Elias M. Stein

Hypoelliptic differential operators and nilpotent groups

2. Sufficient condi t ions for hypoe l l ip t i c i ty . . . . . . . . . . . . . . . . . . . . . 251 8. Graded a n d free Lie a lgebras . . . . . . . . . . . . . . . . . . . . . . . . . . 255 4. H a r m o n i c analysis on iV a n d the p roof of T h eo rem 2 . . . . . . . . . . . . . . . 257 5. Di la t ions a n d h o m o g e n e i t y on groups . . . . . . . . . . . . . . . . . . . . . . 261 6. Smoo th ly va ry ing families of f u n d a m e n t a l solut ions . . . . . . . . . . . . . . . . 265

Problems in harmonic analysis related to curvature

X \R(1 r\\ f /OO # / ( * ) --> e-*0 \B(e9 X)\ JB(t,x) for certain types of «-dimensional sets B(e9 x) shrinking to x as e -* 0. (\B(e9 x)\ is of course the Lebesgue measure of B(e9 x).) Some standard examples of sets B(e9 x) are balls with center x and radius e, and cubes with center x and diameter e. The problem of considering other sets besides balls and cubes for B(e9 x) received much attention in the 1930s. (See for example Buseman and Feller [1934].) This subject again seems to be attracting interest. One of the two main goals of this paper is an exposition of recent developments in which B(e9 x) is replaced by a lower dimensional set. 2

Some New Function Spaces and Their Applications to Harmonic Analysis

Abstract In this paper a family of spaces is introduced which seems well adapted for the study of a variety of questions related to harmonic analysis and its applications. These spaces are the “tent spaces.” They provide the natural setting for the study of such things as maximal functions (the relevant space here is T∞p), and also square functions (where the space T2p is relevant). As such these spaces lead to unifications and simplifications of some basic techniques in harmonic analysis. Thus they are closely related to Lp and Hardy spaces, important parts of whose theory become corollaries of the description of tent spaces. Also, as (“Proc. Conf. Harmonic Analysis, Cortona,” Lect. Notes in Math. Vol. 992, Springer-Verlag, Berlin/New York,1983), already indicated where these spaces first appeared explicitly, the tent spaces can be used to simplify some of the results related to the Cauchy integral on Lipschitz curves, and multilinear analysis. In retrospect one can recognize that various ideas important for tent spaces had been used, if only implicitly, for quite some time. Here one should mention Carlesons inequality, its simplifications and extensions, the theory of Hardy spaces, and atomic decompositions.

Harmonic analysis on nilpotent groups and singular integrals I. Oscillatory integrals

Abstract This paper is devoted to the study of the operator given by Ω where K is a standard Calderon-Zygmund kernel, and P is a real polynomial on R n × R n. We show T is bounded on Lp to itself, when 1

Singular and maximal Radon transforms: analysis and geometry

Part 2. Geometric theory 8. Curvature: Introduction 8.1. Three notions of curvature 8.2. Theorems 8.3. Examples 9. Curvature: Some details 9.1. The exponential representation 9.2. Diffeomorphism invariance 9.3. Curvature condition (CY ) 9.4. Two lemmas 9.5. Double fibration formulation 10. Equivalence of curvature conditions 10.1. Invariant submanifolds and deficient Lie algebras 10.2. Vanishing Jacobians 10.3. Construction of invariant submanifolds

On certain maximal functions and approach regions

If f E L’(iR”) and u(x, y) is the Poisson integral of J; a classical theorem of Fatou [2] asserts that u has nontangential limits almost everywhere on R”. Littlewood [5], and later Zygmund [9], showed that this conclusion is false if nontangential approach is replaced by approach along translates of a curve which approaches the boundary tangentially. In this paper we find a necessary and sufficient condition on an approach region for the associated maximal function to be appropriately bounded. It then turns out that there are many approach regions which are not contained in any nontangential region but for which the conclusions of Fatou’s theorem remain true. Suppose Q c IR:+ ’ has the property that whenever (xi, u,) E B and whenever [x -x, [ < y y, it follows that (x, v) E Q. Then the maximal function associated to such a set is bounded if and only if the cross-sectional measure at height y is bounded by a constant times y”. We also study regions where the cross-sectional area is larger than y”. We introduce a modified maximal function which is again bounded. This result leads to a simple proof of the boundedness of certain tangential maximal functions of Poisson integrals of potentials studied in 161. The plan of this paper is as follows: In Section 1, we motivate our main ideas by discussing two particular examples. The method of proof here is somewhat different from the proofs of the more general results. In Section 2, we obtain the necessary and sufficient conditions for Fatou’s theorem. In Section 3, we study more general approach regions, and the appropriately modified maximal functions. In Section 4, we show how this maximal function controls the tangential boundary behavior of Poisson integrals of potentials.

Models of Degenerate Fourier Integral Operators and Radon Transforms

Fourier integral operators whose Lagrangians project with singularities arise frequently in many areas of analysis and geometry [3], [5], [9], [10], [18], [20]. However, there are as yet few analytic tools available for their study, and even their simplest regularity properties with respect to the singularities of the Lagrangian are still obscure [4], [5], [13], [16]. In this paper we study a class of oscillatory integrals TA and Fourier integral operators R which can be expected to model the higher order singularities of the Lagrangian. They have homogeneous polynomial phases in two variables of order n, and indeed the case n = 3 is the model for Lagrangians which project with Whitney folds [3], [10], [11]. The main difficulty which sets the higher order degeneracy cases n > 4 apart from the the lower ones n = 2, 3 is that the critical varieties which arise there are usually not smooth manifolds. A systematic study of these oscillatory integrals was begun in [14]. The main idea in that work was to treat the critical varieties as smooth manifolds away from a lower-dimensional subvariety, and to keep track of the distance to this lower-dimensional subvariety. To achieve this we introduced a method of stationary phase which exhibited clearly the dependence on the distance between critical points. The method gave sharp bounds on the size of the kernel K(x, y) of TAT*, but no information on its phase and the resulting cancellations. It does not seem possible to refine this approach to the phase level, and this suggests looking instead for decompositions of the operators TA which can incorporate indirectly the required cancellations. The main goal of this paper is to introduce such decompositions. These decompositions, which we will try to describe momentarily, reflect the singular nature of the critical points, and are powerful enough to yield sharp bounds for the above oscillatory integrals and Radon transforms. Although the models under consideration are very

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

On the product theory of singular integrals

We establish Lp-boundedness for a class of product singular integral operators on spaces M = M1 x M2 x . . . x Mn. Each factor space Mi is a smooth manifold on which the basic geometry is given by a control, or Carnot-Caratheodory, metric induced by a collection of vector fields of finite type. The standard singular integrals on Mi are non-isotropic smoothing operators of order zero. The boundedness of the product operators is then a consequence of a natural Littlewood- Paley theory on M. This in turn is a consequence of a corresponding theory on each factor space. The square function for this theory is constructed from the heat kernel for the sub-Laplacian on each factor.

Hilbert integrals, singular integrals, and Radon transforms II

Etude des transformations de Radon singulieres et des integrales de Hilbert, appliquees aux problemes aux valeurs limites (probleme de Neumann)