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Title: On the boundedness of the mapping $f\to |f|$ in Besov spaces (English)
Author: Oswald, P.
Language: English
Journal: Commentationes Mathematicae Universitatis Carolinae
ISSN: 0010-2628 (print)
ISSN: 1213-7243 (online)
Volume: 33
Issue: 1
Year: 1992
Pages: 57-66
Category: math
Summary: For $1\leq p\leq\infty$, precise conditions on the parameters are given under which the particular superposition operator $T:f\to |f|$ is a bounded map in the Besov space $B^s_{p,q}(R^1)$. The proofs rely on linear spline approximation theory. (English)
Keyword: Nemytzki operators
Keyword: Besov spaces
Keyword: moduli of smoothness
Keyword: linear splines
MSC: 35B45
MSC: 41A15
MSC: 46E35
MSC: 47H30
idZBL: Zbl 0766.46018
idMR: MR1173747
Date available: 2009-01-08T17:53:33Z
Last updated: 2012-04-30
Stable URL:
Reference: [AZ] Appell J., Zabrejko P.: Nonlinear superposition operators.Cambr. Univ. Press, Cambridge, 1990. Zbl 1156.47052, MR 1066204
Reference: [BM] Bourdaud G., Meyer Y.: Fonctions qui operent sur les espaces de Sobolev.J. Funct. Anal. 97 (1991), 351-360. Zbl 0737.46011, MR 1111186
Reference: [CW] Cazenave T., Weissler F.B.: The Cauchy problem for the critical nonlinear Schrödinger equation in $H^s$.Nonl. Anal. Th. Meth. Appl. 14 (1990), 807-836. MR 1055532
Reference: [MM1] Marcus M., Mizel V.J.: Absolute continuity on tracks and mappings of Sobolev spaces.Arch. Rat. Mech. Anal. 45 (1972), 294-320. Zbl 0236.46033, MR 0338765
Reference: [MM2] Marcus M., Mizel V.J.: Nemitsky operators on Sobolev spaces.Arch. Rat. Mech. Anal. 51 (1973), 347-370. Zbl 0266.46029, MR 0348480
Reference: [MM3] Marcus M., Mizel V.J.: Every superposition operator mapping one Sobolev space into another is continuous.J. Funct. Anal. 33 (1978), 217-229. MR 0546508
Reference: [N] Nikolskij S.M.: Approximation of functions of several variables and imbedding theorems (2nd edition).Nauka, Moskva, 1977. MR 0506247
Reference: [O1] Oswald P.: On estimates for one-dimensional spline approximation.In: Splines in Numerical Analysis (eds. J.Späth, J.W.Schmidt), Proc. ISAM'89 Wei{ß}ig 1989, Akad. Verl., Berlin, 1989, 111-124. Zbl 0739.41015, MR 1004256
Reference: [O2] Oswald P.: On estimates for hierarchic basis representations of finite element functions.Report N/89/16, FSU Jena, 1989.
Reference: [RS] Runst T., Sickel W.: Mapping properties of $T:f\to |f|$ in Besov-Triebel-Lizorkin spaces and an application to a nonlinear boundary value problem.J. Approx. Th. (submitted).
Reference: [Sch] Schumaker L.L.: Spline functions: basic theory.Wiley, New York, 1981. Zbl 1123.41008, MR 0606200
Reference: [S1] Sickel W.: On boundedness of superposition operators in spaces of Triebel-Lizorkin type.Czech. Math. J. 39 (1989), 323-347. Zbl 0693.46039, MR 0992137
Reference: [S2] Sickel W.: Superposition of functions in Sobolev spaces of fractional order.A survey. Banach Center Publ. (submitted). Zbl 0792.47062
Reference: [T] Triebel H.: Interpolation theory, function spaces, differential operators.Dt. Verlag Wiss., Berlin 1978 - North-Holland, Amsterdam-New York-Oxford, 1978. Zbl 0830.46028, MR 0500580


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