Square functions with general measures II

We continue developing the theory of conical and vertical square functions on $R^{n}$, where $\mu$ is a power bounded measure, possibly non-doubling. We provide new boundedness criteria and construct various counterexamples. First, we prove a general local $Tb$ theorem with tent space $T^{2,\infty}$ type testing conditions to characterise the $L^{2}$ boundedness. Second, we completely answer the question, whether the boundedness of our operators on $L^{2}$ 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 $\mu = dx$. For this, we present a counterexample. Our kernels $s_t$, $t > 0$, 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 $L^{2}$ 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 $\mu$ is doubling.