In this paper we construct a class of homogeneous Hilbert modules over the disc algebra $\mathcal{A}(\mathbb D)$ as quotients of certain natural modules over the function algebra $\mathcal{A}(\mathbb D^2)$. These quotient modules are described using the jet construction for Hilbert modules. We show that the quotient modules obtained this way, belong to the class ${\mathrm B}_k(\mathbb D)$ and that they are mutually inequivalent, irreducible and homogeneous.
Let $ Ω\subset \mathbb{C}^m $ be an open, connected and bounded set and $\mathcal{A}(Ω)$ be a function algebra of holomorphic functions on $Ω$. Suppose that $\mathcal{M}_q$ is the quotient Hilbert module obtained from a submodule of functions in a Hilbert module $\mathcal{M}$ vanishing to order $k$ along a smooth irreducible complex analytic set $\mathcal{Z}\subsetΩ$ of codimension at least $2$. In this article, we prove that the compression of the multiplication operators onto $\mathcal{M}_q$ is homogeneous with respect to a suitable subgroup of the automorphism group Aut$(Ω)$ of $Ω$ dependin…
This paper is the third of a series giving a self-contained way from the Neveu-Schwarz algebra to a new series of irreducible subfactors. Here we introduce the local von Neumann algebra of the Neveu-Schwarz algebra, to obtain Jones-Wassermann subfactors for each representation of the discrete series. Then using primary fields we prove the irreducibility of these subfactors; to next compute the Connes fusion ring and obtain the explicit formula of the subfactors indices.
A complete list of homogeneous operators in the Cowen-Douglas class $B_n(D)$ is given. This classification is obtained from an explicit realization of all the homogeneous Hermitian holomorphic vector bundles on the unit disc under the action of the universal covering group of the bi-holomorphic automorphism group of the unit disc.
In a recent paper, the authors have constructed a large class of operators in the Cowen-Douglas class Cowen-Douglas class of the unit disc $\mathbb D$ which are {\em homogeneous} with respect to the action of the group Möb -- the Möbius group consisting of bi-holomorphic automorphisms of the unit disc $\mathbb{D}$. The {\em associated representation} for each of these operators is {\em multiplicity free}. Here we give a different independent construction of all homogeneous operators in the Cowen-Douglas class with multiplicity free associated representation and verify that they are exactly the…
In this paper we construct a large class of multiplication operators on reproducing kernel Hilbert spaces which are {\em homogeneous} with respect to the action of the Möbius group consisting of bi-holomorphic automorphisms of the unit disc $\mathbb D$. For every $m \in \N$ we have a family of operators depending on $m+1$ positive real parameters. The kernel function is calculated explicitly. It is proved that each of these operators is bounded, lies in the Cowen-Douglas class of $\mathbb D$ and is irreducible. These operators are shown to be mutually pairwise unitarily inequivalent.
We consider compact group actions on C*- and W*- algebras. We prove results that relate the duality property of the action (as defined in the Introduction) with other relevant properties of the system such as the relative commutant of the fixed point algebras being trivial (called the irreducibility of the inclusion) and also to the Galois correspondence between invariant C*-subalgebras containing the fixed point algebra and the class of closed normal subgroups of the compact group.
We introduce and study in a general setting the concept of homogeneity of an operator and, in particular, the notion of homogeneity of an integral operator. In the latter case, homogeneous kernels of such operators are also studied. The concept of homogeneity is associated with transformations of a measure - measure dilations, which are most natural in the context of our general research scheme. For the study of integral operators, the notions of weak and strong homogeneity of the kernel are introduced. The weak case is proved to generate a homogeneous operator in the sense of our definition, …
In this paper we study the Besov continuity of pseudo-differential operators on compact homogeneous manifolds $M=G/K.$ We use the global quantization of these operators in terms of the representation theory of compact homogeneous manifolds.
The main features of homogeneous Cowen-Douglas operators, well-known for the unit disk, are generalized to the setting of hermitian bounded symmetric domains of arbitrary rank.
We study the mapping properties of the Hardy--Littlewood fractional maximal operator between Lorentz spaces of the homogeneous tree and discuss the optimality of all the results.
Using an integral formula on a homogeneous Siegel domain, we show a necessary and sufficient condition for composition operators on the weighted Bergman space of a minimal bounded homogeneous domain to be compact. To describe the compactness of composition operators, we see a boundary behavior of the Bergman kernel.
In this paper we study quasi-homogeneous operators, which include the homogeneous operators, in the Cowen-Douglas class. We give two separate theorems describing canonical models (with respect to equivalence under unitary and invertible operators, respectively) for these operators using techniques from complex geometry. This considerably extends the similarity and unitary classification of homogeneous operators in the Cowen-Douglas class obtained recently by the last author and A. Korányi. Specifically, the complex geometric invariants used for our classification are the curvature and the seco…
An explicit construction of all the homogeneous holomorphic Hermitian vector bundles over the unit disc $\mathbb D$ is given. It is shown that every such vector bundle is a direct sum of irreducible ones. Among these irreducible homogeneous holomorphic Hermitian vector bundles over $\mathbb D$, the ones corresponding to operators in the Cowen-Douglas class ${\mathrm B}_n(\mathbb D)$ are identified. The classification of homogeneous operators in ${\mathrm B}_n(\mathbb D)$ is completed using an explicit realization of these operators. We also show how the homogeneous operators in ${\mathrm B}_n(…
We present many examples of Banach spaces of linear operators and homogeneous polynomials without the approximation property, thus improving results of Dineen and Mujica [11] and Godefroy and Saphar [13].
Necessary and sufficient conditions for positive Toeplitz operators on the Bergman space of a minimal bounded homogeneous domain to be bounded or compact are described in terms of the Berezin transform, the averaging function and the Carleson property.
We study the bounded and the compact weighted composition operators from the Bloch space into the weighted Banach spaces of holomorphic functions on bounded homogeneous domains, with particular attention to the unit polydisk. For bounded homogeneous domains, we characterize the bounded weighted composition operators and determine the operator norm. In addition, we provide sufficient conditions for compactness. For the unit polydisk, we completely characterize the compact weighted composition operators, as well as provide computable estimates on the operator norm.
The aim of this note is to give the boundedness conditions for Hausdorff operators on Hardy spaces $H^{1}$ with the norm defined via $(1,q)$ atoms over homogeneous spaces of Lie groups with doubling property and to apply results we obtain to generalized Delsarte operators and to Hausdorff operators over multidimensional spheres.
Hausdorff operators on the real line and multidimensional Euclidean spaces originated from some classical summation methods. Now it is an active research area. Hausdorff operators on general groups were defined and studied by the author since 2019. The purpose of this paper is to define and study Hausdorff operators on Lebesgue and real Hardy spaces over homogeneous spaces of locally compact groups. We introduce in particular an atomic Hardy space over homogeneous spaces of locally compact groups and obtain conditions for boundedness of Hausdorff operators on such spaces. Several corollaries a…
We give estimates for the essential norms of a positive Toeplitz operator on the Bergman space of a minimal bounded homogeneous domain in terms of the Berezin transform or the averaging function of the symbol. Using these estimates, we also give necessary and sufficient conditions for the positive Toeplitz operators to be compact.
We generalize and extend results on decay rates of singular values or eigenvalues of positive integral operators from unit spheres to two-point homogeneous spaces. The rates we present depend upon the order of the Laplace-Beltrami operator used to define the smoothness conditions on generating kernels, the Schatten class containing the integral operator generated by the derivative of the generating kernel and the dimension of the space.
We give conditions for boundedness of Hausdorff operators on real Hardy spaces $H^1$ over homogeneous spaces of locally compact groups with local doubling property. The special case of the hyperbolic plane is considered.
Following previous works for the unit ball, we define quasi-radial pseudo-homogeneous symbols on the projective space and obtain the corresponding commutativity results for Toeplitz operators. A geometric interpretation of these symbols in terms of moment maps is developed. This leads us to the introduction of a new family of symbols, extended pseudo-homogeneous, that provide larger commutative Banach algebras generated by Toeplitz operators. This family of symbols provides new commutative Banach algebras generated by Toeplitz operators on the unit ball.
In this paper, the fractional integral operator on non-homogeneous metric measure spaces is introduced, which contains the classic fractional integral operator, fractional integral with non-doubling measures and fractional integral with fractional kernel of order $α$ and regularity $ε$ introduced by García-Cuerva and Gatto as special cases. And the $(L^{p}(μ),L^{q}(μ))$-boundedness for fractional integral operators on non-homogeneous metric measure spaces is established. From this, the $(L^{p}(μ),L^{q}(μ))$-boundedness for commutators and multilinear commutators generated by fractional integra…
We prove a new dyadic representation theorem with applications to the $T(1)$ and $A_2$ theorems. In particular, we obtain the non-homogeneous $T(1)$ theorem under weaker kernel regularity than the earlier approaches.
Let $(X,d,μ)$ be a metric measure space satisfying both the geometrically doubling and the upper doubling measure conditions, which is called non-homogeneous metric measure space. In this paper, via a sharp maximal operator, the boundedness of commutators generated by multilinear singular integral with $RBMO(μ)$ function on non-homogeneous metric measure spaces in $m$-multiple Lebesgue spaces is obtained.
In this paper, we introduce homogeneous mixed Herz-Morrey spaces $M\dot{K}_{p,\vec{q}}^{α,λ}(\mathbb{R}^n)$ and show it's some properties. Firstly, the boundedness of sublinear operators, fractional type operators in homogeneous mixed Herz-Morrey spaces is investigated. In particular, the above results are still valid for Calder$\acute{o}$n-Zygmund operators and fractional maximal operators. Lastly, the boundedness of their commutators in homogeneous mixed Herz-Morrey spaces is obtained.
Let (X,d,μ) be a space of homogeneous type and E a UMD Banach space. Under the assumption mu({x})=0 for all x in X, we prove a representation theorem for singular integral operators on (X,d,mu) as a series of simple shifts and rearrangements plus two paraproducts. This gives a T(1) Theorem in this setting.
In this paper, we study the multiplication operators on the Bloch space of a bounded homogeneous domain in $\mathbb{C}^n$. Specifically, we characterize the bounded and the compact multiplication operators, establish estimates on the operator norm, and determine the spectrum. We prove that the only bounded multiplication operators on the Bloch space of the polydisk are those whose symbol is constant. Furthermore, we prove that for a large class of bounded symmetric domains, the isometric multiplication operators are those whose symbol is a constant of modulus one.
Let $\x$ be the homogeneous tree of degree $q+1~(q\geq2)$. In this article, we present symbolic criteria for boundedness of pseudo-differential operators on homogeneous tree $\x$. A sufficient condition for weak type $(p, q)$ boundedness is also given. We present necessary and sufficient conditions on the symbols $σ$ such that the corresponding pseudo-differential operators $T_σ$ from $L^{p_1} (\x)$ into $L^{p_2} (\x)$ to be nuclear for $1\leq p_1, p_2<\infty$. We show that the adjoint of the nuclear operator from $L^{p_2'}(\x)$ into $L^{p_1'}(\x)$ is again a nuclear operator. Explicit formula…
The aim of the present paper is to obtain a Sjölin-type maximal estimate for pseudo-differential operators with homogeneous symbols. The crux of the proof is to obtain a phase decomposition formula which does not involve the time traslation. The proof is somehow parallel to the paper by Pramanik and Terwilleger (P. Malabika and E. Terwilleger, A weak $L^2$ estimate for a maximal dyadic sum operator on $R^n$, Illinois J. Math, {\bf 47} (2003), no. 3, 775--813). In the present paper, we mainly concentrate on our new phase decomposition formula and the results in the Cotlar type estimate, which a…
Let $D$ be a bounded homogeneous domain in $\mathbb{C}^n$. In this paper, we study the bounded and the compact weighted composition operators mapping the Hardy space $H^\infty(D)$ into the Bloch space of $D$. We characterize the bounded weighted composition operators, provide operator norm estimates, and give sufficient conditions for compactness. We prove that these conditions are necessary in the case of the unit ball and the polydisk. We then show that if $D$ is a bounded symmetric domain, the bounded multiplication operators from $H^\infty(D)$ to the Bloch space of $D$ are the operators wh…
We study some dynamical properties of composition operators defined on the space $\mathcal{P}(^m X)$ of $m$-homogeneous polynomials on a Banach space $X$ when $\mathcal{P}(^m X)$ is endowed with two different topologies: the one of uniform convergence on compact sets and the one defined by the usual norm. The situation is quite different for both topologies: while in the case of uniform convergence on compact sets every power bounded composition operator is uniformly mean ergodic, for the topology of the norm there is no relation between the latter properties. Several examples are given.
The authors study Hardy spaces, of arbitrary order, on a space of homogeneous type. This extends earlier work that treated only $H^p$ for $p$ near 1. Applications are given to the boundedness of certain singular integral operators, especially on domains in complex space.
By using commutator methods, we show uniform resolvent estimates and obtain globally smooth operators for self-adjoint injective homogeneous operators $H$ on graded groups, including Rockland operators, sublaplacians and many others. Left or right invariance is not required. Typically the globally smooth operator has the form $T=V|H|^{1/2}$, where $V$ only depends on the homogeneous structure of the group through Sobolev spaces, the homogeneous dimension and the minimal and maximal dilation weights. For stratified groups improvements are obtained, by using a Hardy-type inequality. Some of the …
In this paper, we present the current results in the study of weighted composition operators on the Bloch space of bounded homogeneous domains in $\mathbb{C}^n$ with particular emphasis on the issues of boundedness and compactness. We also discuss the bounded and the compact weighted composition operators from the Bloch space to the Hardy space $H^\infty$.
We use methods of harmonic analysis and group representation theory to study the spectral properties of the abstract parabolic operator $\mathscr L = -d/dt+A$ in homogeneous function spaces. We provide sufficient conditions for invertibility of such operators in terms of the spectral properties of the operator $A$ and the semigroup generated by $A$. We introduce a homogeneous space of functions with absolutely summable spectrum and prove a generalization of the Gearhart-Prüss Theorem for such spaces. We use the results to prove existence and uniqueness of solutions of a certain class of non-li…
In this paper we study the mapping properties of the averaging operator over a variety given by a system of homogeneous equations over a finite field. We obtain optimal results on the averaging problems over two dimensional varieties whose elements are common solutions of diagonal homogeneous equations. The proof is based on a careful study of algebraic and geometric properties of such varieties. In particular, we show that they are not contained in any hyperplane and are complete intersections. We also address partial results on averaging problems over arbitrary dimensional homogeneous variet…
In this paper, we first construct the homogeneous $q$-shift operator $\widetilde{E}(a,b;D_{q})$ and the homogeneous $q$-difference operator $\widetilde{L}(a,b; θ_{xy})$. We then apply these operators in order to represent and investigate generalized Cauchy and a general form of Hahn polynomials. We derive some $q$-identities such as: generating functions, extended generating functions, Mehler's formula and Roger's formula for these $q$-polynomials.
The differential expression $L_m=-\partial_x^2 +(m^2-1/4)x^{-2}$ defines a self-adjoint operator H_m on L^2(0;\infty) in a natural way when $m^2 \geq 1$. We study the dependence of H_m on the parameter m, show that it has a unique holomorphic extension to the half-plane Re(m) > -1, and analyze spectral and scattering properties of this family of operators.
