Gelfand pairs on the Heisenberg group and Schwartz functions

Let Hn be the (2n+1)-dimensional Heisenberg group and K a compact group of automorphisms of Hn such that (K?Hn,K) is a Gelfand pair. We prove that the Gelfand transform is a topological isomorphism between the space of K-invariant Schwartz functions on Hn and the space of Schwartz function on a closed subset of Rs homeomorphic to the Gelfand spectrum of the Banach algebra of K-invariant integrable functions on Hn.     

Gelfand transforms of polyradial Schwartz functions on the Heisenberg group

We prove that the Gelfand transform is a topological isomorphism between the space of polyradial Schwartz functions on the Heisenberg group and the space of Schwartz functions on the Heisenberg brush. We obtain analogous results for radial Schwartz functions on Heisenberg type groups.     

Fourier multipliers for Sobolev spaces on the Heisenberg group

In this paper, it is shown that the class of right Fourier multipliers for the Sobolev space W ^k,p (H ⁿ ) coincides with the class of right Fourier multipliers for L ^p (H ⁿ ) for k ∈ ?, 1 < p < ∞. Towards this end, it is shown that the operators R _j $ \bar R $ _j ?^?1 and $ \bar R $ _j R _j ?^?1 are bounded on L ^p (H ⁿ ), 1 < p < ∞, where $$ R_j = \frac{\partial } {{\partial z_j }} - \frac{i} {4}\bar z_j \frac{\partial } {{\partial t}}, \bar R_j = \frac{\partial } {{\partial \bar z_j }} + \frac{i} {4}z_j \frac{\partial } {{\partial t}} $$ and ? is the sublaplacian on H ⁿ . This proof is based on the Calderon-Zygmund theory on the Heisenberg group. It is also shown that when p = 1, the class of right multipliers for the Sobolev space W ^k,1(H ⁿ ) coincides with the dual space of the projective tensor product of two function spaces.     

Fourier transform of symmetric gauss measures on the Heisenberg group

An explicit form is derived for the Fourier transform of symmetric Gauss measures on the Heisenberg group at the Schr&ouml;dinger representation. Using this explicit formula, necessary and sufficient conditions are given for the convolution of two symmetric Gauss measures to be a symmetric Gauss measure and for commutability of two symmetric Gauss measures. Moreover, necessary and sufficient conditions are presented for the convolution of two symmetric Gauss convolution semigroups to be a convolution semigroup.     

The group Fourier transforms and multipiers of the Hardy spaces on the Heisenberg group

We give the estimate for the growth of the group Fourier transforms of the Hardy spaces and establish and H^p-multipleir theorem satisfying Michlin condition on the Heisenberg group.     

Shannon multiresolution analysis on the Heisenberg group

We present a notion of frame multiresolution analysis on the Heisenberg group, abbreviated by FMRA, and study its properties. Using the irreducible representations of this group, we shall define a sinc-type function which is our starting point for obtaining the scaling function. Further, we shall give a concrete example of a wavelet FMRA on the Heisenberg group which is analogous to the Shannon MRA on R.     

Quantum Heisenberg group and reflection equation. I

The quantum group of upper triangular matrices and the Hopf algebra associated with the reflection equation are constructed for the R-matrix corresponding to the quantum Heisenberg group. Bibliography: 12 titles. Dedicated to P. P. Kulish on the occasion of his 50th birthday Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 209, 1994, pp. 28–36. Translated by E. V. Damanskinsky.     

Tempered distributions on Heisenberg groups whose convolution with Schwartz class functions is Schwartz class

Let N be a nilpotent Lie group and Q a tempered distribution on N. We say that Q is a left $\text{J}$-multiplier if convolution on the left by Q takes Schwartz class functions to Schwartz class functions; there is a similar definition for right $\text{J}$-multipliers. We show that if ? is an irreducible unitary representation of N, then one can define ρ(Q):$\text{H}$^∞:(ρ)→$\text{H}$^∞:(ρ) whenever Q is a left $\text{J}$-multiplier. The main results of the paper characterize left $\text{J}$-multipliers Q on Heisenberg groups in terms of the transform operators ?(Q) and show how this characterization can be used to find fundamental solutions of some left invariant differential operators. There is also an example of a left $\text{J}$-multiplier which is not a right $\text{J}$-multiplier.     

Solvability on the Heisenberg group

We solve in various spaces the linear equations L_αg = f , where L_α belongs to a class of transversally elliptic second order differential operators on the Heisenberg group with double characteristics and complex‐valued coefficients, not necessarily locally solvable. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Hypoellipticity on the Heisenberg group

Let P be a left-invariant differential operator on the Heisenberg group Hⁿ, P homogeneous with respect to the dilations on Hⁿ. We show that a necessary and sufficient condition for the hypoellipticity of P is that π(P) be an injective operator for every irreducible unitary representation π of Hⁿ (except the trivial representation). Furthermore, hypoellipticity is preserved if the homogeneous operator P is perturbed by terms of lower order of homogeneity. (Homogeneity means homogeneity with respect to dilations of Hⁿ.) It is also shown that if P is homogeneous, left-invariant and hypoelliptic on Hⁿ, then its formal adjoint is hypoelliptic.     

Boundedness of the fractional maximal operator in generalized Morrey space on the Heisenberg group

In this paper we study the fractional maximal operator M _α , 0 ≤ α < Q on the Heisenberg group ? _n in the generalized Morrey spaces M _{p, ?}(? _n ), where Q = 2n + 2 is the homogeneous dimension of ? _n . We find the conditions on the pair (? ₁, ? ₂) which ensures the boundedness of the operator M _α from one generalized Morrey space M _{p, ?1}(? _n ) to another M _{q, ?2}(? _n ), 1 < p < q < ∞, 1/p?1/q = α/Q, and from the space M _{1, ?1}(? _n ) to the weak space WM _{q, ?2}(? _n ), 1 < q < ∞, 1 ? 1/q = α/Q. We also find conditions on the φ which ensure the Adams type boundedness of M _α from $M_{p,\phi ^{\tfrac{1} {p}} } \left( {\mathbb{H}_n } \right)$ to $M_{q,\phi ^{\tfrac{1} {q}} } \left( {\mathbb{H}_n } \right)$ for 1 < p < q < ∞ and from M _{1, ?}(? _n ) to $WM_{q,\phi ^{\tfrac{1} {q}} } \left( {\mathbb{H}_n } \right)$ for 1 < q < ∞. As applications we establish the boundedness of some Schrödinger type operators on generalized Morrey spaces related to certain nonnegative potentials V belonging to the reverse Hölder class B _∞(” _n ).     

On the Schwartz space of the basic affine space

Let G be the group of points of a split reductive algebraic group G over a local field k and let X = G / U where U is the group of k-points of a maximal unipotent subgroup of G. In this paper we construct a certain canonical G-invariant space (called the Schwartz space of X) of functions on X, which is an extension of the space of smooth compactly supported functions on X. We show that the space of all elements of , which are invariant under the Iwahori subgroup I of G, coincides with the space generated by the elements of the so called periodic Lusztig basis, introduced recently by G. Lusztig (cf. [10] and [11]). We also give an interpretation of this space in terms of a certain equivariant K-group (this was also done by G. Lusztig — cf. [12]). Finally we present a global analogue of , which allows us to give a somewhat non-traditional treatment of the theory of the principal Eisenstein series.     

The Dirichlet problem for the Kohn Laplacian on the Heisenberg group,I

For (x,y,t)∈$\text{R}$ⁿ × $\text{R}$ⁿ × $\text{R}$, denote $\text{X}{}_{\mathrm{j}}\text{=}\text{?}\text{?x}{}_{\mathrm{j}}\text{+ 2y}{}_{\mathrm{j}}\text{?}\text{?t}\text{, y}{}_{\mathrm{j}}\text{=}\text{?}\text{?y}{}_{\mathrm{j}}\text{? 2x}{}_{\mathrm{j}}\text{?}\text{?t}$ and $\text{L}{}_{\mathrm{\alpha }}\text{=?}\text{1}\text{4}\text{∑}\text{j=1}\text{n}\text{X}{}_{\mathrm{j}}{}^2\text{+ Y}{}_{\mathrm{j}}{}^2\text{+}\text{?}\text{?t}$. When α = n ? 2q, $\text{L}$_a represents the action of the Kohn Laplacian □_b on q-forms on the Heisenberg group. For ?n < α < n, we construct a parametrix for the Dirichlet problem in smooth domains D near non-characteristic points of ?D. A point w of ?D is non-characteristic if one of X₁,…, X_n, Y₁,…, Y_n is transverse to ?D at w. This yields sharp local estimates in the Dirichlet problem in the appropriate non-isotropic Lipschitz classes. The main new tool is a “convolution calculus” of pseudo-differential operators that can be applied to the relevant layer potentials, for which the usual asymptotic composition formula is false. Characteristic points are treated in Part II.     

Convex functions on the Heisenberg group

Convex functions in Euclidean space can be characterized as universal viscosity subsolutions of all homogeneous fully nonlinear second order elliptic partial differential equations. This is the starting point we have chosen for a theory of convex functions on the Heisenberg group.Received: 5 July 2002, Accepted: 24 October 2002, Published online: 6 June 2003Mathematics Subject Classification (1991): 49L25, 35J70, 35J67, 22E30Guozhen Lu: First author supported by US NSF grant DMS-9970352Juan J. Manfredi: Second author supported by US NSF grant DMS-0100107Bianca Stroffolini: Third author was supported by G.N.A.M.P.A. and by the 2002 projectPartial Differential Equations and Control Theory     

Hausdorff operators on the Heisenberg group

This paper is devoted to the high-dimensional and multilinear Hausdorff operators on the Heisenberg group H n. The sharp bounds for the strong type(p, p)(1 ≤ p ≤∞) estimates of n-dimensional Hausdorff operators on H n are obtained. The sharp bounds for strong(p, p) estimates are further extended to multilinear cases. As an application, we derive the sharp constant for the multilinear Hardy operator on H n. The weak type(p, p)(1 ≤ p ≤∞) estimates are also obtained.     

Hardy-Hausdorff spaces on the Heisenberg group

In this paper, by using the tent spaces on the Siegel upper half space, which are defined in terms of Choquet integrals with respect to Hausdorff capacity on the Heisenberg group, the Hardy-Hausdorff spaces on the Heisenberg group are introduced. Then, by applying the properties of the tent spaces on the Siegel upper half space and the Sobolev type spaces on the Heisenberg group, the atomic decomposition of the Hardy-Hausdorff spaces is obtained. Finally, we prove that the predual spaces of Q spaces on the Heisenberg group are the Hardy-Hausdorff spaces.     

Stable distributions on the Heisenberg group

Translated from Problemy Ustoichivosti Stokhasticheskikh Modelei, Trudy Seminara, pp. 128–131, 1987.