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This chapter discusses a few special cases where a theory of multi-parameter singular integral operators has already been developed. These include the product theory of singular integrals, convolution with flag kernels on graded groups, convolution with both the left and right invariant Calderón–Zygmund singular integral operators on stratified Lie groups, and composition of standard pseudodifferential operators with certain singular integrals corresponding to non-Euclidean geometries. The chapter outlines these examples and their applications and relates them to the trichotomy discussed in Chapter 1.

This chapter turns to a general theory which generalizes and unifies all of the examples in the preceding chapters. A main issue is that the first definition from the trichotomy does not generalize to the multi-parameter situation. To deal with this, strengthened cancellation conditions are introduced. This is done in two different ways, resulting in four total definitions for singular integral operators (the first two use the strengthened cancellation conditions, while the later two are generalizations of the later two parts of the trichotomy). Thus, we obtain four classes of singular integral operators, denoted by A₁, A₂, A₃, and A₄. The main theorem of the chapter is A₁ = A₂ = A₃ = A₄; i.e., all four of these definitions are equivalent. This leads to many nice properties of these singular integral operators.

This chapter remains in the single-parameter case and turns to the case when the metric is a Carnot–Carathéodory (or sub-Riemannian) metric. It defines a class of singular integral operators adapted to this metric. The chapter has two major themes. The first is a more general reprise of the trichotomy described in Chapter 1 (Theorem 2.0.29). The second theme is a generalization of the fact that Euclidean singular integral operators are closely related to elliptic partial differential equations. The chapter also introduces a quantitative version of the classical Frobenius theorem from differential geometry. This “quantitative Frobenius theorem” can be thought of as yielding “scaling maps” which are well adapted to the Carnot–Carathéodory geometry, and is of central use throughout the rest of the monograph.

Solutions to interface problems may be developed using analytic elements with mathematical solutions to the Laplace equation developed by singular integral equations. This formulation leads to solutions with discontinuities occurring across line segments, where the potential or stream function is discontinuous across double layer elements in Section 5.2, and the normal or tangential component of the vector field is discontinuous across single layer elements in Section 5.3. Examples illustrate a broad range of solutions to interface conditions possible with these elements. Series expansions are used to represent the far-field at larger distances from elements in Section 5.4, which leads to higher-order elements with nearly exact solutions and also provides a simpler representation for contiguous strings of adjacent elements. Such strings of elements are used with polygon elements in 5.5 to solve conditions along the interfaces of heterogeneities, and to provide a common series expansion to represent the far-field for a group of neighboring elements. Methods are extended to analytic elements with curvilinear geometry using conformal mappings (Section 5.6) and to three-dimensional fields in Section 5.7.

This chapter develops the theory of multi-parameter Carnot–Carathéodory geometry, which is needed to study singular integral operators. In the case when the balls are of product type, all of the results are simple variants of results in the single-parameter theory. When the balls are not of product type, these ideas become more difficult. What saves the day is the quantitative Frobenius theorem given in Chapter 2. This can be used to estimate certain integrals, as well as develop an appropriate maximal function and an appropriate Littlewood–Paley square function, all of which are essential to our study of singular integral operators.

This chapter discusses a case for single-parameter singular integral operators, where ρ‎ is the usual distance on ℝⁿ. There, we obtain the most classical theory of singular integrals, which is useful for studying elliptic partial differential operators. The chapter defines singular integral operators in three equivalent ways. This trichotomy can be seen three times, in increasing generality: Theorems 1.1.23, 1.1.26, and 1.2.10. This trichotomy is developed even when the operators are not translation invariant (many authors discuss such ideas only for translation invariant, or nearly translation invariant operators). It also presents these ideas in a slightly different way than is usual, which helps to motivate later results and definitions.