Adaptive Risk Control (ARC) is an online calibration strategy based on set prediction that offers worst-case deterministic long-term risk control, as well as statistical marginal coverage guarantees. ARC adjusts the size of the prediction set by varying a single scalar threshold based on feedback from past decisions. In this work, we introduce Localized Adaptive Risk Control (L-ARC), an online calibration scheme that targets statistical localized risk guarantees ranging from conditional risk to marginal risk, while preserving the worst-case performance of ARC. L-ARC updates a threshold function within a reproducing kernel Hilbert space (RKHS), with the kernel determining the level of localization of the statistical risk guarantee. The theoretical results highlight a trade-off between localization of the statistical risk and convergence speed to the long-term risk target. Thanks to localization, L-ARC is demonstrated via experiments to produce prediction sets with risk guarantees across different data subpopulations, significantly improving the fairness of the calibrated model for tasks such as image segmentation and beam selection in wireless networks.