88 research outputs found
Schwartz correspondence for real motion groups in low dimensions
No full textFor a Gelfand pair (G, K) with G a Lie group of polynomial growth and K a compact subgroup, the Schwartz correspondence states that the spherical transform maps the bi-K-invariant Schwartz space S(K\G/K) isomorphically onto the space S(Sigma_D), where Sigma_D is an embedded copy of the Gelfand spectrum in R^ell, canonically associated to a generating system D of G-invariant differential operators on G/K, and S(Sigma_D) consists of restrictions to Sigma_D of Schwartz functions on R^ell. Schwartz correspondence is known to hold for a large variety of Gelfand pairs of polynomial growth. In this paper we prove that it holds for the strong Gelfand pair (Mn,SOn) with n=3,4. The rather trivial case n=2 is included in previous work by the same authors
Schwartz correspondence for the complex motion group on C-2
No full textIf (G, K) is a Gelfand pair, with G a Lie group of polynomial growth and K a compact subgroup of G, the Gelfand spectrum & sigma; of the bi-K-invariant algebra L1(K\G/K) admits natural embeddings into Rt spaces as a closed subset.For any such embedding, define S(& sigma;) as the space of restrictions to & sigma; of Schwartz functions on Rt. We call Schwartz correspondence for (G, K) the property that the spherical transform is an isomorphism of S(K\G/K) onto S(& sigma;).In all the cases studied so far, Schwartz correspondence has been proved to hold true. These include all pairs with G = K H and K abelian and a large number of pairs with G = K & alpha; H and H nilpotent.We prove Schwartz correspondence for the pair (U2 IX M2(C), U2), where M2(C) is the complex motion group and U2 = K acts on it by conjugation. Our proof goes through a detailed analysis of (M2(C), U2) as a strong Gelfand pair and reduction of the problem to Schwartz correspondence for each K-type & tau; & ISIN; K ⠂ with appropriate control of the estimates in terms of & tau;
The Cayley transform and uniformly bounded representations
No full textLet G be a simple Lie group of real rank one, with Iwasawa decomposition KA \bar N and Bruhat
big cell NMA\bar N: Then the space G/MA \bar N may be (almost) identified with N and with K /M,
and these identifications induce the (generalised) Cayley transform C : N \to K /M. We show
that C is a conformal map of Carnot–Caratheodory manifolds, and that composition with the
Cayley transform, combined with multiplication by appropriate powers of the Jacobian,
induces isomorphisms of Sobolev spaces on N
and on K/M. We use this to construct
uniformly bounded and slowly growing representations of G
On the Schwartz correspondence for Gelfand pairs of polynomial growth
No full textLet (G;K) be a Gelfand pair, with G a Lie group of polynomial growth, and let Σ ⊂ Rl be a homeomorphic image of the Gelfand spectrum, obtained by choosing a generating system D1; . . . ;Dl of G-invariant di erential operators on G=K and associating to a bounded spherical function φ the l-tuple of its eigenvalues under the action of the Dj 's. We say that property (S) holds for (G;K) if the spherical transform maps the bi-K-invariant Schwartz space S(KnG=K) isomorphically onto S(Σ), the space of restrictions to Σ of the Schwartz functions on Rl. This property is known to hold for many nilpotent pairs, i.e., Gelfand pairs where G = K⋊N, with N nilpotent. In this paper we enlarge the scope of this analysis outside the range of nilpotent pairs, stating the basic setting for general pairs of polynomial growth and then focussing on strong Gelfand pairs
The Schwartz correspondence for the complex motion group on
No full textIf is a Gelfand pair, with a Lie group of polynomial growth and
a compact subgroup of , the Gelfand spectrum of the
bi--invariant algebra admits natural embeddings into
spaces as a closed subset.
For any such embedding, define as the space of
restrictions to of Schwartz functions on . We call
Schwartz correspondence for the property that the spherical transform
is an isomorphism of onto .
In all the cases studied so far, Schwartz correspondence has been proved to
hold true. These include all pairs with and abelian and a
large number of pairs with and nilpotent.
In this paper we study what is probably the simplest of the pairs with
, non-abelian and non-nilpotent, with , the complex motion group, and acting on it by inner
automorphisms
Utility of serological screening for measles, mumps and rubella in immunocompromised patients
No full textMarchi et al. in their article (Measles in pregnancy: a threat for Italian women? Hum Vaccin Immunother. 2019 Jun 20:1–3) observed that 96.9% of pregnant women were positive for anti-measles IgG (with a higher risk of contracting measles in those aged 19–29 years) emphasizing the importance of serological screening before pregnancy. We evaluated seroprotection/seropositivity rates to Measles, Mumps and Rubella in 324 adults with an acquired immune-deficiency needing an immunization program. We found that younger patients (20–29 years) had a seroprevalence below 85%. Overall, a relevant proportion (21.6%, 54/250) of patients was susceptible to at least one infection needing immunization. Our results confirm the usefulness of proper strategies for identifying individuals susceptible to vaccine-preventable infections and protecting them through vaccination
Uniformly bounded representations and completely bounded multipliers of Sl(2,ℝ)
No full textWe estimate the norms of many matrix coefficients of irreducible uniformly bounded representations of SL(2,ℝ) as completely bounded multipliers of the Fourier algebra. Our results suggest that the known inequality relating the uniformly bounded norm of a representation and the completely bounded norm of its coefficients may not be optimal
- …
