Sub-elliptic boundary value problems in flag domains

Abstract

A flag domain in R 3 is a subset of R 3 of the form tpx, y, tq : y ă Apxqu, where A : R Ñ R is a Lipschitz function. We solve the Dirichlet and Neumann problems for the sub-elliptic Kohn-Laplacian △ 5 " X 2 `Y 2 in flag domains Ω Ă R 3 , with L 2 -boundary values. We also obtain improved regularity for solutions to the Dirichlet problem if the boundary values have first order L 2 -Sobolev regularity. Our solutions are obtained as sub-elliptic single and double layer potentials, which are best viewed as integral operators on the first Heisenberg group. We develop the theory of these operators on flag domains, and their boundaries. CONTENTS1. Introduction 1.1. Dirichlet and Neumann problems 1.2. The regularity problem 1.3. Uniqueness in the Dirichlet problem 1.4. Previous work 1.5. Open problems 1.6. Acknowledgements 2. Preliminaries 2.1. The Heisenberg group 2.2. Surface measures, normal and tangent vectors 2.3. Summary of key operators 3. Principal values of the Riesz transform 3.1. Existence of R t pf νq 3.2. Existence of R t pf τ q 3.3. Existence of Rpf νq and Rpf τ q 3.4. The choice of truncations is irrelevant 4. Various maximal functions 4.1. Heisenberg cones and intrinsic Lipschitz graphs 4.2. Non-tangential and radial limits and maximal functions 5. Boundary behaviour of the Riesz transform 6. The divergence theorem and some corollaries 7. Vertical distributions and T -multipliers

Schauder estimates up to the boundary on H-type groups: an approach via the double layer potential

“…In the present situation, as mentioned above, the situation is more dramatic since the double layer potential is singular even on smooth domains. Fortunately in the present work we do not need to develop an L 2 theory for singular integrals in the sub-Riemannian setting (as Orponen and Villa did in [19]), but we only need to prove that K has small C 2,α norm with respect to 1 2 in order to use the continuity method developed by [22] (see also [16, page 56]). Our argument develops a method of reflection, special to the sub-Riemannian setting, which we believe will be proved to be useful for other problems when the lack of commutativity is critical.…”