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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).
##### 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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##### References:
 [1] V. I. Bogachev, A. V. Kolesnikov, and K. V. Medvedev, Mat. Sb. 196, 3–30 (2005). · doi:10.4213/sm1271 [2] R. I. Zhdanov and Yu. V. Ovsienko, Vestn. Mosk. Univ., Ser. 1: Mat. Mekh., No. 1, 3–6 (2007). [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 [7] D. E. Aleksandrova, Teor. Veroyatn. Ee Primen. 50(1), 145–150 (2005). · doi:10.4213/tvp162
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