On the boundedness of the mapping $f\to |f|$ in Besov spaces

Patrick Oswald

Commentationes Mathematicae Universitatis Carolinae (1992)

  • Volume: 33, Issue: 1, page 57-66
  • ISSN: 0010-2628

Abstract

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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.

How to cite

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Oswald, Patrick. "On the boundedness of the mapping $f\to |f|$ in Besov spaces." Commentationes Mathematicae Universitatis Carolinae 33.1 (1992): 57-66. <http://eudml.org/doc/247418>.

@article{Oswald1992,
abstract = {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.},
author = {Oswald, Patrick},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {scale of Besov spaces; linear spline approximation},
language = {eng},
number = {1},
pages = {57-66},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {On the boundedness of the mapping $f\to |f|$ in Besov spaces},
url = {http://eudml.org/doc/247418},
volume = {33},
year = {1992},
}

TY - JOUR
AU - Oswald, Patrick
TI - On the boundedness of the mapping $f\to |f|$ in Besov spaces
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1992
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 33
IS - 1
SP - 57
EP - 66
AB - 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.
LA - eng
KW - scale of Besov spaces; linear spline approximation
UR - http://eudml.org/doc/247418
ER -

References

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  2. Bourdaud G., Meyer Y., Fonctions qui operent sur les espaces de Sobolev, J. Funct. Anal. 97 (1991), 351-360. (1991) Zbl0737.46011MR1111186
  3. 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. (1990) MR1055532
  4. Marcus M., Mizel V.J., Absolute continuity on tracks and mappings of Sobolev spaces, Arch. Rat. Mech. Anal. 45 (1972), 294-320. (1972) Zbl0236.46033MR0338765
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  6. Marcus M., Mizel V.J., Every superposition operator mapping one Sobolev space into another is continuous, J. Funct. Anal. 33 (1978), 217-229. (1978) MR0546508
  7. Nikolskij S.M., Approximation of functions of several variables and imbedding theorems (2nd edition), Nauka, Moskva, 1977. MR0506247
  8. 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. Zbl0739.41015MR1004256
  9. Oswald P., On estimates for hierarchic basis representations of finite element functions, Report N/89/16, FSU Jena, 1989. 
  10. 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). 
  11. Schumaker L.L., Spline functions: basic theory, Wiley, New York, 1981. Zbl1123.41008MR0606200
  12. Sickel W., On boundedness of superposition operators in spaces of Triebel-Lizorkin type, Czech. Math. J. 39 (1989), 323-347. (1989) Zbl0693.46039MR0992137
  13. Sickel W., Superposition of functions in Sobolev spaces of fractional order, A survey. Banach Center Publ. (submitted). Zbl0792.47062
  14. Triebel H., Interpolation theory, function spaces, differential operators, Dt. Verlag Wiss., Berlin 1978 - North-Holland, Amsterdam-New York-Oxford, 1978. Zbl0830.46028MR0500580

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