Maximally subelliptic partial differential equations (PDE) are a far-reaching generalization of elliptic PDE. Elliptic PDE hold a special place: sharp results are known for general linear and even fully nonlinear elliptic PDE. Over the past half-century, important results for elliptic PDE have been generalized to maximally subelliptic PDE. This text presents this theory and generalizes the sharp, interior regularity theory for general linear and fully nonlinear elliptic PDE to the maximally subelliptic setting.

This research monograph develops a new theory of "multi-parameter" singular integrals associated with Carnot-Carathéodory balls. The first two chapters give an introduction to the classical theory of Calderón-Zygmund singular integrals and applications to linear partial differential equations. The third chapter outlines the theory of mutli-parameter Carnot-Carathéodory geometry, where the main tool is a quantitative version of the classical theorem of Frobenius, as developed in the paper Multi-parameter Carnot-Carathéodory balls and the theorem of Frobenius listed below. The fourth chapter gives several examples of multi-parameter singular integrals which arise naturally in various problems. The fifth, and final, chapter develops a general theory of singular integrals which generalizes and unifies the examples in the fourth chapter.

We study fractional variants of the quasi-norms introduced by Brezis, Van Schaftingen, and Yung in the study of the Sobolev space Ẇ^1,p. The resulting spaces are identified as a special class of real interpolation spaces of Sobolev-Slobodeckiĭ spaces. We establish the equivalence between Fourier analytic definitions and definitions via difference operators acting on measurable functions. We prove various new results on embeddings and non-embeddings, and give applications to harmonic and caloric extensions. For suitable wavelet bases we obtain a characterization of the approximation spaces for best n-term approximation from a wavelet basis via smoothness conditions on the function; this extends a classical result by DeVore, Jawerth and Popov.

Let α>0, β>α, and let X₁,..., X_q be C^α vector fields on a C^α+1 manifold which span the tangent space at every point, where C^s denotes the Zygmund-Hölder space of order s. We give necessary and sufficient conditions for when there is a C^β+1 structure on the manifold, compatible with its C^α+1 structure, with respect to which X₁,..., X_q are C^β. This strengthens previous results of the first author which dealt with the setting α>1, β>max{α, 2}.

Given a finite collection of C¹ complex vector fields on a C² manifold M such that they and their complex conjugates span the complexified tangent space at every point, the classical Newlander-Nirenberg theorem gives conditions on the vector fields so that there is a complex structure on M with respect to which the vector fields are T^0,1. In this paper, we give intrinsic, diffeomorphic invariant, necessary and sufficient conditions on the vector fields so that they have a desired level of regularity with respect to this complex structure (i.e., smooth, real analytic, or have Zygmund regularity of some finite order). By addressing this in a quantitative way we obtain a holomorphic analog of the quantitative theory of sub-Riemannian geometry initiated by Nagel, Stein, and Wainger. We call this sub-Hermitian geometry. Moreover, we proceed more generally and obtain similar results for manifolds which have an associated formally integrable elliptic structure. This allows us to introduce a setting which generalizes both the real and complex theories.

As part of his celebrated Complex Frobenius Theorem, Nirenberg showed that given a smooth elliptic structure (on a smooth manifold), the manifold is locally diffeomorphic to an open subset of ℝ^rxℂⁿ (for some r and n) in such a way that the structure is locally the span of ^∂⁄_∂t1, …, ^∂⁄_∂tr, ^∂⁄_∂z1, …, ^∂⁄_∂zn; where ℝ^rxℂⁿ has coordinates (t₁, …, t_r, z₁, …, z_n). In this paper, we give optimal regularity for the coordinate charts which achieve this realization. Namely, if the manifold has Zygmund regularity of order s+2 and the structure has Zygmund regularity of order s+1 (for some s>0), then the coordinate chart may be taken to have Zygmund regularity of order s+2. We do this by generalizing Malgrange's proof of the Newlander-Nirenberg Theorem to this setting.

Given a finite collection of C¹ vector fields on a C² manifold which span the tangent space at every point, we consider the question of when there is locally a coordinate system in which these vector fields have a higher level of smoothness. We give necessary and sufficient conditions for when there is a coordinate system in which the vector fields are smooth, or real analytic, or have Zygmund regularity of some finite order. By addressing this in a quantitative way, we strengthen and generalize previous works on the quantitative theory of sub-Riemannian (aka Carnot-Carathéodory) geometry due to Nagel, Stein, and Wainger, Tao and Wright, and others (including the below paper Multi-parameter Carnot-Carathéodory balls and the theorem of Frobenius). Furthermore, we provide a diffeomorphism invariant version of these theories. In the first paper, we study a particular coordinate system adapted to a collection of vector fields (sometimes called canonical coordinates) and present results related to the above questions which are not quite sharp; these results from the backbone of the series. The methods of the first paper are based on techniques from ODEs. In the second paper, we use additional methods from PDEs to obtain the sharp results. In the third paper, we prove results concerning real analyticity and use methods from ODEs.

We establish optimal Lebesgue estimates for a class of generalized Radon transforms defined by averaging functions along polynomial-like curves. The presence of an essentially optimal weight allows us to prove uniform estimates, wherein the Lebesgue exponents are completely independent of the curves and the operator norms depend only on the polynomial degree. Moreover, our weighted estimates possess rather strong diffeomorphism invariance properties, allowing us to obtain uniform bounds for averages on curves satisfying a natural nilpotency hypothesis.

We prove a result related to Bressan's mixing problem. We establish an inequality for the change of Bianchini semi-norms of characteristic functions under the flow generated by a divergence free time dependent vector field. The approach leads to a bilinear singular integral operator for which we prove bounds on Hardy spaces. We include additional observations about the approach and a discrete toy version of Bressan's problem.

We prove L^p₁(ℝ^d)x···xL^p_n+2(ℝ^d) polynomial growth estimates for the Christ-Journé multilinear singular integral forms and suitable generalizations.

We study a trilinear singular integral form acting on two-dimensional functions and possessing invariances under arbitrary matrix dilations and linear modulations. One part of the motivation for introducing it lies in its large symmetry groups acting on the Fourier side. Another part of the motivation is that this form stands between the bilinear Hilbert transforms and the first Calderón commutator, in the sense that it can be reduced to a superposition of the former, while it also successfully encodes the latter. As the main result we determine the exact range of exponents in which the L^p estimates hold for the considered form.

The purpose of this paper is to study a class of ill-posed differential equations. In some settings, these differential equations exhibit uniqueness but not existence, while in others they exhibit existence but not uniqueness. An example of such a differential equation is, for a polynomial P and continuous functions f(t,x):[0,1]x[0,1]→ℝ,

These differential equations are related to inverse problems.

The purpose of this paper is to study the smoothing properties (in L^p Sobolev spaces) 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 fractional kernel" supported on a small neighborhood of 0∈ ℝ^N. When K is a Calderón-Zygmund kernel these operators were studied by Christ, Nagel, Stein, and Wainger, and when K is a multi-parameter singular kernel they were studied by the author and Stein. In both of these situations, conditions on γ were given under which the above operator is bounded on L^p (1 < p < ∞). Under these same conditions, we introduce non-isotropic L^p Sobolev spaces associated to γ. Furthermore, when K is a fractional kernel which is smoothing of an order which is close to 0 (i.e., very close to a singular kernel) we prove mapping properties of the above operators on these non-isotropic Sobolev spaces. As a corollary, under the conditions introduced on γ by Christ, Nagel, Stein, and Wainger, we prove optimal smoothing properties in isotropic L^p Sobolev spaces for the above operator when K is a fractional kernel which is smoothing of very low order.

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.

A contribution to the paper Analysis and applications: The mathematical work of Elias Stein, by Charles Fefferman, Alex Ionescu, Terence Tao and Stephen Wainger.