Continuity and differentiability of triangular mappings.

*(English. Russian original)*Zbl 1308.46082
Dokl. Math. 82, No. 2, 741-745 (2010); translation from Dokl. Akad. Nauk., Ross. Akad. Nauk. 434, No. 3, 304-308 (2010).

The author studies continuity and differentiability of the so-called increasing triangular mappings. The studies of various aspects of a triangular mapping \(T_{\mu,\nu}\) for any pair of probability measures \(\mu\) and \(\nu\) on \(\mathbb{R}^n\) for which \(\nu=\mu \circ T_{\mu,\nu}^{-1}\) were initiated by V. I. Bogachev et al. [Sb. Math. 196, No. 3, 309–335 (2005); translation from Mat. Sb. 196, No. 3, 3–30 (2005; Zbl 1072.28010)]. The present work is a continuation of their and the author’s research in this field. In particular, the author describes some sufficient conditions under which \(T\) belongs to various Sobolev classes, where \(T\) denotes the increasing triangular mapping sending the measure \(\mu\) to the Lebesgue measure on \([0,1]^n\). For example, Theorem 1 describes a certain sufficient condition on the measure \(\mu\) under which \(T \in W_{\mathrm{loc}}^{1,\alpha}(\mathbb{R}^n)\) for \(\alpha>1\). Theorem 2 describes a certain sufficient condition on the measure \(\mu\) under which \(T \in W_{\mathrm{loc}}^{0,\gamma}(\mathbb{R}^n)\) for \(\gamma=(1-\frac{n}{\beta})^{n-1}\) with \(\beta > n\).

Under the assumptions that \(\mu\) and \(\nu\) are probability measures on the cube \(\Omega=[0,1]^n\) with positive probability densities \(\rho_{\mu}\) and \(\rho_{\nu}\), for which \(\rho_{\nu}\) is bounded away from zero and bounded, and \(\mu_n\) and \(\nu_n\) are probability measures on \(\Omega\) whose densities converge uniformly to \(\rho_{\mu}\) and \(\rho_{\nu}\), respectively, it is shown that the canonical triangular transformations \(T_{\mu_n,\nu_n}\) converge uniformly to \(T_{\mu,\nu}\) (cf. Theorem 4).

Under the assumptions that \(\mu\) and \(\nu\) are probability measures on the cube \(\Omega=[0,1]^n\) with positive probability densities \(\rho_{\mu}\) and \(\rho_{\nu}\), for which \(\rho_{\nu}\) is bounded away from zero and bounded, and \(\mu_n\) and \(\nu_n\) are probability measures on \(\Omega\) whose densities converge uniformly to \(\rho_{\mu}\) and \(\rho_{\nu}\), respectively, it is shown that the canonical triangular transformations \(T_{\mu_n,\nu_n}\) converge uniformly to \(T_{\mu,\nu}\) (cf. Theorem 4).

Reviewer: Gogi Pantsulaia (Tbilisi)

##### MSC:

46T20 | Continuous and differentiable maps in nonlinear functional analysis |

28C20 | Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.) |

46G12 | Measures and integration on abstract linear spaces |

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\textit{R. I. Zhdanov}, Dokl. Math. 82, No. 2, 741--745 (2010; Zbl 1308.46082); translation from Dokl. Akad. Nauk., Ross. Akad. Nauk. 434, No. 3, 304--308 (2010)

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##### References:

[1] | V. I. Bogachev, A. V. Kolesnikov, and K. V. Medvedev, Mat. Sb. 196, 3–30 (2005). · doi:10.4213/sm1271 |

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[3] | R. A. Adams and J. J. Fournier, Sobolev Spaces, 2nd ed. (Academic, Amsterdam, 2003), Vol. 140. |

[4] | V. I. Bogachev, Foundations of Measure Theory, 2nd ed. (Regulyarnaya i Khaoticheskaya Dinamika, Izhevsk), 2006) [in Russian], Vol. 2. |

[5] | V. I. Bogachev, Differentiable Measures and Malliavin Calculus (Moscow) (Regulyarnaya i Khaoticheskaya Dinamika, Izhevsk, 2008) [in Russian]. |

[6] | T. Radó and P. V. Reichelderfer, Continuous Transformations in Analysis (Springer-Verlag, Berlin, 1955). · Zbl 0067.03506 |

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