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On stability of classes of affine mappings. (English. Russian original) Zbl 0865.30025

Sib. Math. J. 36, No. 5, 930-942 (1995); translation from Sib. Mat. Zh. 36, No. 5, 1081-1095 (1995).
A class \(\mathcal A\) consists of all affine mappings \(g:\mathbb{R}^n\to\mathbb{R}^m\) of the form \(g(x)=\alpha(x)+b\), \(x\in\mathbb{R}^n\), restricted to a domain \(\Delta\subset\mathbb{R}^n\), where \(b\in\mathbb{R}^m\), \(\alpha\) belongs to a given set \(A\) of the space \(L(\mathbb{R}^n,\mathbb{R}^m)\) of linear mappings of \(\mathbb{R}^n\) into \(\mathbb{R}^m\), with the operator norm \(|\lambda|_{n,m}=\sup_{|x|\leq 1}|\lambda(x)|\), \(\lambda\in L(\mathbb{R}^n,\mathbb{R}^m)\) and represents the differential \(Dg\) of the affine mapping \(g\). Let be \(f:\Delta\to\mathbb{R}^m\), the ball \(B(x,r)\subset\Delta\) and let us denote \[ \omega_B(f,{\mathcal G}):=\inf_{g:B\to\mathbb{R}^n,g\in{\mathcal G}}\Biggl\{r^{-1}\sup_{y\in B}|f(y)-g(y)|\Biggr\}, \]
\[ \Omega(f,{\mathcal G}):=\sup_{x\in\Delta} \Omega(x,f,{\mathcal G})=\sup_{x\in\Delta} \Biggl\{\varlimsup_{r\to 0} \omega_{B(x,r)}(f,{\mathcal G})\Biggr\}. \] Starting from F. Jones’s [Commun. Pure Appl. Math. 14, 391-413 (1961; Zbl 0102.17404); ibid. 21, 77-110 (1968; Zbl 0157.45803); ibid. 22, 265-278 (1969; Zbl 0174.17402)] stability theory of the class \(I_n\) of isometric mappings from the space \(\mathbb{R}^n\) into itself, A. P. Kopylov [Sib. Mat. Zh. 25, No. 2(144), 132-144 (1984; Zbl 0546.30019)] proposed a general approach for the study of the stability problem of Lipschitz mappings, called by him \(\omega\)-stability of the class of this mappings.
The author, in the paper under review, establishes the following result in the case of affine mappings: Let \(m,n\in\mathbb{N}\) and \(\mathcal A\) a \(K^*\)-normal class of mappings (cf. A. P. Kopylov’s paper quoted above) with \(D{\mathcal A}=0\). Then \(\mathcal A\) is \(\omega\)-stable and there exists a function \(\sigma:[0,+\infty[\to[0,+\infty[\) such that 1) \(\sigma(\varepsilon)\to\sigma(0)=0\) if \(\varepsilon\to 0\) and 2) for each domain \(\Delta\subset\mathbb{R}^n\), star-like with respect to one of its points, for each mapping \(f:\Delta\to\mathbb{R}^m\) with \(\Omega(f,{\mathcal G})<\infty\), there exists a mapping \(g:\Delta\to\mathbb{R}^m\) of the class \(\mathcal A\) such that \(\sup_{y\in\Delta}|f(y)-g(y)|\leq\sigma[\Omega(f,{\mathcal A})]\text{diam }\Delta\).
In the particular case \(m=n=1\), \(\mathcal A\) is \(\omega\)-stable iff \(D{\mathcal A}=0\). If \(\mathcal A\) is such that \(D{\mathcal A}=\{t\alpha+\beta,t\in J\}\), where \(\alpha,\beta\in L(\mathbb{R}^n,\mathbb{R}^m)\), \(\alpha\neq 0\), \(\text{rank }\alpha\geq 2\) and \(J\) is a compact set in \(\mathbb{R}\), then, \(\mathcal A\) is \(\omega\)-stable. In the particular case \(J=\{t\in\mathbb{R}^n: a\leq t\leq b\}\), \(a,b\in\mathbb{R}\), \(a<b\), then, \(\text{rank }\alpha\geq 2\) iff \(\mathcal A\) is \(\omega\)-stable.
Finally, he establishes also a criterium of \(\omega\)-saturation [cf. A. P. Kopylov, “Stability in the \(C\)-norm of classes of mappings” (1990; Zbl 0772.30023)].
Reviewer: P.Caraman (Iaşi)

MSC:

30C62 Quasiconformal mappings in the complex plane
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References:

[1] F. John, ”Rotation and strain,” Comm. Pure Appl. Math.,14, No. 3, 391–413 (1961). · Zbl 0102.17404 · doi:10.1002/cpa.3160140316
[2] F. John, ”On quasiisometric mappings. I,” Comm. Pure Appl. Math.,21, No. 1, 77–110 (1968). · Zbl 0157.45803 · doi:10.1002/cpa.3160210107
[3] F. John, ”On quasiisometric mappings. II,” Comm. Pure Appl. Math.,22, No. 2, 265–278 (1969). · Zbl 0174.17402 · doi:10.1002/cpa.3160220209
[4] A. P. Kopylov, ”On stability of isometric mappings,” Sibirsk. Mat. Zh.,25, No. 2, 132–144 (1984). · Zbl 0546.30019
[5] Yu. G. Reshetnyak, Space Mappings with Bounded Distortion, Amer. Math. Soc., Providence, R. I. (1989) (Math. Monogr.;73). · Zbl 0667.30018
[6] A. P. Kopylov, Stability in theC-Norm of Classes of Mappings [in Russian], Nauka, Novosibirsk (1990). · Zbl 0772.30023
[7] A. A. Egorov, ”On stability of classes of affine mappings,” Dokl. RAN,325, No. 3, 425–427 (1992). · Zbl 0821.30015
[8] L. G. Gurov, ”On stability of Lorentz mappings,” Dokl. Akad. Nauk SSSR,213, No. 2, 267–269 (1973). · Zbl 0316.58011
[9] L. G. Gurov, ”On stability of pseudoisometries,” in: Contemporary Problems of Geometry and Analysis [in Russian], Nauka, Novosibirsk, 1989, pp. 89–98 (Trudy Inst. Mat. (Novosibirsk);14).
[10] T. V. Sokolova, Stability of Homothety Mappings Relative to the SpaceW p 1 [in Russian], Dis. Kand. Fiz.-Mat. Nauk, Novosibirsk (1991).
[11] K. Kuratowski, Topology. Vol. 1 and 2 [Russian translation], Mir, Moscow (1966, 1969).
[12] L. Hörmander, The Analysis of Linear Partial Differential Equations. Vol. 1 [Russian translation], Mir, Moscow (1986).
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