New local T1 theorems on non-homogeneous spaces

We develop new local T1 theorems to characterize Calderon-Zygmund operators that extend boundedly or compactly on L^p (R^n, ) with a measure of power growth. The results, whose proofs do not require random grids, allow the use of a countable collection of testing functions. As a corollary, we describe the measures of the complex plane for which the Cauchy integral defines a compact operator on Lᵖ (C, ).