Traces of functions with a dominating mixed derivative in 3

Jan Vybíral; Winfried Sickel

Czechoslovak Mathematical Journal (2007)

  • Volume: 57, Issue: 4, page 1239-1273
  • ISSN: 0011-4642

Abstract

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We investigate traces of functions, belonging to a class of functions with dominating mixed smoothness in 3 , with respect to planes in oblique position. In comparison with the classical theory for isotropic spaces a few new phenomenona occur. We shall present two different approaches. One is based on the use of the Fourier transform and restricted to p = 2 . The other one is applicable in the general case of Besov-Lizorkin-Triebel spaces and based on atomic decompositions.

How to cite

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Vybíral, Jan, and Sickel, Winfried. "Traces of functions with a dominating mixed derivative in $\mathbb {R}^3$." Czechoslovak Mathematical Journal 57.4 (2007): 1239-1273. <http://eudml.org/doc/31191>.

@article{Vybíral2007,
abstract = {We investigate traces of functions, belonging to a class of functions with dominating mixed smoothness in $\{\mathbb \{R\}\}^3$, with respect to planes in oblique position. In comparison with the classical theory for isotropic spaces a few new phenomenona occur. We shall present two different approaches. One is based on the use of the Fourier transform and restricted to $p=2$. The other one is applicable in the general case of Besov-Lizorkin-Triebel spaces and based on atomic decompositions.},
author = {Vybíral, Jan, Sickel, Winfried},
journal = {Czechoslovak Mathematical Journal},
keywords = {Sobolev spaces of dominating mixed smoothness; Besov and Lizorkin-Triebel classes of dominating mixed smoothness; Fourier analytic characterizations; atomic decompositions; traces on hyperplanes in oblique position; Sobolev spaces of dominating mixed smoothness},
language = {eng},
number = {4},
pages = {1239-1273},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Traces of functions with a dominating mixed derivative in $\mathbb \{R\}^3$},
url = {http://eudml.org/doc/31191},
volume = {57},
year = {2007},
}

TY - JOUR
AU - Vybíral, Jan
AU - Sickel, Winfried
TI - Traces of functions with a dominating mixed derivative in $\mathbb {R}^3$
JO - Czechoslovak Mathematical Journal
PY - 2007
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 57
IS - 4
SP - 1239
EP - 1273
AB - We investigate traces of functions, belonging to a class of functions with dominating mixed smoothness in ${\mathbb {R}}^3$, with respect to planes in oblique position. In comparison with the classical theory for isotropic spaces a few new phenomenona occur. We shall present two different approaches. One is based on the use of the Fourier transform and restricted to $p=2$. The other one is applicable in the general case of Besov-Lizorkin-Triebel spaces and based on atomic decompositions.
LA - eng
KW - Sobolev spaces of dominating mixed smoothness; Besov and Lizorkin-Triebel classes of dominating mixed smoothness; Fourier analytic characterizations; atomic decompositions; traces on hyperplanes in oblique position; Sobolev spaces of dominating mixed smoothness
UR - http://eudml.org/doc/31191
ER -

References

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  1. Function spaces and potential theory, Springer, Berlin, 1996. (1996) MR1411441
  2. Spaces of differentiable functions with dominating mixed derivatives, Nauka Kaz. SSR, Alma-Ata, 1976. (Russian) (1976) MR0860038
  3. Characterizations of the Nikol’skii-Besov and Lizorkin-Triebel function spaces of mixed smoothness, Trudy Mat. Inst. Steklov 243 (2003), 53-65. (2003) Zbl1090.46025MR2049462
  4. Sparse grids, Acta Numerica (2004), 1–123. (2004) MR2249147
  5. 10.1512/iumj.1985.34.34041, Indiana Univ. Math. J. 34 (1985), 777–799. (1985) MR0808825DOI10.1512/iumj.1985.34.34041
  6. 10.1016/0022-1236(90)90137-A, J. Funct. Anal. 93 (1990), 34–170. (1990) MR1070037DOI10.1016/0022-1236(90)90137-A
  7. Abschätzungen für Differentialoperatoren im Halbraum, Akademie-Verlag, Berlin, 1981. (1981) MR0644480
  8. 10.1007/s002110050450, Numer. Math 83 (1999), 279–312. (1999) MR1712687DOI10.1007/s002110050450
  9. Commutativité de deux fonctuers d’interpolation et applications, J. Math. Pures Appl. 45 (1966), 143–290. (1966) MR0221309
  10. An axiomatic approach to function spaces, spectral synthesis, and Luzin approximation, Memoirs of the AMS (to appear). MR2326315
  11. 10.1007/BF03322075, Result. Math. 33 (1998), 106–119. (1998) Zbl0905.46017MR1610139DOI10.1007/BF03322075
  12. Classification of differentiable functions on the basis of spaces with dominating mixed smoothness, Trudy Mat. Inst. Steklov 77 (1965), 143–167. (1965) MR0192223
  13. On boundary properties of differentiable functions of several variables, Dokl. Akad. Nauk SSSR 146 (1962), 542–545. (1962) MR0143064
  14. On stable boundary values of differentiable functions of several variables, Mat. Sbornik 61 (1963), 224–252. (1963) MR0156182
  15. Sparse approximation of singularity functions, Constr. Approx. 21 (2005), 63–81. (2005) Zbl1073.65118MR2105391
  16. Über die Spur von Funktionen mit dominierenden gemischten Glattheitseigenschaften auf der Diagonale, Ph.D-thesis, Jena, 1997. (1997) 
  17. Vector-valued Sobolev and Besov spaces, Teubner-Texte zur Math. 96 (1987), 4-44. (1987) Zbl0666.46039MR0932287
  18. Topics in Fourier analysis and function spaces, Wiley, Chichester, 1987. (1987) MR0891189
  19. 10.1007/BF02413728, Ann. Mat. Pura Appl. 99 (1974), 247–316. (1974) Zbl0282.46022MR0372641DOI10.1007/BF02413728
  20. A diagonal embedding theorem for function spaces with dominating mixed smoothness properties, Banach Center Publications, Warsaw 22 (1989), 475–486. (1989) Zbl0707.46020MR1097217
  21. Fractals and Spectra, Birkhäuser, Basel, 1997. (1997) Zbl0898.46030MR1484417
  22. The Structure of Functions, Birkhäuser, Basel, 2001. (2001) Zbl0984.46021MR1851996
  23. Characterisations of function spaces with dominating mixed smoothness properties, Jenaer Schriften zur Mathematik und Informatik, Math/Inf/15/03, 2003. (2003) 
  24. Function spaces with dominating mixed smoothness, Diss. Math. 436 (2006), 1–73. (2006) MR2231066
  25. A diagonal embedding theorem for function spaces with dominating mixed smoothness, Funct. et Approx. 33 (2005), 101–120. (2005) MR2274153
  26. 10.1007/s00211-003-0498-1, Numer. Math. 98 (2004), 731–759. (2004) Zbl1062.35100MR2099319DOI10.1007/s00211-003-0498-1
  27. 10.1007/s00211-005-0581-x, Numer. Math. 101 (2005), 381–389. (2005) Zbl1084.65125MR2195349DOI10.1007/s00211-005-0581-x

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