# Multi-parameter Singular Integrals

Brian Street
Pages: 416
https://www.jstor.org/stable/j.ctt6wq1bt

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1. Front Matter
(pp. i-iv)
(pp. v-viii)
3. Preface
(pp. ix-xvi)
4. Chapter One The Calderón-Zygmund Theory I: Ellipticity
(pp. 1-38)

Our story begins with a classical situation: convolution with homogeneous, Calderón-Zygmund kernels on${{\mathbb{R}}^{n}}$. Let${{S}^{n-1}}\hookrightarrow {{\mathbb{R}}^{n}}$denote the unit sphere in${{\mathbb{R}}^{n}}$. We let$C_{z}^{\infty }\left( {{S}^{n-1}} \right)\subset {{C}^{\infty }}\left( {{S}^{n-1}} \right)$denote those$k\in {{C}^{\infty }}\left( {{S}^{n-1}} \right)$with$\int_{{{S}^{n-1}}}{k(\omega )}\ d\omega =0$(whereωdenotes the surface area measure onSn-1).

For a function$k\in C_{z}^{\infty }\left( {{S}^{n-1}} \right)$, and a complex number$c\in \mathbb{C}$, we define a distribution$K=K(k,c)\in C_{0}^{\infty }{{\left( {{\mathbb{R}}^{n}} \right)}^{\prime }}$by$\left\langle K,f \right\rangle =cf(0)+\underset{\epsilon \to 0}{\mathop{\lim }}\,\int_{\left| x \right|>\epsilon }{k(x/\left| x \right|)\frac{1}{{{\left| x \right|}^{n}}}f(x)\ dx,\quad f\in C_{0}^{\infty }({{\mathbb{R}}^{n}}).}\caption {(1.1)}$

Lemma 1.0.1. (1.1)defines a distribution

Proof. FixM> 0, and let$\Gamma :=\overline{{{B}^{n}}(M)}$, where${{B}^{n}}(M)\subset {{\mathbb{R}}^{n}}$is the ball of radiusMin${{\mathbb{R}}^{n}}$, centered at 0. Note that Γ is compact. LetC(Γ) denote the Fréchet space of...

5. Chapter Two The Calderón-Zygmund Theory II: Maximal Hypoellipticity
(pp. 39-197)

In Chapter 1, we were concerned with${{\mathbb{R}}^{n}}$endowed with the usual Euclidean metric. As we saw in Section 1.4, singular integrals associated to the Euclidean metric play an important role in understandingellipticpartial differential operators. If a partial differential operator fails to be elliptic, then there is no immediate analog of the theory developed above. However, in certain circumstances, many of the theorems in Chapter 1 do have analogs. In this chapter, we discuss one such situation, where we study a generalization of ellipticity, known as maximal hypoellipticity.

We begin with some notation. Associated tornon-commuting indeterminates,...

6. Chapter Three Multi-parameter Carnot-Carathéodory Geometry
(pp. 198-222)

In the previous chapters, we focused on “single-parameter” singular integrals. By this, we mean that the singular integrals are defined in terms of an underlying family of ballsB(x, δ) whereδ> 0. The main focus of this monograph is a more general setting, where the underlying balls have many “parameters”B(x,δ1, …,δν). We focus exclusively on the case of Carnot-Carathéodory balls. In this chapter, we develop the theory necessary to deal with these multi-parameter balls. Much as in the single-parameter case, the quantitative Frobenius theorem (Theorem 2.2.22) is a key tool.

We begin by introducing...

7. Chapter Four Multi-parameter Singular Integrals I: Examples
(pp. 223-267)

In Chapters 1 and 2, we focused on “single-parameter” singular integrals. I.e., the singular integrals are defined in terms of an underlying family of ballsB(x,δ) whereδ> 0. The main focus of this monograph is a more general setting, where the underlying balls have many “parameters,”B(x,δ1, …,δν), whereB(x,δ1, …,δν) is the sort of Carnot-Carathéodory ball studied in Chapter 3. In Chapter 5 we introduce a general theory of these multi-parameter singular integrals. In this chapter, we present several examples (many of which already appear in the literature) which fall...

8. Chapter Five Multi-parameter Singular Integrals II: General Theory
(pp. 268-362)

In Chapter 4, we presented several examples of multi-parameter singular integrals that arise naturally. We now turn to the main purpose of this monograph: to present an intrinsically defined algebra of “multi-parameter singular integral operators” which includes and generalizes all of the ideas from Chapter 4.

The setting is the same as Chapter 3. That is, we are givenνlists of vector fields on an open set$U\subseteq {{\mathbb{R}}^{\nu }}$with single-parameter formal degrees: for each 1 ≤μν, we have$\left( {{X}^{\mu }},{{{\hat{d}}}^{\mu }} \right):=\left( X_{1}^{\mu },\hat{d}_{1}^{\mu } \right),\ldots ,\left( X_{{{q}_{\mu }}}^{\mu },\hat{d}_{{{q}_{\mu }}}^{\mu } \right);0\ne \hat{d}_{j}^{\mu }\in \mathbb{N}$. We assume that$X_{1}^{1},\ldots ,X_{{{q}_{1}}}^{1},\ldots ,\ldots X_{1}^{\nu },\ldots X_{{{q}_{\nu }}}^{\nu }$span the tangent space at every point ofU. From theseν...

9. Appendix A Functional Analysis
(pp. 363-375)
10. Appendix B Three Results from Calculus
(pp. 376-379)
11. Appendix C Notation
(pp. 380-382)
12. Bibliography
(pp. 383-392)
13. Index
(pp. 393-395)