Function spaces generated by blocks associated with spheres, Lie groups and spaces of homogeneous type

The geometric quantization on a homogeneous manifold is studied. For any quantizable function f , the stochastical expression for the unitary group exp(i tQ λ(f)) generated by the quantized operator Q λ( f ) is established. As an application, a Feynman-Kac formula for the compact semisimple Lie group is rederived.

The authors introduce a class of generalized Riesz potentials with kernels having weak regularity on spaces of homogeneous type in the sense of Coifman and Weiss and establish their boundedness on Lebesgue spaces and Hardy spaces. As applications, the authors obtain the boundedness on Lebesgue spaces and Hardy spaces of commutators generated by Lipschitz functions and generalized Riesz potentials or Calderon-Zygmund operators with kernels having weak regularity on spaces of homogeneous type. Mathematics subject classification (2010): 31C15, 42B20, 47B47, 42B30, 43A99.

Hardy space theory on spaces of homogeneous type via orthonormal wavelet bases

Let [Formula: see text] be a compact Lie group acting on a smooth manifold [Formula: see text]. In this paper, we consider Meinrenken’s [Formula: see text]-equivariant bundle gerbe connections on [Formula: see text] as objects in a 2-groupoid. We prove this 2-category is equivalent to the 2-groupoid of gerbe connections on the differential quotient stack associated to [Formula: see text], and isomorphism classes of [Formula: see text]-equivariant gerbe connections are classified by degree 3 differential equivariant cohomology. Finally, we consider the existence and uniqueness of conjugation-equivariant gerbe connections on compact semisimple Lie groups.

An efficient construction of lattice points FM of any density and any admissible symmetry is described in a finite region F⊃FM of a real n‐dimensional Euclidean space. The shape of F and the lattice symmetry of FM is determined by a compact semisimple Lie group of rank n underlying the construction. The density of FM is fixed by our choice of a positive integer M, where 1⩽M<∞. The Lie group allows one to introduce systems of special functions discretely orthogonal on FM together with the possibility of expanding digital data on FM into finite series of such special functions.

Flat connections on punctured surfaces and geodesic polygons in a Lie group

If K is a connected compact Lie group with simple Lie algebra and if k is an integer relatively prime to the order of the Weyl group W of K then the number ν ( K , k ) \nu (K,k) of conjugacy classes of K consisting of elements x satisfying x k = 1 {x^k} = 1 is given by \[ ν ( K , k ) = ∏ i = 1 l m i + k m i + 1 , \nu (K,k) = \prod \limits _{i = 1}^l {\frac {{{m_i} + k}}{{{m_i} + 1}},} \] where l is the rank of K and m 1 , … , m l {m_1}, \ldots ,{m_l} are the exponents of W. If G is the complexification of K then we have ν ( G , k ) = ν ( K , k ) \nu (G,k) = \nu (K,k) without any restriction on k.

Fractional integrals, potential operators and two-weight, weak type norm inequalities on spaces of homogeneous type

In this note we combine the dyadic families introduced by M. Christ in (Colloq. Math. 60/61(2):601–628, 1990) and the discrete partitions introduced by J.M. Wu in (Proc. Am. Math. Soc. 126(5):1453–1459, 1998) to get approximation of a compact space of homogeneous type by a uniform sequence of finite spaces of homogeneous type. The convergence holds in the sense of a metric built on the Hausdorff distance between compact sets and on the Kantorovich-Rubinshtein metric between measures.

In this article, we obtain a local Tb theorem for singular integral operators on spaces of homogeneous type by using tree selection algorithm of the dyadic model and the BCR algorithm, which extends an earlier result of M. Christ [Colloq. Math. 60/61 (1990), 601–628].

Boundedness of the Commutators of Fractional Maximal Operator on Morrey-Herz Space on Spaces of Homogeneous Type

By applying the remarkable orthonormal basis constructed recently by Ausher and Hytönen on spaces of homogeneous type in the sense of Coifman and Weiss, pointwise multipliers of inhomogeneous Besov and Triebel-Lizorkin spaces are obtained. We make no additional assumptions on the quasi-metric or the doubling measure. Hence, the results of this paper extend earlier related results to a more general setting.

Weighted norm inequalities for the maximal singular integral operators on spaces of homogeneous type

In this note the classical embedding theorem for the Besov and Triebel-Lizorkin spaces on R n {R^n} is generalized to the Besov and Triebel-Lizorkin spaces on spaces of homogeneous type. The proof is new even for R n {R^n} case.

Let (𝒳, d,μ) be a space of homogeneous type in the sense of Coifman and Weiss. Assuming that μ satisfies certain estimates from below and there exists a suitable Calderón reproducing formula in L 2(𝒳), the authors establish a Lusin‐area characterization for the atomic Hardy spaces H p at(𝒳) of Coifman and Weiss for p ∈ (p 0, 1], where p 0 = n /(n + ε 1) depends on the “dimension” n of 𝒳 and the “regularity” ε 1 of the Calderón reproducing formula. Using this characterization, the authors further obtain a Littlewood–Paley g *λ ‐function characterization for Hp (𝒳) when λ &gt; n + 2n /p and the boundedness of Calderón–Zygmund operators on Hp (𝒳). The results apply, for instance, to Ahlfors n ‐regular metric measure spaces, Lie groups of polynomial volume growth and boundaries of some unbounded model domains of polynomial type in ℂN . (© 2006 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)