Square functions with general measures II Henri MartikainenMihalis MourgoglouTuomas Orponen 42B20Square functionnon-homogeneous analysisRBMOlocal Tb We continue developing the theory of conical and vertical square functions on $\mathbb{R}^n,\mu)$, where $\mu$ can be non-doubling. We provide new boundedness criteria and construct various counterexamples. First, we prove a general local $Tb$ theorem with tent space $T^{2,\infty}$-type testing conditions to characterise the $L^2$ boundedness. Second, we completely answer whether or not 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 $\mu=\mathrm{d}x$. 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. Indiana University Mathematics Journal 2014 text pdf 10.1512/iumj.2014.63.5379 10.1512/iumj.2014.63.5379 en Indiana Univ. Math. J. 63 (2014) 1249 - 1279 state-of-the-art mathematics http://iumj.org/access/