By Brian Street

This publication develops a brand new conception of multi-parameter singular integrals linked to Carnot-Carathéodory balls. Brian road first information the classical concept of Calderón-Zygmund singular integrals and purposes to linear partial differential equations. He then outlines the idea of multi-parameter Carnot-Carathéodory geometry, the place the most device is a quantitative model of the classical theorem of Frobenius. road then provides numerous examples of multi-parameter singular integrals bobbing up certainly in a number of difficulties. the ultimate bankruptcy of the ebook develops a normal concept of singular integrals that generalizes and unifies those examples. this can be one of many first normal theories of multi-parameter singular integrals that is going past the product concept of singular integrals and their analogs. *Multi-parameter Singular Integrals* will curiosity graduate scholars and researchers operating in singular integrals and similar fields.

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Wr satisfy H¨ormander’s condition, then ρ is a metric. That is, ρ (x, y) < ∞, for every x, y ∈ M . 5 Suppose M is endowed with a Riemannian metric, and denote by R (x, y) the distance between x and y in this metric. Suppose, also, that W1 , . . , Wr satisfy H¨ormander’s condition of order m on M . 2). Thus, the topology induced by ρ is the same as the topology on the manifold, however the metrics are not necessarily equivalent. In Chapter 1, to define our singular integrals, we worked on Rn and used Lebesgue measure on Rn .

1. We say T : C0∞ (Rn ) → C ∞ (Rn ) is a Calder´on-Zygmund operator of order t ∈ (−n, ∞) if: (i) (Growth Condition) For every multi-indices α and β, −n−t−|α|−|β| ∂xα ∂yβ T (x, y) ≤ Cα,β |x − y| In particular, we assume T (x, y) is a C ∞ function for x = y. 20 CHAPTER 1 (ii) (Cancellation Condition) For all bounded sets B ⊂ C0∞ (Rn ) and for all φ ∈ B, R > 0, and z ∈ Rn , define φR,z (x) = φ (R (x − z)). We assume, for every multi-index α, sup sup sup R−t−|α| |∂xα T φR,z (x)| ≤ CB,α , φ∈B R>0 x,z∈Rn with the same estimates for T replaced by T ∗ , the formal L2 adjoint of T .

23, 0 we may use this to characterize Calder´on-Zygmund kernels in other ways, as the next theorem shows. We state this theorem without proof. 26. Fix t ∈ (−n, ∞), and let K ∈ S0 (Rn ) . The following are equivalent: (i) K is a Calder´on-Zygmund kernel of order t. (ii) Op (K) : S0 (Rn ) → S0 (Rn ) and for any bounded set B ⊂ S0 (Rn ), the set g ∈ S0 (Rn ) ∃R > 0, f ∈ B, g (R) = R−t Op (K) f (R) ⊂ S0 (Rn ) is a bounded set. (iii) For each j ∈ Z there is a function ςj ∈ S0 (Rn ) with ςj j ∈ Z ⊂ S0 (Rn ) a bounded set and such that (2j ) 2jt ςj .