In offline reinforcement learning (RL) we have no opportunity to explore so we must make assumptions that the data is sufficient to guide picking a good policy, taking the form of assuming some coverage, realizability, Bellman completeness, and/or hard margin (gap). In this work we propose value-based algorithms for offline RL with PAC guarantees under just partial coverage, specifically, coverage of just a single comparator policy, and realizability of soft (entropy-regularized) Q-function of the single policy and a related function defined as a saddle point of certain minimax optimization problem. This offers refined and generally more lax conditions for offline RL. We further show an analogous result for vanilla Q-functions under a soft margin condition. To attain these guarantees, we leverage novel minimax learning algorithms to accurately estimate soft or vanilla Q-functions with $L^2$-convergence guarantees. Our algorithms' loss functions arise from casting the estimation problems as nonlinear convex optimization problems and Lagrangifying.