We consider the regular Lagrangian flow X associated to a bounded divergence-free vector field b with variation. prove Lusin-Lipschitz regularity result for and we show that Lipschitz constant grows at most linearly in time. As consequence deduce both geometric analytical mixing have lower bound of order $t^{-1}$ as $t \to \infty$.

Bar Induction occupies a central place in Brouwerian mathematics. This note is concerned with the strength of Bar Induction on the basis of Constructive ZermeloFraenkel Set Theory, CZF. It is shown that CZF augmented by decidable Bar Induction proves the 1-consistency of CZF. This answers a question of P. Aczel who used Bar Induction to give a proof of the Lusin Separation Theorem in the constr...

4. -, Les valeurs asymptotiques de quelques fonctions miromorphes dans le cercle-uniti, C. R. Acad. Sci. Paris vol. 237 (1953) pp. 16-18. 5. N. Lusin and J. Privaloff, Sur I'unicitl et la multipliciti des fonctions analytiques, Ann. ficole Norm. (13) vol. 42 (1925) pp. 143-191. 6. R. Nevanlinna, Eindeutige analytische Funktionen, Berlin, Springer, 1936. 7. F. Riesz, Uber die Randwerte einer ana...

We show that the Lusin area integral or the square function on the unit ball of C, regarded as an operator in weighted space L(w) has a linear bound in terms of the invariant A2 characteristic of the weight. We show a dimension-free estimate for the “area-integral” associated to the weighted L(w) norm of the square function. We prove the equivalence of the classical and the invariant A2 classes.

In a paper of 1929, Banach and Kuratowski proved—assuming the continuum hypothesis—a combinatorial theorem which implies that there is no nonvanishing σ-additive finite measure μ on R which is defined for every set of reals. It will be shown that the combinatorial theorem is equivalent to the existence of a K-Lusin set of size 20 and that the existence of such sets is independent of ZFC + ¬CH.