Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-28T13:39:18.439Z Has data issue: false hasContentIssue false
  • English
  • Français

Extension Theorems on Weighted Sobolev Spaces and Some Applications

Published online by Cambridge University Press:  20 November 2018

Seng-Kee Chua
Seng-Kee Chua*
Affiliation:
National University of Singapore, Department of Mathematics 2, Science Drive 2 Singapore 117543matcsk@nus.edu.sg
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We extend the extension theorems to weighted Sobolev spaces $L_{w,k}^{p}\left( \mathcal{D} \right)$ on $(\varepsilon ,\delta )$ domains with doubling weight $w$ that satisfies a Poincaré inequality and such that ${{w}^{-1/p}}$ is locally ${{L}^{{{p}'}}}$. We also make use of the main theorem to improve weighted Sobolev interpolation inequalities.

Keywords

46E35Poincaré inequalitiesAp weightsdoubling weights(ε, δ) and (ε,∞) domains
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 58 , Issue 3 , 01 June 2006 , pp. 492 - 528
Copyright
Copyright © Canadian Mathematical Society 2006

References

[1] Adams, R. and Fournier, J., Sobolev Spaces. Academic Press, New York 2003.Google Scholar
[2] Bojarski, B., Remarks on Sobolev imbedding inequalities. In: Complex Analysis. Lecture Notes in Math. 1351, Springer-Verlag, 1989, pp. 52–68.Google Scholar
[3] Brown, R. C. and Hinton, D. B., Sufficient conditions for weighted inequalities of sum form. J. Math. Anal. Appl. 112(1985), no. 2, 563–578.Google Scholar
[4] Brown, R. C. and Hinton, D. B., Weighted interpolation inequalities of sum and product form in ℝ n. Proc. LondonMath. Soc. 56(1988), no. 3, 261–280.Google Scholar
[5] Brown, R. C. and Hinton, D. B., Weighted interpolation inequalities and embeddings in ℝ n . Canad. J. Math. 62(1990), no. 6, 959–980.Google Scholar
[6] Calderón, A. P., Lebesgue spaces of differentiable functions and distributions. Proc. Symp. Pure Math. 4, American Mathematical Society, Providence, RI, 1961, pp. 33–49.Google Scholar
[7] Chanillo, S. and Wheeden, R. L., Poincaré inequalities for a class of non-Ap weights. Indiana Math. J. 41(1992), no. 3, 605–623.Google Scholar
[8] Chiarenza, F. and Frasca, M., A note on a weighted Sobolev inequality. Proc. Amer. Math. Soc. 93(1985), no. 4, 703–704.Google Scholar
[9] Christ, M., The extension problem for certain function spaces involving fractional orders of differentiability. Ark. Mat. 22(1984), no. 1, 63–81.Google Scholar
[10] Chua, S.-K., Extension theorems on weighted Sobolev spaces. Indiana Math. J. 41(1992), no. 4, 1027–1076.Google Scholar
[11] Chua, S.-K., Weighted Sobolev's inequalities on domains satisfying the chain condition. Proc. Amer. Math. Soc. 117(1993), 449–457.Google Scholar
[12] Chua, S.-K., Some remarks on extension theorems for weighted Sobolev spaces. Illinois J. Math. 38(1994), 95–126.Google Scholar
[13] Chua, S.-K., On weighted Sobolev interpolation inequalities. Proc. Amer. Math. Soc. 121(1994), no. 2, 441–449.Google Scholar
[14] Chua, S.-K., Weighted Sobolev interpolation inequalities on certain domains. J. London Math. Soc. (2) 51(1995), no. 3, 532–544.Google Scholar
[15] Chua, S.-K., On weighted Sobolev spaces. Canad. J. Math. 48(1996), 527–541.Google Scholar
[16] Chua, S.-K., Weighted inequalities on John Domains. Jour. Math. Anal. Appl. 258(2001), 763–776.Google Scholar
[17] Chua, S.-K., Sharp conditions for weighted interpolation inequalities. ForumMath. 17(2005), 461–478.Google Scholar
[18] Coifman, R. R. and Fefferman, C., Weighted norm inequalities for maximal functions and singular integrals. Studia Math. 51(1974), 241–250.Google Scholar
[19] Edmunds, D. E., Kufner, A., and Sun, J., Extension of functions in weighted Sobolev spaces. Rend. Accad. Naz. Sci. XL Mem. Mat. (5) 14(1990), 327–339.Google Scholar
[20] Fabes, E. B., Kenig, C. E., and Serapioni, R. P., The local regularity of solutions of degenerate elliptic equations. Comm. Partial Differential Equations 7(1982), no. 1, 77–116.Google Scholar
[21] Franchi, B., Pérez, C., and Wheeden, R. L., Self-improving properties of John-Nirenberg and Poincaré inequalities on spaces of homogeneous type. J. Funct. Anal. 153(1998), 67–97.Google Scholar
[22] Gol’dshtein, V. M. and Reshetnyak, Yu. G., Quasiconformal Mappings and Sobolev Spaces. Kluwer Academic Publishers, Dordrecht, 1990.Google Scholar
[23] Gutierrez, C. E. and Wheeden, R. L., Sobolev interpolation inequalities with weights. Trans. Amer. Math. Soc. 323(1991), no. 1, 263–281.Google Scholar
[24] Hajłasz, P. and Koskela, P., Isoperimetric inequalities and imbedding theorems in irregular domain. J. London Math. Soc. (2) 58(1998), no. 2, 425–450.Google Scholar
[25] Hajłasz, P. and Koskela, P., Sobolev met Poincaré. Mem. Amer. Math. Soc. 145(2000), no. 688.Google Scholar
[26] Hunt, R. A., Kurtz, D. S., and Neugebauer, C. I., A note on the equivalence of Ap and Sawyer's condition for equal weights. In: Conference on Harmonic Analysis in Honor of A. Zymund. Wadsworth, Belmont, CA, 1983, pp. 156–158.Google Scholar
[27] Jones, P., Quasiconformal mappings and extendability of functions in Sobolev spaces. Acta Math. 147(1981), no. 1-2, 71–88.Google Scholar
[28] Kufner, A., Weighted Sobolev Spaces. John Wiley, New York, 1985.Google Scholar
[29] Kufner, A., How to define reasonable Sobolev spaces. Comment. Math. Univ. Carolin. 25(1984), no. 3, 537–554.Google Scholar
[30] Martio, O. and Sarvas, J., Injectivity theorems in plane and space. Ann. Acad. Sci. Fenn. Ser. A / Math. 4(1979), no. 2, 383–401.Google Scholar
[31] Maz’ja, V. G., Sobolev Spaces. Springer Series in Soviet Mathematics, Springer-Verlag, New York 1985.Google Scholar
[32] Muckenhoupt, B., Weighted norm inequalities for the Hardy maximal function. Trans. Amer. Math. Soc. 165(1972), 207–226.Google Scholar
[33] Muckenhoupt, B. and Wheeden, R., On the dual of weighted H1 of the half-space. Studia Math. 63(1978), no. 1, 57–79.Google Scholar
[34] Sawyer, E., A characterization of a two-weight norm inequality for maximal operators. Studia Math. 75(1982), no. 1, 1–11.Google Scholar
[35] Sawyer, E. and Wheeden, R. L., Weighted inequalities for fractional integrals on Euclidean and homogeneous spaces. Amer. J. Math. 114(1992), no. 4, 813–874.Google Scholar
[36] Stein, E. M., Singular Integrals and Differentiability Properties of Functions. Princeton Mathematical Series 30, Princeton University Press, Princeton, NJ, 1970.Google Scholar
[37] Strömberg, J.-O. and Torchinsky, A., Weighted Hardy Spaces. Lecture Notes in Mathematics 1381, Springer-Verlag, Berlin, 1989.Google Scholar
[38] Torchinsky, A., Real-Variable Methods in Harmonic Analysis. Pure and Applied Mathematics 123, Academic Press, Orlando, FL, 1986.Google Scholar