We prove quantitative estimates for flows of vector fields subject to anisotropic regularity
conditions: some derivatives of some components are (singular integrals of) measures, while
the remaining derivatives are (singular integrals of) integrable functions. This is motivated by the
regularity of the vector field in the Vlasov-Poisson equation with measure density. The proof
exploits an anisotropic variant of the argument in [20, 14] and suitable estimates for the difference
quotients in such anisotropic context. In contrast to regularization methods, this approach gives
quantitative estimates in terms of the given regularity bounds. From such estimates it is possible
to recover the well posedness for the ordinary differential equation and for Lagrangian solutions to
the continuity and transport equations.

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