Multipliers and singular integrals on exponential growth groups
The authors introduce an abstract version of the Calderón-Zygmund theory and show that it is applicable to many situations. For example, if \(T\) is a homogeneous tree with the natural distance, they define a measure \(\mu\) on \(T\) and show that with this measure and distance, \(T\) has the Calderón-Zygmund property. They also define a maximal function \(M\) and show that it is of weak type \((1,1)\). The case when \(G\) is a Lie group is also discussed. The left-invariant Haar measure is used as the basic measure and they work always with right-invariant vector fields and differential operators. Suppose on \(G\) there are given vector fields \(X_i\) and the distance \(d\) on \(G\) is the distance adapted to these vector fields. Assume also that there is a given Laplacian \(L \geq 0\) on \(G\), and that its heat kernel is \(p_t(\cdot)\). It is shown that if \(G\) satisfies the Calderón-Zygmund condition with an additional hypothesis then the Riesz transforms \(XL^{- 1/2}\) are of weak type \((1,1)\) and bounded on \(L^p\), \(1< p \leq 2\).
scientific article; zbMATH DE number 2018243