singular integrals and commutators
Categories:35K20, 35B65, 35R05

34. CJM 2009 (vol 61 pp. 1262)

Dong, Z.
On the Local Lifting Properties of Operator Spaces
In this paper, we mainly study operator spaces which have the locally lifting property (LLP). The dual of any ternary ring of operators is shown to satisfy the strongly local reflexivity, and this is used to prove that strongly local reflexivity holds also for operator spaces which have the LLP. Several homological characterizations of the LLP and weak expectation property are given. We also prove that for any operator space $V$, $V^{**}$ has the LLP if and only if $V$ has the LLP and $V^{*}$ is exact.

Keywords:operator space, locally lifting property, strongly locally reflexive

35. CJM 2009 (vol 61 pp. 190)

Lu, Yufeng; Shang, Shuxia
Bounded Hankel Products on the Bergman Space of the Polydisk
We consider the problem of determining for which square integrable functions $f$ and $g$ on the polydisk the densely defined Hankel product $H_{f}H_g^\ast$ is bounded on the Bergman space of the polydisk. Furthermore, we obtain similar results for the mixed Haplitz products $H_{g}T_{\bar{f}}$ and $T_{f}H_{g}^{*}$, where $f$ and $g$ are square integrable on the polydisk and $f$ is analytic.

Keywords:Toeplitz operator, Hankel operator, Haplitz products, Bergman space, polydisk
Categories:47B35, 47B47

36. CJM 2008 (vol 60 pp. 1010)

Galé, José E.; Miana, Pedro J.
$H^\infty$ Functional Calculus and Mikhlin-Type Multiplier Conditions
Let $T$ be a sectorial operator. It is known that the existence of a bounded (suitably scaled) $H^\infty$ calculus for $T$, on every sector containing the positive half-line, is equivalent to the existence of a bounded functional calculus on the Besov algebra $\Lambda_{\infty,1}^\alpha(\R^+)$. Such an algebra includes functions defined by Mikhlin-type conditions and so the Besov calculus can be seen as a result on multipliers for $T$. In this paper, we use fractional derivation to analyse in detail the relationship between $\Lambda_{\infty,1}^\alpha$ and Banach algebras of Mikhlin-type. As a result, we obtain a new version of the quoted equivalence.

Keywords:functional calculus, fractional calculus, Mikhlin multipliers, analytic semigroups, unbounded operators, quasimultipliers
Categories:47A60, 47D03, 46J15, 26A33, 47L60, 47B48, 43A22

37. CJM 2008 (vol 60 pp. 241)

Alexandrova, Ivana
Semi-Classical Wavefront Set and Fourier Integral Operators
Here we define and prove some properties of the semi-classical wavefront set. We also define and study semi-classical Fourier integral operators and prove a generalization of Egorov's theorem to manifolds of different dimensions.

Keywords:wavefront set, Fourier integral operators, Egorov theorem, semi-classical analysis
Categories:35S30, 35A27, 58J40, 81Q20

38. 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

39. CJM 2007 (vol 59 pp. 966)

Forrest, Brian E.; Runde, Volker; Spronk, Nico
Operator Amenability of the Fourier Algebra in the $\cb$-Multiplier Norm
Let $G$ be a locally compact group, and let $A_{\cb}(G)$ denote the closure of $A(G)$, the Fourier algebra of $G$, in the space of completely bounded multipliers of $A(G)$. If $G$ is a weakly amenable, discrete group such that $\cstar(G)$ is residually finite-dimensional, we show that $A_{\cb}(G)$ is operator amenable. In particular, $A_{\cb}(\free_2)$ is operator amenable even though $\free_2$, the free group in two generators, is not an amenable group. Moreover, we show that if $G$ is a discrete group such that $A_{\cb}(G)$ is operator amenable, a closed ideal of $A(G)$ is weakly completely complemented in $A(G)$ if and only if it has an approximate identity bounded in the $\cb$-multiplier norm.

Keywords:$\cb$-multiplier norm, Fourier algebra, operator amenability, weak amenability
Categories:43A22, 43A30, 46H25, 46J10, 46J40, 46L07, 47L25

40. CJM 2007 (vol 59 pp. 638)

MacDonald, Gordon W.
Distance from Idempotents to Nilpotents
We give bounds on the distance from a non-zero idempotent to the set of nilpotents in the set of $n\times n$ matrices in terms of the norm of the idempotent. We construct explicit idempotents and nilpotents which achieve these distances, and determine exact distances in some special cases.

Keywords:operator, matrix, nilpotent, idempotent, projection
Categories:47A15, 47D03, 15A30

41. CJM 2007 (vol 59 pp. 614)

Labuschagne, C. C. A.
Preduals and Nuclear Operators Associated with Bounded, $p$-Convex, $p$-Concave and Positive $p$-Summing Operators
We use Krivine's form of the Grothendieck inequality to renorm the space of bounded linear maps acting between Banach lattices. We construct preduals and describe the nuclear operators associated with these preduals for this renormed space of bounded operators as well as for the spaces of $p$-convex, $p$-concave and positive $p$-summing operators acting between Banach lattices and Banach spaces. The nuclear operators obtained are described in terms of factorizations through classical Banach spaces via positive operators.

Keywords:$p$-convex operator, $p$-concave operator, $p$-summing operator, Banach space, Banach lattice, nuclear operator, sequence space
Categories:46B28, 47B10, 46B42, 46B45

42. CJM 2007 (vol 59 pp. 276)

Bernardis, A. L.; Martín-Reyes, F. J.; Salvador, P. Ortega
Weighted Inequalities for Hardy--Steklov Operators
We characterize the pairs of weights $(v,w)$ for which the operator $Tf(x)=g(x)\int_{s(x)}^{h(x)}f$ with $s$ and $h$ increasing and continuous functions is of strong type $(p,q)$ or weak type $(p,q)$ with respect to the pair $(v,w)$ in the case $0
Keywords:Hardy--Steklov operator, weights, inequalities
Categories:26D15, 46E30, 42B25

43. CJM 2007 (vol 59 pp. 311)

Christianson, Hans
Growth and Zeros of the Zeta Function for Hyperbolic Rational Maps
This paper describes new results on the growth and zeros of the Ruelle zeta function for the Julia set of a hyperbolic rational map. It is shown that the zeta function is bounded by $\exp(C_K |s|^{\delta})$ in strips $|\Real s| \leq K$, where $\delta$ is the dimension of the Julia set. This leads to bounds on the number of zeros in strips (interpreted as the Pollicott--Ruelle resonances of this dynamical system). An upper bound on the number of zeros in polynomial regions $\{|\Real s | \leq |\Imag s|^\alpha\}$ is given, followed by weaker lower bound estimates in strips $\{\Real s > -C, |\Imag s|\leq r\}$, and logarithmic neighbourhoods $\{ |\Real s | \leq \rho \log |\Imag s| \}$. Recent numerical work of Strain--Zworski suggests the upper bounds in strips are optimal.

Keywords:zeta function, transfer operator, complex dynamics

44. CJM 2005 (vol 57 pp. 897)

Berezhnoĭ, Evgenii I.; Maligranda, Lech
Representation of Banach Ideal Spaces and Factorization of Operators
Representation theorems are proved for Banach ideal spaces with the Fatou property which are built by the Calder{\'o}n--Lozanovski\u\i\ construction. Factorization theorems for operators in spaces more general than the Lebesgue $L^{p}$ spaces are investigated. It is natural to extend the Gagliardo theorem on the Schur test and the Rubio de~Francia theorem on factorization of the Muckenhoupt $A_{p}$ weights to reflexive Orlicz spaces. However, it turns out that for the scales far from $L^{p}$-spaces this is impossible. For the concrete integral operators it is shown that factorization theorems and the Schur test in some reflexive Orlicz spaces are not valid. Representation theorems for the Calder{\'o}n--Lozanovski\u\i\ construction are involved in the proofs.

Keywords:Banach ideal spaces, weighted spaces, weight functions,, Calderón--Lozanovski\u\i\ spaces, Orlicz spaces, representation of, spaces, uniqueness problem, positive linear operators, positive sublinear, operators, Schur test, factorization of operators, f
Categories:46E30, 46B42, 46B70

45. CJM 2005 (vol 57 pp. 598)

Kornelson, Keri A.
Local Solvability of Laplacian Difference Operators Arising from the Discrete Heisenberg Group
Differential operators $D_x$, $D_y$, and $D_z$ are formed using the action of the $3$-dimensional discrete Heisenberg group $G$ on a set $S$, and the operators will act on functions on $S$. The Laplacian operator $L=D_x^2 + D_y^2 + D_z^2$ is a difference operator with variable differences which can be associated to a unitary representation of $G$ on the Hilbert space $L^2(S)$. Using techniques from harmonic analysis and representation theory, we show that the Laplacian operator is locally solvable.

Keywords:discrete Heisenberg group,, unitary representation,, local solvability,, difference operator
Categories:43A85, 22D10, 39A70

46. CJM 2005 (vol 57 pp. 61)

Binding, Paul; Strauss, Vladimir
On Operators with Spectral Square but without Resolvent Points
Decompositions of spectral type are obtained for closed Hilbert space operators with empty resolvent set, but whose square has closure which is spectral. Krein space situations are also discussed.

Keywords:unbounded operators, closed operators,, spectral resolution, indefinite metric
Categories:47A05, 47A15, 47B40, 47B50, 46C20

47. CJM 2003 (vol 55 pp. 1264)

Havin, Victor; Mashreghi, Javad
Admissible Majorants for Model Subspaces of $H^2$, Part II: Fast Winding of the Generating Inner Function
This paper is a continuation of Part I [6]. We consider the model subspaces $K_\Theta=H^2\ominus\Theta H^2$ of the Hardy space $H^2$ generated by an inner function $\Theta$ in the upper half plane. Our main object is the class of admissible majorants for $K_\Theta$, denoted by Adm $\Theta$ and consisting of all functions $\omega$ defined on $\mathbb{R}$ such that there exists an $f \ne 0$, $f \in K_\Theta$ satisfying $|f(x)|\leq\omega(x)$ almost everywhere on $\mathbb{R}$. Firstly, using some simple Hilbert transform techniques, we obtain a general multiplier theorem applicable to any $K_\Theta$ generated by a meromorphic inner function. In contrast with [6], we consider the generating functions $\Theta$ such that the unit vector $\Theta(x)$ winds up fast as $x$ grows from $-\infty$ to $\infty$. In particular, we consider $\Theta=B$ where $B$ is a Blaschke product with ``horizontal'' zeros, i.e., almost uniformly distributed in a strip parallel to and separated from $\mathbb{R}$. It is shown, among other things, that for any such $B$, any even $\omega$ decreasing on $(0,\infty)$ with a finite logarithmic integral is in Adm $B$ (unlike the ``vertical'' case treated in [6]), thus generalizing (with a new proof) a classical result related to Adm $\exp(i\sigma z)$, $\sigma>0$. Some oscillating $\omega$'s in Adm $B$ are also described. Our theme is related to the Beurling-Malliavin multiplier theorem devoted to Adm $\exp(i\sigma z)$, $\sigma>0$, and to de Branges' space $\mathcal{H}(E)$.

Keywords:Hardy space, inner function, shift operator, model, subspace, Hilbert transform, admissible majorant
Categories:30D55, 47A15

48. CJM 2003 (vol 55 pp. 1231)

Havin, Victor; Mashreghi, Javad
Admissible Majorants for Model Subspaces of $H^2$, Part I: Slow Winding of the Generating Inner Function
A model subspace $K_\Theta$ of the Hardy space $H^2 = H^2 (\mathbb{C}_+)$ for the upper half plane $\mathbb{C}_+$ is $H^2(\mathbb{C}_+) \ominus \Theta H^2(\mathbb{C}_+)$ where $\Theta$ is an inner function in $\mathbb{C}_+$. A function $\omega \colon \mathbb{R}\mapsto[0,\infty)$ is called an admissible majorant for $K_\Theta$ if there exists an $f \in K_\Theta$, $f \not\equiv 0$, $|f(x)|\leq \omega(x)$ almost everywhere on $\mathbb{R}$. For some (mainly meromorphic) $\Theta$'s some parts of Adm $\Theta$ (the set of all admissible majorants for $K_\Theta$) are explicitly described. These descriptions depend on the rate of growth of $\arg \Theta$ along $\mathbb{R}$. This paper is about slowly growing arguments (slower than $x$). Our results exhibit the dependence of Adm $B$ on the geometry of the zeros of the Blaschke product $B$. A complete description of Adm $B$ is obtained for $B$'s with purely imaginary (``vertical'') zeros. We show that in this case a unique minimal admissible majorant exists.

Keywords:Hardy space, inner function, shift operator, model, subspace, Hilbert transform, admissible majorant
Categories:30D55, 47A15

49. CJM 2002 (vol 54 pp. 1280)

Skrzypczak, Leszek
Besov Spaces and Hausdorff Dimension For Some Carnot-Carathéodory Metric Spaces
We regard a system of left invariant vector fields $\mathcal{X}=\{X_1,\dots,X_k\}$ satisfying the H\"ormander condition and the related Carnot-Carath\'eodory metric on a unimodular Lie group $G$. We define Besov spaces corresponding to the sub-Laplacian $\Delta=\sum X_i^2$ both with positive and negative smoothness. The atomic decomposition of the spaces is given. In consequence we get the distributional characterization of the Hausdorff dimension of Borel subsets with the Haar measure zero.

Keywords:Besov spaces, sub-elliptic operators, Carnot-Carathéodory metric, Hausdorff dimension
Categories:46E35, 43A15, 28A78

50. CJM 2002 (vol 54 pp. 1121)

Bao, Jiguang
Fully Nonlinear Elliptic Equations on General Domains
By means of the Pucci operator, we construct a function $u_0$, which plays an essential role in our considerations, and give the existence and regularity theorems for the bounded viscosity solutions of the generalized Dirichlet problems of second order fully nonlinear elliptic equations on the general bounded domains, which may be irregular. The approximation method, the accretive operator technique and the Caffarelli's perturbation theory are used.

Keywords:Pucci operator, viscosity solution, existence, $C^{2,\psi}$ regularity, Dini condition, fully nonlinear equation, general domain, accretive operator, approximation lemma
Categories:35D05, 35D10, 35J60, 35J67
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