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 Lᵖ( Rⁿ,μ) with μ 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 μ of the complex plane for which the Cauchy integral defines a compact operator on Lᵖ( C,μ).