Characterization of traces of smooth functions on Ahlfors regular sets. (English) Zbl 1293.46022
Summary: We extend the results of P. Shvartsman [Math. Nachr. 279, No. 11, 1212–1241 (2006; Zbl 1108.46030)] on characterizing the traces of Besov and Triebel-Lizorkin spaces on Ahlfors \(n\)-regular sets to the case of \(d\)-regular sets, \(n - 1 < d < n\). The characterizations of trace spaces are given in terms of local polynomial approximations.
MSC:
| 46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
Keywords:
Triebel-Lizorkin space; Besov space; Ahlfors \(d\)-regular set; trace theorem; local polynomial approximationCitations:
Zbl 1108.46030References:
| [1] | Adams, D. R.; Hedberg, L. I., Function Spaces and Potential Theory, Grundlehren Math. Wiss., vol. 314 (1996), Springer-Verlag: Springer-Verlag Berlin |
| [2] | Besov, O. V.; Ilʼin, V. P.; Nikolʼskii, S. M., Integral Representations of Functions and Embedding Theorems, vol. 1 (1978), J. Wiley and Sons: J. Wiley and Sons New York, vol. 2, 1979 · Zbl 0392.46022 |
| [3] | Bojarski, B., Remarks on Sobolev Imbedding Inequalities, Lecture Notes in Math., vol. 1351, 52-68 (1988), Springer: Springer New York · Zbl 0662.46037 |
| [4] | Brudnyi, Yu., Spaces defined by means of local approximations, Trudy Moskov. Math. Obshch.. Trudy Moskov. Math. Obshch., Trans. Moscow Math. Soc., 24, 73-139 (1974), English transl.: · Zbl 0295.46055 |
| [5] | Brudnyi, Yu., Piecewise Polynomial Approximation, Embedding Theorem and Rational Approximation, Lecture Notes in Math., vol. 556, 73-98 (1976), Springer-Verlag: Springer-Verlag Berlin · Zbl 0347.41006 |
| [6] | Brudnyi, Yu., Sobolev spaces and their relatives: local polynomial approximation approach, (Sobolev Spaces in Mathematics II. Sobolev Spaces in Mathematics II, Int. Math. Ser., vol. 9 (2009), Springer: Springer New York), 31-68 · Zbl 1175.46020 |
| [7] | Brudnyi, A.; Brudnyi, Yu., Remez type inequalities and Morrey-Campanato spaces on Ahlfors regular sets, Contemp. Math., 445, 19-44 (2007) · Zbl 1132.41313 |
| [8] | DeVore, R. A.; Sharpley, R. C., Maximal functions measuring smoothness, Mem. Amer. Math. Soc., 47, 1-115 (1984) · Zbl 0529.42005 |
| [9] | Dorronsoro, J. R., Poisson integrals of regular functions, Trans. Amer. Math. Soc., 297, 669-685 (1986) · Zbl 0613.31005 |
| [10] | Frazier, M.; Jawerth, B., A discrete transform and decompositions of distribution spaces, J. Funct. Anal., 93, 34-170 (1990) · Zbl 0716.46031 |
| [11] | Hedberg, L. I.; Netrusov, Y. V., An axiomatic approach to function spaces, spectral synthesis, and Luzin approximation, Mem. Amer. Math. Soc., 188, 1-97 (2007) · Zbl 1186.46028 |
| [12] | Ihnatsyeva, L.; Korte, R., Smoothness spaces of higher order on lower dimensional subsets of the Euclidean space (2011), preprint |
| [13] | Iwaniec, T.; Nolder, C. A., Hardy-Littlewood inequality for quasiregular mappings in certain domains in \(R^n\), Ann. Acad. Sci. Fenn. Ser. A I Math., 10, 267-282 (1985) · Zbl 0588.30023 |
| [14] | Jonsson, A., Atomic decomposition of Besov spaces on closed sets, (Function Spaces, Differential Operators and Non-Linear Analysis. Function Spaces, Differential Operators and Non-Linear Analysis, Teubner-Texte Math., vol. 133 (1993), Teubner: Teubner Leipzig), 285-289 · Zbl 0831.46028 |
| [15] | Jonsson, A.; Wallin, H., Function Spaces on Subsets of \(R^n\), Math. Rep., vol. 2, Part 1 (1984), Harwood Acad. Publ.: Harwood Acad. Publ. London · Zbl 0875.46003 |
| [16] | Leindler, L., Generalization of inequalities of Hardy and Littlewood, Acta Sci. Math. (Szeged), 31, 279-285 (1970) · Zbl 0203.06103 |
| [17] | Luukkainen, J., Assouad dimension: antifractal metrization, porous sets, and homogeneous measures, J. Korean Math. Soc., 35, 23-76 (1998) · Zbl 0893.54029 |
| [18] | Mamedov, G. A., On traces of functions from anisotropic Nikolʼskii-Besov and Lizorkin-Triebel spaces on subsets of Euclidean space, Proc. Steklov Inst. Math., 194, 165-185 (1993) · Zbl 0806.46034 |
| [19] | Mattila, P., Geometry of Sets and Measures in Euclidean Spaces: Fractals and Rectifiability, Cambridge Stud. Adv. Math., vol. 44 (1995), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0819.28004 |
| [20] | Mazʼya, V. G.; Poborchi, S. V., Differentiable Functions on Bad Domains (1997), World Scientific: World Scientific Singapore · Zbl 0918.46033 |
| [21] | Netrusov, Y. V., Sets of singularities of functions in spaces of Besov and Lizorkin-Triebel type, Proc. Steklov Inst. Math., 187, 185-203 (1990) · Zbl 0719.46018 |
| [22] | Saka, K., The trace theorem for Triebel-Lizorkin spaces and Besov spaces on certain fractal sets I. The restriction theorem, Mem. College Ed. Akita Univ. Natur. Sci., 48, 1-17 (1995) · Zbl 0866.46014 |
| [23] | Saka, K., The trace theorem for Triebel-Lizorkin spaces and Besov spaces on certain fractal sets II. The extension theorem, Mem. College Ed. Akita Univ. Natur. Sci., 49, 1-27 (1996) · Zbl 0866.46015 |
| [24] | Shvartsman, P., Extension theorem for a class of spaces defined by local approximations, (Studies in the Theory of Functions of Several Real Variables (1978), Yaroslavl State University: Yaroslavl State University Yaroslavl), 215-242, (in Russian) · Zbl 0463.41025 |
| [25] | Shvartsman, P., Local approximations and intrinsic characterization of spaces of smooth functions on regular subsets of \(R^n\), Math. Nachr., 279, 1212-1241 (2006) · Zbl 1108.46030 |
| [26] | Stein, E. M., Singular Integrals and Differentiability Properties of Functions (1970), Princeton University Press: Princeton University Press Princeton · Zbl 0207.13501 |
| [27] | Triebel, H., Local approximation spaces, Z. Anal. Anwend., 8, 261-288 (1989) · Zbl 0695.46015 |
| [28] | Triebel, H., Theory of Function Spaces (1983), Birkhäuser: Birkhäuser Basel · Zbl 0546.46028 |
| [29] | Triebel, H., Theory of Function Spaces II (1992), Birkhäuser: Birkhäuser Basel · Zbl 0763.46025 |
| [30] | Triebel, H., Fractals and Spectra (1997), Birkhäuser: Birkhäuser Basel · Zbl 0898.46030 |
| [31] | Triebel, H., The Structure of Functions (2001), Birkhäuser: Birkhäuser Basel · Zbl 0984.46021 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
