Geodesic
Traces of functions with a dominating mixed derivative in $\Bbb R^3$
Czechoslovak Mathematical Journal, Tome 57 (2007) no. 4, pp. 1239-1273 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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.
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.
Classification : 42B35, 46E35
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 
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Vybíral, Jan; Sickel, Winfried. Traces of functions with a dominating mixed derivative in $\Bbb R^3$. Czechoslovak Mathematical Journal, Tome 57 (2007) no. 4, pp. 1239-1273. http://geodesic.mathdoc.fr/item/CMJ_2007_57_4_a9/

[1] D. R. Adams and L. I. Hedberg: Function spaces and potential theory. Springer, Berlin, 1996. | MR

[2] T. I. Amanov: Spaces of differentiable functions with dominating mixed derivatives. Nauka Kaz. SSR, Alma-Ata, 1976. (Russian) | MR

[3] D. B. Bazarkhanov: Characterizations of the Nikol’skii-Besov and Lizorkin-Triebel function spaces of mixed smoothness. Trudy Mat. Inst. Steklov 243 (2003), 53-65. | MR | Zbl

[4] H.-J. Bungartz and M. Griebel: Sparse grids. Acta Numerica (2004), 1–123. | MR

[5] M. Frazier and B. Jawerth: Decomposition of Besov spaces. Indiana Univ. Math. J. 34 (1985), 777–799. | DOI | MR

[6] M. Frazier and B. Jawerth: A discrete transform and decomposition of distribution spaces. J. Funct. Anal. 93 (1990), 34–170. | DOI | MR

[7] I. W. Gelman and W. G. Maz’ya: Abschätzungen für Differentialoperatoren im Halbraum. Akademie-Verlag, Berlin, 1981. | MR

[8] M. Griebel, P. Oswald and T. Schiekofer: Sparse grids for boundary integral equations. Numer. Math 83 (1999), 279–312. | DOI | MR

[9] P. Grisvard: Commutativité de deux fonctuers d’interpolation et applications. J. Math. Pures Appl. 45 (1966), 143–290. | MR

[10] L. I. Hedberg and Y. Netrusov: An axiomatic approach to function spaces, spectral synthesis, and Luzin approximation. Memoirs of the AMS (to appear). | MR

[11] R. Hochmuth: A $\varphi $-transform result for spaces with dominating mixed smoothness properties. Result. Math. 33 (1998), 106–119. | DOI | MR | Zbl

[12] P. I. Lizorkin and S. M. Nikol’skij: Classification of differentiable functions on the basis of spaces with dominating mixed smoothness. Trudy Mat. Inst. Steklov 77 (1965), 143–167. | MR

[13] S. M. Nikol’skij: On boundary properties of differentiable functions of several variables. Dokl. Akad. Nauk SSSR 146 (1962), 542–545. | MR

[14] S. M. Nikol’skij: On stable boundary values of differentiable functions of several variables. Mat. Sbornik 61 (1963), 224–252. | MR

[15] P.-A. Nitsche: Sparse approximation of singularity functions. Constr. Approx. 21 (2005), 63–81. | MR | Zbl

[16] M. C. Rodríguez Fernández: Über die Spur von Funktionen mit dominierenden gemischten Glattheitseigenschaften auf der Diagonale, Ph.D-thesis, Jena. 1997.

[17] H.-J. Schmeisser: Vector-valued Sobolev and Besov spaces. Teubner-Texte zur Math. 96 (1987), 4-44. | MR | Zbl

[18] H.-J. Schmeisser and H. Triebel: Topics in Fourier analysis and function spaces. Wiley, Chichester, 1987. | MR

[19] G. Sparr: Interpolation of several Banach spaces. Ann. Mat. Pura Appl. 99 (1974), 247–316. | DOI | MR | Zbl

[20] H. Triebel: A diagonal embedding theorem for function spaces with dominating mixed smoothness properties. Banach Center Publications, Warsaw 22 (1989), 475–486. | MR | Zbl

[21] H. Triebel: Fractals and Spectra. Birkhäuser, Basel, 1997. | MR | Zbl

[22] H. Triebel: The Structure of Functions. Birkhäuser, Basel, 2001. | MR | Zbl

[23] J. Vybíral: Characterisations of function spaces with dominating mixed smoothness properties. Jenaer Schriften zur Mathematik und Informatik, Math/Inf/15/03, 2003.

[24] J. Vybíral: Function spaces with dominating mixed smoothness. Diss. Math. 436 (2006), 1–73. | MR

[25] J. Vybíral: A diagonal embedding theorem for function spaces with dominating mixed smoothness. Funct. et Approx. 33 (2005), 101–120. | MR

[27] H. Yserentant: On the regularity of the electronic Schrödinger equation in Hilbert spaces of mixed derivatives. Numer. Math. 98 (2004), 731–759. | DOI | MR | Zbl

[28] H. Yserentant: Sparse grid spaces for the numerical solution of the electronic Schrödinger equation. Numer. Math. 101 (2005), 381–389. | DOI | MR | Zbl