Fourier analysis on the Heisenberg group. I. Schwartz space

We derive a usable characterization of the group FT (Fourier Transform) of Schwartz space on the Heisenberg group $\text{H}$ⁿ, in terms of certain asymptotic series. To accomplish this we study in detail the FT of multiplication and differentiation operators on $\text{H}$ⁿ, the relation of multiple Fourier series to the FT, and the process of group contraction on $\text{H}$ⁿ. We use our characterization to solve a form of the division problem for convolution of $\text{H}$ⁿ, which has application to Hardy space theory.     

Quantum dissipative systems. I. Canonical quantization and quantum Liouville equation

Sedov's variational principle, which is a generalization of the principle of least action to dissipative processes, is used to generalize canonical quantization and the von Neumann equations to dissipative systems. The example of a harmonic oscillator with friction is considered.Institute of Nuclear Physics, State University, Moscow. Translated from Teoreticheskaya i Matematicheskaya Fizika Vol. 100, No. 3, pp. 402–417, September, 1994.     

The prescribed p-mean curvature equation of low regularity in the Heisenberg group

This work reports on the author’s recent study about regularity and the singular set of a C ¹ smooth surface with prescribed p (or H)-mean curvature in the 3-dimensional Heisenberg group. As a differential equation, this is a degenerate hyperbolic and elliptic PDE of second order, arising from the study of CR geometry. Assuming only the p-mean curvature H ∈ C ⁰, it is shown that any characteristic curve is C ² smooth and its (line) curvature equals ?H. By introducing special coordinates and invoking the jump formulas along characteristic curves, it is proved that the Legendrian (horizontal) normal gains one more derivative. Therefore the seed curves are C ² smooth. This work also obtains the uniqueness of characteristic and seed curves passing through a common point under some mild conditions, respectively. In an on-going project, it is shown that the p-area element is in fact C ² smooth along any characteristic curve and satisfies a certain ordinary differential equation of second order. Moreover, this ODE is analyzed to study the singular set.     

An equation for the characteristic function of a stable distribution on the Heisenberg group

Translated fromProblemy Ustoichivosti Stokhasticheskikh Modelei. Trudy Seminara, 1988, pp. 137–140.     

Compensated compactness and the Heisenberg group

Work partially supported by the National Science Foundation     

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)     

Quasiconvexity in the Heisenberg group

We show that if A is a closed subset of the Heisenberg group whose vertical projections are nowhere dense, then the complement of A is quasiconvex. In particular, closed sets which are null sets for the cc-Hausdorff 3-measure have quasiconvex complements. Conversely, we exhibit a compact totally disconnected set of Hausdorff dimension three whose complement is not quasiconvex.     

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.     

Wiener measure for Heisenberg group

We build Wiener measure for the path space on the Heisenberg group by using of the heat kernel corresponding to the sub-Laplacian and give the definition of the Wiener integral. Then we give the Feynman-Kac formula.     

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.     

Quantum Heisenberg antiferromagnet on low-dimensional frustrated lattices

We investigate low-temperature magnetization processes for two frustrated quantum Heisenberg antiferromagnets using the lattice gas model. The emerging discrete degrees of freedom indicate a close relation of the frustrated quantum Heisenberg antiferromagnet to the classical lattice gas with a finite nearestneighbor repulsion or, equivalently, to the Ising antiferromagnet in a uniform magnetic field. Using this relation, we obtain analytic results for thermodynamically large systems in the one-dimensional case. In the two-dimensional case, we simulate systems of the size up to 100×100 sites using the classical Monte Carlo method.     

Rectifiability and perimeter in the Heisenberg group

In this paper, we fully extend to the Heisenberg group endowed with its intrinsic Carnot-Carathéodory metric and perimeter the classical De Giorgi's rectifiability divergence theorems. Received: 27 March 2000 / Revised version: 13 December 2000 / Published online: 24 September 2001