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We prove quantitative estimates for flows of vector fields subject toanisotropic regularity conditions: some derivatives of some components are(singular integrals of) measures, while the remaining derivatives are (singularintegrals of) integrable functions. This is motivated by the regularity of thevector field in the Vlasov-Poisson equation with measure density. The proofexploits an anisotropic variant of the argument of Crippa and De Lellis andsuitable estimates for the difference quotients in such anisotropic context. Incontrast to regularization methods, this approach gives quantitative estimatesin terms of the given regularity bounds. From such estimates it is possible torecover the well posedness for the ordinary differential equation and forLagrangian solutions to the continuity and transport equations.

Lagrangian flows for vector fields with anisotropic regularity