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Cavity optomechanics with frequency-independent high-reflectivity mirrors shows different operation regimes distinguished by the ratio of the mechanical frequency and the photon loss rate. Working in the resolved-sideband regime thus enables cooling or amplification of the mechanical motion while optomechanical systems in the bad-cavity limit can be used to efficiently measure the mechanical motion. The use of mirrors with frequency-dependent reflectivity can bring new, interesting effects, such as Doppler cooling of the mechanical motion or modification of the sideband ratio. Here, we develop a full quantum theory of cavity optomechanics where the mechanically compliant mirror has reflectivity that strongly depends on the frequency of the incident light and identify regimes where these new optomechanical effects can be observed. These results are relevant for mirrors formed by self-assembled two-dimensional atomic layers, where the reflectivity is sharply peaked around the internal resonance of the atoms, or for structured membranes with engineered spatial defects.