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Linear operators with off‐diagonal decay appear in many areas of mathematics including harmonic and numerical analysis, and their stability is one of the basic assumptions. In this paper, we consider a family of localized integral operators in the Beurling algebra with kernels having mild singularity near the diagonal and certain Hölder continuity property, and prove that their weighted stabilities for different exponents and Muckenhoupt weights are equivalent to each other on a space of homogeneous type with Ahlfors regular measure. [ABSTRACT FROM AUTHOR]