| 1. |
- Alaghmandan, Mahmood, 1983, et al.
(författare)
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Weighted discrete hypergroups
- 2016
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Ingår i: Indiana University Mathematics Journal. - : Indiana University Mathematics Journal. - 0022-2518. ; 65:2, s. 423-451
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Tidskriftsartikel (refereegranskat)abstract
- Weighted group algebras have been studied extensively in Abstract Harmonic Analysis, where complete characterizations have been found for some important properties of weighted group algebras, namely, amenability and Arens regularity. One of the generalizations of weighted group algebras is weighted hypergroup algebras. Defining weighted hypergroups, analogous to weighted groups, we study Arens regularity and isomorphism to operator algebras for them. We also examine our results on three classes of discrete weighted hypergroups constructed by conjugacy classes of FC groups, the dual space of compact groups, and the hypergroup structure defined by orthogonal polynomials. We observe some unexpected examples regarding Arens regularity and operator isomorphisms of weighted hypergroup algebras.
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| 2. |
- Aleman, Alexandru, et al.
(författare)
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Integration operators on Bergman spaces
- 1997
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Ingår i: Indiana University Mathematics Journal. - : Indiana University Mathematics Journal. - 0022-2518. ; 46:2, s. 337-356
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Tidskriftsartikel (refereegranskat)abstract
- Let ${\bold D}$ denote the unit disk in the complex plane and let $m$ be the area Lebesgue measure on ${\bold D}$. Given a positive integrable function $w$ (a weight) on ${\bold D}$, let $L^p_{\rm a}(w)$ denote the collection of analytic functions $f$ on ${\bold D}$ such that $|f|^pw$ is integrable. Given an analytic function $g$ on ${\bold D}$, the operator $T_g$ is defined by $T_g f(z) = \int_0^{z} f(\zeta)g'(\zeta)\,d\zeta$. The authors consider conditions on $g$ such that $T_g$ is bounded on $L^p_{\rm a}(w)$. In many cases the derivative $D$ is an isomorphism between the subspace of $L^p_{\rm a}(w)$ consisting of functions vanishing at $0$ and some space $L^p_{\rm a}(v)$ with another weight $v$. Thus, the question of boundedness or compactness of $T_g$ becomes the corresponding question for the operator of multiplication by $g'$ acting from $L^p_{\rm a}(w)$ to $L^p_{\rm a}(v)$. The authors consider only $w$ which are radial: $w(re^{i\theta}) = w(r)$. In the first part of the paper it is shown that $\int |f|^pw\,dm \le C\int |f'(z)|^p v(|z|) \,dm(z)$, $p \ge 1$, where $v(r) = \int_r^1 w(u)\,du$. Under the assumption that $v(r) \le C(1 - r)w(r)$, which is valid in particular for $w(r) \equiv (1 - r)^\alpha$, $\alpha > -1$, it follows that $T_g$ is bounded when $g'(z)(1 - |z|)$ is bounded. The converse can be proved in a more general setting. It is obtained by estimating the norm of the linear functional $D_\lambda\colon f \mapsto f'(\lambda)$ in terms of that of the evaluation functional $L_\lambda\colon f \mapsto f(\lambda)$. Rather general hypotheses on a Banach space of analytic functions are obtained in order that $\| D_\lambda \|(1 - |\lambda|) \le C\| L_\lambda \|$. This leads immediately to the converse: if the operator $T_g$ is bounded, then $g'(z)(1 - |z|)$ is bounded. Several classes of weights are shown to satisfy the hypotheses required for both the necessity and the sufficiency of the condition. In all cases, the problem of compactness of $T_g$ is also considered, and the solution involves the little-oh versions of the same conditions for boundedness. In the special case where $p = 2$ and $w(r) = (1 - r)^\alpha$ the Schatten class of $T_g$ is determined. The techniques are briefly applied to the weight $w(r) = \exp[ -\beta(1 - r)^{-\alpha}]$, $\alpha > 0$. In this case $v(r) \le C(1 - r)^{\alpha + 1}w(r)$, and that leads to the conclusion that $T_g$ is bounded when $g'(z)(1 - |z|)^{\alpha + 1}$ is bounded. The necessity of this condition is left open. A theorem of V. L. Oleinik [Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 47 (1974), 120--137, 187, 192--193; MR0369705 (51 #5937) (Theorem 3.3)] shows that it is necessary at least when $\alpha > 1$.
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| 3. |
- Aleman, Alexandru, et al.
(författare)
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Pseudocontinuations and the backward shift
- 1998
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Ingår i: Indiana University Mathematics Journal. - : Indiana University Mathematics Journal. - 0022-2518. ; 47:1, s. 223-276
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Tidskriftsartikel (refereegranskat)abstract
- Beurling's theorem characterizes the forward shift invariant subspaces in the Hardy space $H^2$ on the open unit disk $\bold D$. The description is in terms of an inner function, that is, a function in $H^2$ whose nontangential boundary values have modulus $1$ almost everywhere. If $S$ stands for the forward shift $Sf(z)=zf(z)$, then the adjoint $L=S^*$ is the backward shift, $Lf(z)=\break (f(z)-f(0))/z$. The annihilator of a forward shift invariant subspace is then backward shift invariant, and Beurling's theorem leads to a description also of the backward shift invariant subspaces, as noted by R. G. Douglas, H. S. Shapiro and A. L. Shields [Ann. Inst. Fourier (Grenoble) 20 (1970), fasc. 1, 37--76; MR0270196 (42 #5088)]. Whereas the forward invariant subspaces are described primarily in terms of zeros, the backward invariant subspaces are characterized in terms of pseudocontinuations. To be concrete, take $I$ to be the forward invariant subspace of all functions in $H^2$ that vanish along a given finite sequence $A$ of distinct points in $\bold D$. Its annihilator $I^\perp$ is finite-dimensional, and consists of all rational functions with simple poles along the sequence $A^*$ obtained by reflecting $A$ in the unit circle. Then, if we let the finite sequence $A$ ``grow'' to become in the limit a Blaschke sequence plus a negative singular mass on the circle, the annihilator will increase as well, but there will remain a ``connection'' between the behavior inside $\bold D$ and the behavior outside in the exterior disk ${\bold D}_{\rm e}$, the complement of the closed unit disk on the Riemann sphere. The connection is furnished by the pseudocontinuation across the circle: we have a holomorphic Nevanlinna class function on the inside, and a meromorphic Nevanlinna class function on the outside, and they have the same nontangential boundary values almost everywhere on the unit circle. The issue at hand is whether the Hardy space situation is typical of backward invariant subspaces in Banach spaces $\scr B$ of analytic functions on the disk. A dichotomy appears: if $\scr B$ is bigger than the corresponding Hardy space, then the backward invariant subspaces possess pseudocontinuations across the unit circle, whereas if $\scr B$ is smaller, this is no longer generally the case. What happens is best understood in terms of forward invariant subspaces. With the standard Cauchy duality (the extension of the $H^2$-self-duality), we can think of the dual ${\scr B}^*$ of $\scr B$ as a space of holomorphic functions on $\bold D$, and study the forward shift invariant subspaces on ${\scr B}^*$. Let us concentrate on the case when $\scr B$ is a Hilbert space, of Dirichlet or Bergman type; then ${\scr B}^*$ falls into the same category, too. Every forward invariant subspace $\scr M$ of Dirichlet type has index $1$, which means that $S\scr M$ has codimension $1$ in $\scr M$; this is analogous to the $H^2$ case. Apparently, this means that the annihilator $\scr M^\perp$ (which is a backward invariant subspace of a Bergman space) consists of pseudocontinuable functions. However, there are plenty of forward invariant subspaces of a Bergman space which have index bigger than $1$ [see, e.g., H. Hedenmalm, J. Reine Angew. Math. 443 (1993), 1--9; MR1241125 (94k:30092)]. The annihilator of such a forward invariant subspace is a backward invariant subspace of a Dirichlet space, and some playing around with the formulas for pseudocontinuations suggests that in this case, it should not be unique (and hence not exist as a pseudocontinuation). This is then worked out rigorously in the paper.
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| 4. |
- Aleman, Alexandru, et al.
(författare)
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Single point extremal functions in Bergman-type spaces
- 2002
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Ingår i: Indiana University Mathematics Journal. - : Indiana University Mathematics Journal. - 0022-2518. ; 51:3, s. 581-605
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Tidskriftsartikel (refereegranskat)abstract
- Let A be a zero sequence for the Bergman space L-a(2) of the unit disc D, and let phi(A) be the corresponding canoniacal zero divisor. In this paper we consider quotients of the type phi(Au {alpha})/phi(A), alpha is an element of D. By use of methods from the theory of reproducing kernels we shall show that the modulus of such functions is always bounded by 3, and that they can be written as a product of a single Blaschke factor and a function whose real part is greater than 1. Our methods apply in somewhat larger generality. In particular, our results lead to a new proof of the contractive zero-divisor property in weighted Bergman spaces with logarithmically subharmonic weights. For the unweighted Bergman spaces L-a(p), 0 < p < infinity, we show that the canonical zero divisor phi(A) for a zero sequence with n elements can be written as a product of n starlike functions.
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| 5. |
- Aleman, Alexandru, et al.
(författare)
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Spectra of Integration Operators and Weighted Square Functions
- 2012
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Ingår i: Indiana University Mathematics Journal. - 0022-2518. ; 61:2, s. 775-793
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Tidskriftsartikel (refereegranskat)abstract
- Motivated by the study of the spectrum of integration operators T(g)f(z) = integral(z)(0) f(xi)g'(xi) d xi, acting on the Hardy spaces H-p, we prove weighted versions of the classical estimates due to Fefferman-Stein and Littlewood-Paley which express the H-p-norm of an analytic function with help of its derivative.
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| 6. |
- Aleman, Alexandru, et al.
(författare)
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Weak products of complete pick spaces
- 2021
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Ingår i: Indiana University Mathematics Journal. - : Indiana University Mathematics Journal. - 0022-2518. ; 70:1, s. 325-352
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Tidskriftsartikel (refereegranskat)abstract
- Let H be the Drury-Arveson or Dirichlet space of the unit ball of Cd. The weak product H ☉ H of H is the collection of all functions h that can be written as h =∑∞n=1 fngn, where ∑∞n=1 ||fn|| ||gn|| < ∞. We show that H ☉ H is contained in the Smirnov class of H; that is, every function in H ☉ H is a quotient of two multipliers of H, where the function in the denominator can be chosen to be cyclic in H . As a consequence, we show that the map N → closH ☉H N establishes a one-to-one and onto correspondence between the multiplier invariant subspaces of H and of H ☉ H . The results hold for many weighted Besov spaces H in the unit ball of Cd provided the reproducing kernel has the complete Pick property. One of our main technical lemmas states that, for weighted Besov spaces H that satisfy what we call the multiplier inclusion condition, any bounded column multiplication operator H → ⊕∞n=1 H induces a bounded row multiplication operator ⊕∞n=1 H → H . For the Drury-Arveson space Hd2 this leads to an alternate proof of the characterization of interpolating sequences in terms of weak separation and Carleson measure conditions.
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| 7. |
- Andersson, Mats, 1957, et al.
(författare)
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On a Monge-Ampere Operator for Plurisubharmonic Functions with Analytic Singularities
- 2019
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Ingår i: Indiana University Mathematics Journal. - : Indiana University Mathematics Journal. - 0022-2518. ; 68:4, s. 1217-1231
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Tidskriftsartikel (refereegranskat)abstract
- We study continuity properties of generalized Monge-Ampere operators for plurisubharmonic functions with analytic singularities. In particular, we prove continuity for a natural class of decreasing approximating sequences. We also prove a formula for the total mass of the Monge-Ampere measure of such a function on a compact Kahler manifold.
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| 8. |
- Andreasson, Håkan, 1966
(författare)
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On global existence for the spherically symmetric Einstein-Vlasov system in Schwarzschild coordinates
- 2007
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Ingår i: Indiana University Mathematics Journal. - : Indiana University Mathematics Journal. - 0022-2518. ; 56, s. 523-552
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Tidskriftsartikel (refereegranskat)abstract
- The spherically symmetric Einstein-Vlasov system in Schwarzschild coordinates (i.e., polar slicing and areal radial coordinate) is considered. An improved continuation criterion for global existence of classical solutions is given. Two other types of criteria which prevent finite time blow-up are also given.
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| 9. |
- Aronsson, Gunnar, et al.
(författare)
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An asymptotic model for compression molding
- 2002
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Ingår i: Indiana University Mathematics Journal. - 0022-2518 .- 1943-5258. ; 51:1
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Tidskriftsartikel (refereegranskat)abstract
- We discuss an idealized model for compression molding, had by taking an asymptotic limit for highly non-Newtonian materials. We interpret the changing pressure distributions as being dictated by a Monge-Kantorovich mass transfer on a fast time scale, and thereby derive a nonlocal geometric law of motion for the air/plastic interface.
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| 10. |
- Bergman, Alex
(författare)
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On cyclicity in de Branges-Rovnyak spaces
- 2023
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Ingår i: Indiana University Mathematics Journal. - 0022-2518.
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Tidskriftsartikel (refereegranskat)abstract
- We study the problem of characterizing the cyclic vec-tors in de Branges-Rovnyak spaces. Based on a description of theinvariant subspaces we show that the difficulty lies entirely in un-derstanding the subspace (aH2)⊥ and give a complete function the-oretic description of the cyclic vectors in the case dim(aH2)⊥ < ∞.Incidentally, this implies analogous results for certain generalizedDirichlet spaces D(μ). Most of our attention is directed to the in-finite case where we relate the cyclicity problem to describing theexposed points of H1 and provide several sufficient conditions. Anecessary condition based on the Aleksandrov-Clark measures of bis also presented.
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