Square functions with general measures II

We continue developing the theory of conical and vertical square functions on (ℝⁿ, μ), where μ can be non-doubling. We provide new boundedness criteria and construct various counterexamples. First, we prove a general local Tb theorem with tent space T^2,∞-type testing conditions to characterise the L² boundedness. Second, we completely answer whether or not the boundedness of our operators on L² implies boundedness on other L^p spaces, including the endpoints. For the conical square function, the answers are generally affirmative, but the vertical square function can be unbounded on L^p for p > 2, even if μ = dx. For this, we present a counterexample. Our kernels st , t > 0, do not necessarily satisfy any continuity in the first variable - a point of technical importance throughout the paper. Third, we construct a non-doubling Cantor-type measure and an associated conical square function operator, whose L² boundedness depends on the exact aperture of the cone used in the definition. Thus, in the non-homogeneous world, the 'change of aperture' technique - widely used in classical tent space literature - is not available. Fourth, we establish the sharp A_p-weighted bound for the conical square function under the assumption that μ is doubling.