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Singular integrals and a problem on mixing flows. (English) Zbl 1387.37010

Summary: We prove a result related to Bressan’s mixing problem. We establish an inequality for the change of Bianchini semi-norms of characteristic functions under the flow generated by a divergence free time dependent vector field. The approach leads to a bilinear singular integral operator for which we prove bounds on Hardy spaces. We include additional observations about the approach and a discrete toy version of Bressan’s problem.

MSC:

37A45 Relations of ergodic theory with number theory and harmonic analysis (MSC2010)
37C10 Dynamics induced by flows and semiflows
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
47G10 Integral operators
37A25 Ergodicity, mixing, rates of mixing
82C70 Transport processes in time-dependent statistical mechanics
30H10 Hardy spaces
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[1] Alberti, G.; Crippa, G.; Mazzucato, A., Exponential self-similar mixing and loss of regularity for continuity equations · Zbl 1302.35241
[2] Bianchini, S., On Bressan’s conjecture on mixing properties of vector fields, (Self-Similar Solutions of Nonlinear PDE. Self-Similar Solutions of Nonlinear PDE, Banach Cent. Publ., vol. 74 (2006), Polish Acad. Sci.: Polish Acad. Sci. Warsaw), 13-31 · Zbl 1108.35028
[3] Bownik, M., Boundedness of operators on Hardy spaces via atomic decompositions, Proc. Am. Math. Soc., 133, 12, 3535-3542 (2005) · Zbl 1070.42006
[4] Bressan, A., A lemma and a conjecture on the cost of rearrangements, Rend. Semin. Mat. Univ. Padova, 110, 97-102 (2003) · Zbl 1114.05002
[5] Bressan, A., Prize offered for a solution of a problem on mixing flows, article posted on the website
[6] Christ, M.; Journé, J. L., Polynomial growth estimates for multilinear singular integral operators, Acta Math., 159, 1-2, 51-80 (1987) · Zbl 0645.42017
[7] Crippa, G.; De Lellis, C., Estimates and regularity results for the DiPerna-Lions flow, J. Reine Angew. Math., 616, 15-46 (2008) · Zbl 1160.34004
[8] David, G.; Journé, J.-L., A boundedness criterion for generalized Calderón-Zygmund operators, Ann. Math. (2), 120, 2, 371-397 (1984) · Zbl 0567.47025
[9] De Lellis, C., ODEs with Sobolev coefficients: the Eulerian and the Lagrangian approach, Discrete Contin. Dyn. Syst., Ser. S, 1, 3, 405-426 (2008) · Zbl 1156.35341
[10] Goldberg, D., A local version of real Hardy spaces, Duke Math. J., 46, 1, 27-42 (1979) · Zbl 0409.46060
[11] Hofmann, S., An off-diagonal T1 theorem and applications, J. Funct. Anal., 160, 2, 581-622 (1998), with an appendix “The Mary Weiss lemma” by Loukas Grafakos and the author · Zbl 0919.42012
[12] Iyer, G.; Kiselev, A.; Xu, X., Lower bounds on the mix norm of passive scalars advected by incompressible enstrophy-constrained flows, Nonlinearity, 27, 5, 973-985 (2014) · Zbl 1293.35248
[13] Léger, F., A new approach to bounds on mixing · Zbl 1474.76024
[14] Lin, Z.; Thiffeault, J.-L.; Doering, C. R., Optimal stirring strategies for passive scalars, J. Fluid Mech., 675, 465-476 (2011) · Zbl 1241.76361
[15] Meda, S.; Sjögren, P.; Vallarino, M., On the \(H^1 - L^1\) boundedness of operators, Proc. Am. Math. Soc., 136, 8, 2921-2931 (2008) · Zbl 1273.42021
[16] Seeger, A., A weak type bound for a singular integral, Rev. Mat. Iberoam., 30, 3, 961-978 (2014) · Zbl 1304.42044
[17] Seeger, A.; Smart, C.; Street, B., Multilinear singular integral forms of Christ-Journé type, Mem. Am. Math. Soc., to appear. See also · Zbl 1445.42011
[18] Stein, E. M., Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, vol. 30 (1970), Princeton University Press: Princeton University Press Princeton, NJ · Zbl 0207.13501
[19] Stein, E. M., Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, (Monographs in Harmonic Analysis, III. Monographs in Harmonic Analysis, III, Princeton Mathematical Series, vol. 43 (1993), Princeton University Press: Princeton University Press Princeton, NJ), with the assistance of Timothy S. Murphy · Zbl 0821.42001
[20] Triebel, H., Theory of Function Spaces, Monographs in Mathematics, vol. 78 (1983), Birkhäuser Verlag: Birkhäuser Verlag Basel · Zbl 0546.46027
[21] Yao, Y.; Zlatoš, A., Mixing and un-mixing by incompressible flows, J. Eur. Math. Soc., 19, 7, 1911-1948 (2017) · Zbl 1369.35071
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