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Remarks on the global Sobolev inequalities in the Minkowski space \({\mathbb{R}}^{n+1}\). (English) Zbl 0686.46019

The inequalities referred to in the title were first introduced by the author [same journal 38, No.3, 321-332 (1985; Zbl 0635.35059)] and developed further by L. Hörmander [“On Sobolev spaces associated with some Lie algebras”, Current topics in partial differential equations. Pap. dedic. Shizohata Occas. 60th birthday, 261-287 (1986; Zbl 0658.46023)] and the author [Nonlinear systems of partial differential equations in applied mathematics, Part 1 (Santa Fe, N.M., 1984), 293-326, Am. Math. Soc., Providence, R.I. (1986; Zbl 0599.35105)]. The present paper contains a simpler derivation of the inequalities based on the classical Sobolev inequality and a scaling argument.

MSC:

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
35L70 Second-order nonlinear hyperbolic equations
58D15 Manifolds of mappings
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References:

[1] Klainerman, Comm. Pure Appl. Math. 38 pp 321– (1985)
[2] On Sobolev spaces associated with some Lie algebras, Report 4, 1985, Inst. Mittag-Leffler.
[3] Klainerman, Lectures in Appl. Math. 23 pp 293– (1986)
[4] Klainerman, Comm. Pure Appl. Math. 38 pp 631– (1985)
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