Interior $h^1$ Estimates for Parabolic Equations with $\operatorname{LMO}$ Coefficients In this paper we establish \emph{a priori} $h^1$-estimates in a bounded domain for parabolic equations with vanishing $\operatorname{LMO}$ coefficients. Keywords:parabolic operator, Hardy space, parabolic, singular integrals and commutatorsCategories:35K20, 35B65, 35R05