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We establish existence, uniqueness, and arbitrary order Sobolev regularityresults for the second order parabolic equations with measurable coefficientsdefined on the conic domains $D$ of the type $$ D(M):=\left\{x\in R^d:\,\frac{x}{|x|}\in M\right\}, \quad \quad M \subset S^{d-1}. $$ We obtain theregularity results by using a system of mixed weights consisting of appropriatepowers of the distance to the vertex and of the distance to the boundary. Wealso provide the sharp ranges of admissible powers of the distance to thevertex and to the boundary.

A weighted Sobolev regularity theory of the parabolic equations with measurable coefficients on conic domains in $R^d$