Der-Chen Chang

Hardy spaces, BMO, and boundary value problems for the Laplacian on a smooth domain in ^{}

We study two different local Hp spaces, 0 < p ≤ 1, on a smooth domain in Rn, by means of maximal functions and atomic decomposition. We prove the regularity in these spaces, as well as in the corresponding dual BMO spaces, of the Dirichlet and Neumann problems for the Laplacian. 0. Introduction Let Ω be a bounded domain in R, with smooth boundary. The L regularity of elliptic boundary value problems on Ω, for 1 < p < ∞, is a classical result in the theory of partial differential equations (see e.g. [ADN]). In the situation of the whole space without boundary, i.e. where Ω is replaced by R, the results for L, 1 < p < ∞, extend to the Hardy spaces H when 0 < p ≤ 1 and to BMO. Thus it is a natural question to ask whether the L regularity of elliptic boundary value problems on a domain Ω has an H and BMO analogue, and what are the H and BMO spaces for which it holds. This question was previously studied in [CKS], where partial results were obtained and were framed in terms of a pair of spaces, hr(Ω) and h p z(Ω). These spaces, variants of those defined in [M] and [JSW], are, roughly speaking, the “largest” and “smallest” h spaces that can be associated to a domain Ω. Our purpose here is to substantially extend the previous results by determining those h spaces on Ω which are particularly applicable to boundary value problems. These spaces allow one to prove sharp results (preservation of the appropriate h spaces) for all values of p, 0 < p ≤ 1, as well as the preservation of corresponding spaces of BMO functions. 0.1. Motivation and statement of results. There are two approaches to defining the appropriate Hardy spaces on Ω. Recall that the spaces H(R), for p < 1, are spaces of distributions. Thus one approach is to look at the problem from the point of view of distributions on Ω. If we denote by D(Ω) the space of smooth functions with compact support in Ω, and by D′(Ω) its dual, we can consider the space of distributions in D′(Ω) which are the restriction to Ω of distributions in H(R) (or in h(R), the local Hardy spaces defined in [G].) These spaces were studied in [M] (for arbitrary open sets) and in [CKS] (for Lipschitz domains), where they were denoted hr(Ω) (the r stands for “restriction”.) While one is able to prove regularity results for the Dirichlet problem for these spaces when p is near 1 (see [CKS]), these spaces are no longer appropriate when p Received by the editors September 5, 1996 and, in revised form, March 20, 1997. 1991 Mathematics Subject Classification. Primary 35J25, 42B25; Secondary 46E15, 42B30. c ©1999 American Mathematical Society

Geometric analysis on quaternion ℍ-type groups

We construct some examples of ℍ-types Carnot groups related to quaternion numbers and study their geometric properties. We involve the Hamiltonian formalism to obtain the equations of geodesics and calculate the cardinality of geodesics joining two different points on these groups. We prove Kepler’s law and give a nice geometric interpretation of the length of geodesies.

A note on weighted Bergman spaces and the Cesáro operator

Abstract. Let B denote the unit ball in C n , and dV (z) normalized Lebesgue measure on B. For α > −1, define dV (z) = (1−|z|)dV (z). Let H(B) denote the space of holomorhic functions on B, and for 0 < p < ∞, let A(dVα) denote L(dVα) ∩ H(B). In this note we characterize A (dVα) as those functions in H(B) whose images under the action of a certain set of differential operators lie in L(dVα). This is valid for 1 ≤ p < ∞. We also show that the Cesàro operator is bounded on A(dVα) for 0 < p < ∞. Analogous results are given for the polydisc.

Heat Kernels for Elliptic and Sub-elliptic Operators

Part I. Traditional Methods for Computing Heat Kernels.- Introduction.- Stochastic Analysis Method.- A Brief Introduction to Calculus of Variations.- The Path Integral Approach.- The Geometric Method.- Commuting Operators.- Fourier Transform Method.- The Eigenfunctions Expansion Method.- Part II. Heat Kernel on Nilpotent Lie Groups and Nilmanifolds.- Laplacians and Sub-Laplacians.- Heat Kernels for Laplacians and Step 2 Sub-Laplacians.- Heat Kernel for Sub-Laplacian on the Sphere S^3.- Part III. Laguerre Calculus and Fourier Method.- Finding Heat Kernels by Using Laguerre Calculus.- Constructing Heat Kernel for Degenerate Elliptic Operators.- Heat Kernel for the Kohn Laplacian on the Heisenberg Group.- Part IV. Pseudo-Differential Operators.- The Psuedo-Differential Operators Technique.- Bibliography.- Index.

FUNDAMENTAL SOLUTIONS FOR HERMITE AND SUBELLIPTIC OPERATORS

In this article, we introduce a geometric method based on multipliers to compute heat kernels for operators with potentials. Using the heat kernel, we compute the fundamental solution for the Hermite operator with singularity at an arbitrary point on Euclidean space and on Heisenberg groups. As a consequence, we obtain the fundamental solutions for the sub-laplacian □J in a family of quadratic submanifolds.

Estimates for powers of sub-Laplacian on the non-isotropic Heisenberg group

AbstractAssume that

Hopf fibration: Geodesics and distances

{\mathcal{L}}_\alpha = - \frac{1}{2}\sum\nolimits_{j = 1}^n {(Z_j \bar Z_j + \bar Z_j Z_j ) + i\alpha T}

Sobolev and Lipschitz estimates for weighted Bergman projections

is the sub-Laplacian on the nonisotropic Heisenberg groupHn;Zj,Zjfor j = 1, 2, …,n andTare the basis of the Lie algebra hn.We apply the Laguerre calculus to obtain the explicit kernel for the fundamental solution of the powers of Lαand the heat kernel exp{−sLα}.Estimates for this kernel in various function spaces can be deduced easily.