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Keywords:
anisotropic Besov and Lizorkin-Triebel spaces; approximation spaces; trace operators; boundary problems; interpolation; atomic decompositions; refined Sobolev embeddings; anisotropic scales
anisotropic Besov and Lizorkin-Triebel spaces; approximation spaces; trace operators; boundary problems; interpolation; atomic decompositions; refined Sobolev embeddings; anisotropic scales
Summary:
Including the previously untreated borderline cases, the trace spaces (in the distributional sense) of the Besov-Lizorkin-Triebel spaces are determined for the anisotropic (or quasi-homogeneous) version of these classes. The ranges of the traces are in all cases shown to be approximation spaces, and these are shown to be different from the usual spaces precisely in the cases previously untreated. To analyse the new spaces, we carry over some real interpolation results as well as the refined Sobolev embeddings of J. Franke and B. Jawerth to the anisotropic scales.
References:
[1] S. Agmon L. Hörmander: Asymptotic properties of differential equations with simple characteristics. J. Anal. Math. 1 (1976), 1-38. DOI 10.1007/BF02786703 | MR 0466902
[2] N. Aronszajn: Boundary values of functions with finite Dirichlet integral. Studies in Eigenvalue Problems, vol. 14, Univ. of Kansas, 1955. Zbl 0068.08201
[3] J. Bergh J. Löfström: Interpolation Spaces. An Introduction. Springer, Berlin, 1976. MR 0482275
[4] O. V. Besov V. P. Ilyin S. M. Nikol'skij: Integral Representations of Functions, Imbedding Theorems. Nauka, Moskva, 1967. (In Russian.)
[5] V. I. Burenkov M. L. Gol'dman: On the extensions of functions of $L_p$. Trudy Mat. Inst. Steklov. 150 (1979), 31-51. English transl. 1981, no. 4, 33-53. MR 0544003
[6] P. Dintelmann: On Fourier multipliers between anisotropic weighted function spaces. Ph.D.Thesis, TH Darmstadt, 1995. (In German.)
[7] P. Dintelmann: Classes of Fourier multipliers and Besov-Nikol'skij spaces. Math. Nachr. 173 (1995). 115-130. DOI 10.1002/mana.19951730108 | MR 1336956
[8] W. Farkas: Atomic and subatomic decompositions in anisotropic function spaces. Math. Nachr. To appear. MR 1734360 | Zbl 0954.46021
[9] C. Fefferman E. M. Stein: Some maximal inequalities. Amer. J. Math. 93 (1971), 107-115. DOI 10.2307/2373450 | MR 0284802
[10] J. Franke: On the spaces $F_{p,q}^s$ of Triebel-Lizorkin type: Pointwise multipliers and spaces on domains. Math. Nachr. 125 (1986), 29-68. DOI 10.1002/mana.19861250104 | MR 0847350
[11] M. Frazier B. Jawerth: Decomposition of Besov spaces. Indiana Univ. Math, J. 34 (1985), 777-799. DOI 10.1512/iumj.1985.34.34041 | MR 0808825
[12] M. Frazier B. Jawerth: A discrete transform and decomposition of distribution spaces. J. Functional Anal. 93 (1990), 34-170. DOI 10.1016/0022-1236(90)90137-A | MR 1070037
[13] E. Gagliardo: Caraterizzazioni della trace sulla frontiera relative ad alcune classi di funzioni in n variabili. Rend. Sem. Mat. Univ. Padova 27 (1957), 284-305. MR 0102739
[14] G. Grubb: Pseudo-differential boundary problems in $L_p$ spaces. Comm. Partial Differential Equations 15 (1990), 289-340. DOI 10.1080/03605309908820688 | MR 1044427
[15] G. Grubb: Functional Calculus of Pseudodifferential Boundary Problems. Birkhäuser, Basel, 1996, second edition. MR 1385196 | Zbl 0844.35002
[16] G. Grubb: Parameter-elliptic and parabolic pseudodifferential boundary problems in global $L_p$ Sobolev spaces. Math. Z. 218 (1995), 43-90. DOI 10.1007/BF02571889 | MR 1312578
[17] L. Hörmander: The Analysis of Linear Partial Differential Operators I-IV. Springer, Berlin, 1983-85. MR 0717035
[18] B. Jawerth: Some observations on Besov and Lizorkin-Triebel spaces. Math. Scand. 40 (1977), 94-104. DOI 10.7146/math.scand.a-11678 | MR 0454618 | Zbl 0358.46023
[19] B. Jawerth: The trace of Sobolev and Besov spaces if 0 < p < 1. Studla Math. 62 (1978), 65-71. MR 0482141 | Zbl 0423.46022
[20] J. Johnsen: Pointwise multiplication of Besov and Triebel-Lizorkin spaces. Math. Nachr. 175 (1995), 85-133. DOI 10.1002/mana.19951750107 | MR 1355014 | Zbl 0839.46026
[21] J. Johnsen: Elliptic boundary problems and the Boutet de Monvel calculus in Besov and Triebel-Lizorkin spaces. Math. Scand. 79 (1996), 25-85. DOI 10.7146/math.scand.a-12593 | MR 1425081 | Zbl 0873.35023
[22] J. Johnsen: Traces of Besov spaces revisited. Submitted 1998.
[23] G. A. Kalyabin: Description of traces for anisotropic spaces of Triebel-Lizorkin type. Trudy Mat. Inst. Steklov. 150 (1979), 160-173. English transl. 1981, no. 4, 169-183. MR 0544009 | Zbl 0417.46040
[24] J. Marschall: Remarks on nonregular pseudo-differential operators. Z. Anal. Anwendungen 15(1996), 109-148. DOI 10.4171/ZAA/691 | MR 1376592
[25] Yu. V. Netrusov: Imbedding theorems of traces of Besov spaces and Lizorkin-Triebel spaces. Dokl. AN SSSR 298 (1988), no. 6. English transl. Soviet Math. Doki. 37 (1988), no. 1, 270-273. MR 0947796
[26] Yu. V. Netrusov: Sets of singularities of functions in spaces of Besov and Lizorkin-Triebel type. Trudy Mat. Inst. Steklov. 187(1989), 162-177. English transl. 199, no. 3, 185-203. MR 1006450
[27] S. M. Nikol'skij: Inequalities for entire analytic functions of finite order and their application to the theory of differentiable functions of several variables. Trudy Mat. Inst. Steklov. 38 (1951), 244-278. Detailed review available in Math. Reviews.
[28] S. M. Nikol'skij: Approximation of Functions of Several Variables end Imbedding Theorems. Springer, Berlin, 1975.
[29] M. Oberguggenberger: Multiplication of distributions and applications to partial differential equations. Pitman notes, vol. 259, Longman Scientific & Technical, England, 1992. Zbl 0818.46036
[30] P. Oswald: Multilevel Finite Element Approximation: Theory and Applications. Teubner, Stuttgart, 1995. MR 1312165
[31] J. Peetre: The trace of Besov spaces-a limiting case. Technical Report, Lund, 1975.
[32] T. Runst W. Sickel: Sobolev Spaces of Fractional Order, Nemytskij Operators and Nonlinear Partial Differential Equations. De Gruyter, Berlin, 1996. MR 1419319
[33] H.-J. Schmeisser H. Triebel: Topics in Fourier Analysis and Function Spaces. Wiley, Chichester, 1987. MR 0891189
[34] A.Seeger: A note on Triebel-Lizorkin spaces. Approximations and Function Spaces, vol. 22, Banach Centre Publ., PWN Polish Sci. Publ., Warszaw, 1989, pp. 391-400. MR 1097208 | Zbl 0698.42008
[35] E. M. Stein: Singular Integrals and Differentiability Properties of Functions. Princeton Univ. Press, Princeton, 1970. MR 0290095 | Zbl 0207.13501
[36] E. M. Stein S. Wainger: Problems in harmonic analysis related to curvature. Bull. Amer. Math. Soc. 84 (1978), J 239-1295. MR 0508453
[37] B. Stöckert H. Triebel: Decomposition methods for function spaces of $B_{p,q}^s$ type and $F_{p,q}^s$ type. Math. Nachr. 89 (1979), 247-267. DOI 10.1002/mana.19790890121 | MR 0546886
[38] H. Triebel: Fourier Analysis and Function Spaces. Teubner-Texte Math., vol. 7, Teubner, Leipzig, 1977. MR 0493311 | Zbl 0345.42003
[39] H. Triebel: Spaces of Besov-Hardy-Sobolev Type. Teubner-Texte Math., vol. 8, Teubner, Leipzig, 1978. MR 0581907 | Zbl 0408.46024
[40] H. Triebel: Theory of Function Spaces. Birkhäuser, Basel, 1983. MR 0781540 | Zbl 0546.46028
[41] H. Triebel: Theory of Function Spaces II. Birkhäuser, Basel, 1992. MR 1163193 | Zbl 0763.46025
[42] H. Triebel: Fractals and Spectra. Birkhäuser, Basel, 1997. MR 1484417 | Zbl 0898.46030
[43] M. Yamazaki: A quasi-homogeneous version of paradifferential operators, I: Boundedness on spaces of Besov type. J. Fac. Sci. Univ. Tokyo, Sect. IA Math. 33 (1986), 131-174. MR 0837335 | Zbl 0608.47058
[44] M. Yamazaki: A quasi-homogeneous version of paradifferential operators, II: A symbolic calculus. J.Fac. Sci. Univ. Tokyo, Sect. IA Math. 33 (1986), 311-345. MR 0866396
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