We study Metropolis-Hastings kernels whose proposal kernels are reversible with respect to a reference measure, and establish sufficient conditions for such kernels to inherit a nonzero right spectral gap from their proposal kernel. Our conditions require the importance weight of the target relative to the reference measure to be bounded above, but do not require it to be bounded away from zero. Our approach combines a localized Holley-Stroock argument, which yields a local Poincare inequality, with a Lyapunov drift condition. As our main application, we analyze the preconditioned Crank-Nicolson (pCN) algorithm for sampling from measures on a Hilbert space that are absolutely continuous with respect to a Gaussian reference measure. Under local boundedness and quasiconvexity assumptions on the potential, we establish a dimension-independent $L^2$-spectral gap for this Metropolis-Hastings algorithm, thereby covering cases in which the local Lipschitz condition used by Hairer et al. (2014) is not satisfied.

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