The author completes work begun in [Stud. Math. 109, No. 1, 23-39 (1994; Zbl 0844.42008)]. Let \(\Omega\) be an open subset of \(R^n\). A cube \(Q\) is always assumed to have faces perpendicular to the coordinate axes, and its length is denoted \(l(Q)\). If \(t>0\), \(tQ\) is the cube concentric with \(Q\) such that \(l(tQ) = tl(Q)\). If \(\sigma >1\) and \(\sigma Q \subseteq \Omega\), we say that \(Q\) is ‘\(\sigma\)-dilatable’. The author considers three types of conditions on positive weights on \(\Omega\), where we assume \(0<q<p\), \(1 \leq \sigma \leq \sigma'\). They all involve \[ \left(\frac 1{| Q| } \int_Q w(x)^p dx\right)^{1/p} \leq K\left(\frac 1{| \sigma Q| } \int_{\sigma Q} w(x)^q dx\right)^{1/q}, \] for all \(\sigma'\)-dilatable \(Q\). If the condition with \(\sigma' = 1\) holds for all \(Q \subseteq \Omega\), we say that \(w \in RH_{p}^{\Omega}\). If there is a \(K > 1\) such that the inequality holds for some \(1 < \sigma \leq \sigma'\) and all \(\sigma'\)-dilatable \(Q\), then we say that \(w \in WRH_{p}^{\Omega}\). If there is a \(K>1\) such that the inequality holds for some \(1 = \sigma < \sigma'\) and all \(\sigma'\)-dilatable \(Q\), then we say that \(w \in RH_{p}^{\Omega, \text{loc}}\). The author notes that the spaces are independent of \(q\), \(\sigma\), and \(\sigma'\) as long as those parameters satisfy the defining equalities or inequalities. He calls the weak reverse Hölder “norm” \(WRH_{p}^{\Omega}(w)\) the smallest \(K\) for which the inequality holds with \(q = p/2\), \(\sigma = 2\), and \(\sigma' = 4\). Similarly, the \(RH_p^{\Omega, \text{loc}}\) “norm” is the smallest constant with \((q, \sigma, \sigma') = (p/2, 1, 4)\) and the \(RH_p^{\Omega}\) “norm” the smallest constant with \((q, \sigma, \sigma') = (p/2, 1, 1)\). The author had given necessary and sufficient conditions for strong-strong, weak-weak, and weak-strong spaces in Part I, but only gave necessary conditions and sufficient conditions for the strong-weak case reconsidered here. If we let \(S_p\) be equal to either \(RH_p^{\Omega}\) or \(RH_p^{\Omega, \text{loc}}\) and if \(f\) is a weight, he shows that if \(f\) multiplies \(S_p\) into \(WRH_q^{\Omega}\), then \(q \leq p\). If \(0 < q \leq p < \infty\), \(f \cdot S_p \subseteq WRH_q^{\Omega}\) if and only if \(f \in \bigcap_{r<s} \;WRH_r^{\Omega}\), where \(s = pq/(p-q) (s = \infty\) if \(p = q)\). If \(0<q \leq \infty\), \(f \cdot S_{\infty} \subseteq WRH_q^{\Omega}\) if and only if \(f \in WRH_q^{\Omega}\).