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We give a simplified proof of recent regularity results of Lewis and Murray, namely, that certain commutators, and the boundary single layer potential for the heat equation in domains in R2 with time dependent boundary, map L p into an appropriate homogeneous Sobolev space. The simplification is achieved by treating directly only the case p = 2, but in a weighted setting

A note on weighted Sobolev spaces, and regularity of commutators and layer potentials associated to the heat equation