Academic literature on the topic 'Cowling-Haagerup type of formula'

We describe the tube algebra and its representations in the cases of diagonal and Bisch–Haagerup subfactors possibly with a scalar [Formula: see text]-cocycle obstruction. We show that these categories are additively equivalent to the direct product over conjugacy classes of representation category of a centralizer subgroup (corresponding to the conjugacy class) twisted by a scalar [Formula: see text]-cocycle obtained from the [Formula: see text]-cocycle obstruction.

We obtain several versions of the Hausdorff–Young and Hardy–Littlewood inequalities for the [Formula: see text]-generalized Fourier transform recently investigated at length by Ben Saïd, Kobayashi, and Ørsted. We also obtain a number of weighted inequalities — in particular Pitt’s inequality — that have application to uncertainty principles. Specifically we obtain several analogs of the Heisenberg–Pauli–Weyl principle for [Formula: see text]-functions, local Cowling–Price-type inequalities, Donoho–Stark-type inequalities and qualitative extensions. We finally use the Hausdorff–Young inequality as a means to obtain entropic uncertainty inequalities.

We define the notions of weak amenability and the Cowling–Haagerup constant for extremal finite index subfactors s of type II1. We prove that the Cowling–Haagerup constant only depends on the standard invariant of the subfactor. Hence, we define the Cowling–Haagerup constant for standard invariants. We explicitly compute the constant for Bisch–Haagerup subfactors and prove that it is equal to the constant of the group involved in the construction. Given a finite family of amenable standard invariants, we prove that their free product in the sense of Bisch–Jones is weakly amenable with constant 1. We show that the Cowling–Haagerup of the tensor product of a finite family of standard invariants is equal to the product of their Cowling–Haagerup constants.

Let A be a symmetrizable affine or hyperbolic generalized Cartan matrix. Let G be a locally compact Kac–Moody group associated to A over a finite field 𝔽q. We suppose that G has type ∞, that is, the Weyl group W of G is a free product of ℤ/2ℤ's. This includes all locally compact Kac–Moody groups of rank 2 and three possible locally compact rank 3 Kac–Moody groups of noncompact hyperbolic type. For every prime power q, we give a sufficient condition for the rank 2 Kac–Moody group G to contain a cocompact lattice [Formula: see text] with quotient a simplex, and we show that this condition is satisfied when q = 2s. If further Mq and [Formula: see text] are abelian, we give a method for constructing an infinite descending chain of cocompact lattices … Γ3 ≤ Γ2 ≤ Γ1 ≤ Γ. This allows us to characterize each of the quotient graphs of groups Γi\\X, the presentations of the Γi and their covolumes, where X is the Tits building of G, a homogeneous tree. Our approach is to extend coverings of edge-indexed graphs to covering morphisms of graphs of groups with abelian groupings. This method is not specific to cocompact lattices in Kac–Moody groups and may be used to produce chains of subgroups acting on trees in a general setting. It follows that the lattices constructed in the rank 2 Kac–Moody group have the Haagerup property. When q = 2 and rank (G) = 3 we show that G contains a cocompact lattice Γ′1 that acts discretely and cocompactly on a simplicial tree [Formula: see text]. The tree [Formula: see text] is naturally embedded in the Tits building X of G, a rank 3 hyperbolic building. Moreover Γ′1 ≤ Λ′ for a non-discrete subgroup Λ′ ≤ G whose quotient Λ′ \ X is equal to G\X. Using the action of Γ′1 on [Formula: see text] we construct an infinite descending chain of cocompact lattices …Γ′3 ≤ Γ′2 ≤ Γ′1 in G. We also determine the quotient graphs of groups [Formula: see text], the presentations of the Γ′i and their covolumes.

This thesis consists of two broad themes. First one revolves around the Hardy's inequalities for the fractional power of Grushin operator $\G$ which is chased via two different approaches. In the first approach, we first prove Hardy's inequality for the generalized sublaplacian defined on $\R\times\R^+$, then using the spherical harmonics, applying Hardy's inequality for individual components, we derive Hardy's inequality for Grushin operator. The techniques used for deriving Hardy's inequality for generalized sublaplacian are in parallel with the ones used in \cite{thangaveluroncal}. We first find Cowling-Haagerup type of formula for the fractional generalised sublaplacian and then using the modified heat kernel, we find integral representations of fractional generalized sublaplacian. Then we derive Hardy's inequality for generalized sublaplacian. In the second approach, we start with an extension problem for Grushin, with initial condition $f\in L^p(\R^{n+1})$. We derive a solution $u(\cdot,\rho)$ to that extension problem and show that solution goes to $f$ in $L^p(\R^{n+1})$ as the extension variable $\rho$ goes to $0$. Further $-\rho^{1-2s}\partial_\rho u $ goes to $B_s\G_s f$ in $L^p(\R^{n+1})$ as $\rho$ goes to $0$, thereby giving us an another way of defining fractional powers of Grushin operator $\G_s$. We also derive trace Hardy inequality for the Grushin operator with the help of extension problem. Finally we prove $L^p$-$L^q$ inequality for fractional Grushin operator, thereby deriving Hardy-Littlewood-Sobolov inequality for the Grushin operator.\\ Second theme consists of Hermite multipliers on modulation spaces $M^{p,q}(\R^n)$. We find a relation between sublaplacian multipliers $m(\tilde{\L})$ on polarised Heisenberg group $\Hb^n_{pol}$ and Hermite multipliers $m(\H)$ on modulation spaces $M^{p,q}(\R^n)$, thereby deriving the conditions on the multipliers $m$ to be Hermite multipliers on modulation spaces. We believe that the conditions on multipliers that we have found are more strict than required. We improve the results for the case the modulation spaces $M^{p,q}(\R^n)$ have $p=q$ by finding a relation between the boundedness of Hermite multipliers on $M^{p,p}$ and the boundedness of Fourier multipliers on torus $\T^n$. We also derive the conditions for boundedness of the solution of wave equation related to Hermite and the solution of Schr\"odinger equation related to Hermite on modulation spaces.
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