| 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:
|
http://hdl.handle.net/10338.dmlcz/118471 |
| . |
| 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:
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[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 |
| . |
