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It is proved that the family of K quasiconformal mappings of the unit ball onto itself satisfying PDE ¢u = g, g 2 L 1 (B n ), u(0) = 0, is a uniformly Lipschitz family. In addition, it is showed that the Lipschitz constant tends to 1 as K ! 1 and jgj1 ! 0. This generalizes a similar two-dimensional case treated in (21) and solves the problem initialized in (16).

On the quasiconformal self-mappings of the unit ball satisfying the Poisson differential equation