The purpose of this series is to study the L^p (1 < p < ∞) boundedness of operators of the form f→ ψ(x) ∫ f(γ_t(x)) K(t) dt, where γ_t(x) is a C^∞ function defined on a neighborhood of the origin in (t,x)∈ R^Nx Rⁿ, satisfying γ₀(x)≡ x, ψ is a C^∞ cutoff function supported on a small neighborhood of 0∈ Rⁿ, and K is a "multi-parameter singular kernel" supported on a small neighborhood of 0∈ R^N. The goal is, given an appropriate class of kernels K, to give conditions on γ such that every operator of the above form is bounded on L^p (1 < p < ∞). Associated maximal functions are also studied. The case when K is a Calderón-Zygmund kernel was studied by Christ, Nagel, Stein, and Wainger; we generalize their conditions to the case when K has a "multi-parameter" structure. For example, when K is given by a "product kernel." Even when K is a Calderón-Zygmund kernel, our methods yield some new results. The first paper deals with the L² theory, the second paper deals with the L^p theory, while the third paper deals with the special case when γ is real analytic. The announcement gives an overview of the theory in a simpler special case.

We consider the problem of recovering the coefficient σ(x) of the elliptic equation ∇ •(σ ∇ u)=0 in a body from measurements of the Cauchy data on possibly very small subsets of its surface. We give a constructive proof of a uniqueness result by Kenig, Sjöstrand, and Uhlmann. We construct a uniquely specified family of solutions such that their traces on the boundary can be calculated by solving an integral equation which involves only the given partial Cauchy data. The construction entails a new family of Green's functions for the Laplacian, and corresponding single layer potentials, which may be of independent interest.

Multi-parameter Carnot-Carathéodory balls are studied, generalizing results due to Nagel, Stein, and Wainger in the single parameter setting. The main technical result is seen as a uniform version of the theorem of Frobenius. In addition, maximal functions associated to certain multi-paramter families of Carnot-Carathéodory balls are also studied.

Kohn constructed examples of sums of squares of complex vector fields satisfying Hörmander's condition that lose derivatives, but are nevertheless hypoelliptic. He also demonstrated optimal L² regularity. In this paper, we construct parametricies for Kohn's operators, which lead to the corresponding L^p (1 < p < ∞) and Lipschitz regularity. In fact, our parametrix construction generalizes to a somewhat larger class of operators, yielding some new examples of operators which are hypoelliptic, but lose derivatives. This is the paper version of my thesis, which was done under the supervision of Eli Stein.

Recently, Nagel and Stein studied the ☐_b heat equation, where ☐_b is the Kohn Laplacian on the boundary of a weakly pseudoconvex domain of finite type in C². They showed that the Schwartz kernel of e^-t☐_b satisfies good "off-diagonal" estimates, while that of e^-t☐_b-π satisfies good "on-diagonal" estimates, where π denotes the Szegö projection. We offer a simple proof of these results, which easily generalizes to other, similar situations. Our methods involve adapting the well-known relationship between the heat equation and the finite propagation speed of the wave equation to this situation. In addition, we apply these methods to study multipliers of the form m(☐_b). In particular, we show that m(☐_b) is an NIS operator, where m satisfies an appropriate Mihlin−Hörmander condition.

We present an intrinsically defined algebra of operators containing the right and left invariant Calderón−Zygmund operators on a stratified group. The operators in our algebra are pseudolocal and bounded on L^p (1<p<∞). This algebra provides an example of an algebra of singular integrals that falls outside of the classical Calderón−Zygmund theory.

This is an announcement of the paper "A parametrix for Kohn's operator" listed above. It contains a simplified example.