Hardy inequalities in Triebel-Lizorkin spaces. II: Aikawa dimension. (English) Zbl 1339.46032
Summary: We prove inequalities of Hardy type for functions in Triebel-Lizorkin spaces \(F^s_{pq}(G)\) on a domain \(G\subset \mathbb {R}^n\), whose boundary has the Aikawa dimension strictly less than \(n-sp\).
For Part I see [the authors, Indiana Univ. Math. J. 62, No. 6, 1785–1807 (2013; Zbl 1306.46036)].
For Part I see [the authors, Indiana Univ. Math. J. 62, No. 6, 1785–1807 (2013; Zbl 1306.46036)].
MSC:
| 46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
| 26D15 | Inequalities for sums, series and integrals |
Citations:
Zbl 1306.46036References:
| [1] | Adams, D.R., Hedberg, L.I.: Function Spaces and Potential Theory. Grundlehren Math. Wiss., vol. 314. Springer, Berlin (1996) · Zbl 0834.46021 · doi:10.1007/978-3-662-03282-4 |
| [2] | Aikawa, H.: Quasiadditivity of Riesz capacity. Math. Scand. 69(1), 15-30 (1991) · Zbl 0748.31002 |
| [3] | Besov, O.V., Il’in, V.P., Nikol’skii, S.M.: Integral Representations of Functions and Embedding Theorems. Wiley, New York, vol. 1 (1978); vol. 2 (1979) · Zbl 0392.46022 |
| [4] | Bojarski, B.: Remarks on Sobolev Imbedding Inequalities. Lecture Notes in Math., vol. 1351, pp. 52-68. Springer, New York (1988) · Zbl 0662.46037 |
| [5] | Brudnyi, Yu.: Spaces defined by means of local approximations. Trudy Moskov. Math. Obshch. 24, 69-132 (1971); English transl.: Trans. Moskow Math Soc. 24, 73-139 (1974) · Zbl 0254.46018 |
| [6] | Caetano, A.M.: Approximation by functions of compact support in Besov-Triebel-Lizorkin spaces on irregular domains. Studia Math. 142(1), 47-63 (2000) · Zbl 0985.46019 |
| [7] | DeVore, R.A., Sharpley, R.C.: Maximal functions measuring smoothness. Mem. Am. Math. Soc. 47, 1-115 (1984) · Zbl 0529.42005 |
| [8] | Dyda, B.: A fractional order Hardy inequality. Ill. J. Math. 48, 575-588 (2004) · Zbl 1068.26014 |
| [9] | Edmunds, D.E., Hurri-Syrjänen, R., Vähäkangas, A.V.: Fractional Hardy-type inequalities in domains with uniformly fat complement. Proc. Am. Math. Soc. (to appear) · Zbl 1296.26053 |
| [10] | Frazier, M., Jawerth, B.: A discrete transform and decompositions of distribution spaces. J. Funct. Anal. 93, 34-170 (1990) · Zbl 0716.46031 · doi:10.1016/0022-1236(90)90137-A |
| [11] | Gehring, F.W.: The \[L^p\] Lp-integrability of the partial derivatives of a quasiconformal mapping. Acta Math. 130, 265-277 (1973) · Zbl 0258.30021 · doi:10.1007/BF02392268 |
| [12] | Hurri-Syrjänen, R., Marola, N., Vähäkangas, A.V.: Aspects of Local to Global Results. arXiv:1301.7273 (2013) · Zbl 1302.42030 |
| [13] | Ihnatsyeva, L., Vähäkangas, A.V.: Characterization of traces of smooth functions on Ahlfors regular sets. J. Funct. Anal. 265, 1870-1915 (2013) · Zbl 1293.46022 · doi:10.1016/j.jfa.2013.07.006 |
| [14] | Ihnatsyeva, L., Vähäkangas, A.V.: Hardy inequalities in Triebel-Lizorkin spaces. Indiana Univ. Math. J. (to appear) · Zbl 1306.46036 |
| [15] | Iwaniec, T., Nolder, C.A.: Hardy-Littlewood inequality for quasiregular mappings in certain domains in Rn. Ann. Acad. Sci. Fenn. Ser. A I Math. 10, 267-282 (1985) · Zbl 0588.30023 · doi:10.5186/aasfm.1985.1030 |
| [16] | Koskela, P., Zhong, X.: Hardy’s inequality and the boundary size. Proc. Am. Math. Soc. 131(4), 1151-1158 (2003) · Zbl 1018.26008 · doi:10.1090/S0002-9939-02-06711-4 |
| [17] | Lehrbäck, J.: Weighted Hardy inequalities and the size of the boundary. Manuscripta Math. 127(2), 249-273 (2008) · Zbl 1181.46025 · doi:10.1007/s00229-008-0208-5 |
| [18] | Lehrbäck, J., Tuominen, H.: A note on the dimensions of Assouad and Aikawa. J. Math. Soc. Jpn. 65(2), 343-356 (2013) · Zbl 1279.54022 · doi:10.2969/jmsj/06520343 |
| [19] | Leindler, L.: Generalization of inequalities of Hardy and Littlewood. Acta Sci. Math. (Szeged) 31, 279-285 (1970) · Zbl 0203.06103 |
| [20] | Lewis, J.L.: Uniformly fat sets. Trans. Am. Math. Soc. 308, 177-196 (1988) · Zbl 0668.31002 · doi:10.1090/S0002-9947-1988-0946438-4 |
| [21] | Luukkainen, J.: Assouad dimension: antifractal metrization, porous sets, and homogeneous measures. J. Korean Math. Soc. 35, 23-76 (1998) · Zbl 0893.54029 |
| [22] | Shvartsman, P.: Local approximations and intrinsic characterization of spaces of smooth functions on regular subsets of \[\mathbb{R}^nRn\]. Math. Nachr. 279, 1212-1241 (2006) · Zbl 1108.46030 · doi:10.1002/mana.200510418 |
| [23] | Sickel, W.: On pointwise multipliers for \[F^s_{pq}(\mathbb{R}^n)\] Fpqs(Rn) in case \[\sigma_{p, q}sn/p\] σp,qsn/p. Ann. Mat. Pura Appl. 176, 209-250 (1999) · Zbl 0956.46027 · doi:10.1007/BF02505997 |
| [24] | Stein, E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton, NJ (1970) · Zbl 0207.13501 |
| [25] | Triebel, H.: Local approximation spaces. Z. Anal. Anwendungen. 8, 261-288 (1989) · Zbl 0695.46015 |
| [26] | Triebel, H.: Hardy inequalities in function spaces. Math. Bohem. 124, 123-130 (1999) · Zbl 0940.46016 |
| [27] | Triebel, H.: Non-smooth atoms and pointwise multipliers in function spaces. Ann. Mat. Pura Appl. 182(4), 457-486 (2003) · Zbl 1223.46037 · doi:10.1007/s10231-003-0076-2 |
| [28] | Triebel, H.: Theory of Function Spaces II. Basel, Birkhäuser (1992) · Zbl 0763.46025 · doi:10.1007/978-3-0346-0419-2 |
| [29] | Triebel, H.: The Structure of Functions. Basel, Birkhäuser (2001) · Zbl 0984.46021 · doi:10.1007/978-3-0348-8257-6 |
| [30] | Triebel, H.: Function Spaces and Wavelets on Domains. European Mathematical Society (2008) · Zbl 1158.46002 |
| [31] | Zhou, Y.: Fractional Sobolev extension and imbedding. Trans. Am. Math. Soc. (to appear) · Zbl 1320.46038 |
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