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On Weighted Sobolev Spaces

Published online by Cambridge University Press:  20 November 2018

Seng-Kee Chua
Seng-Kee Chua*
Affiliation:
National University of Singapore Department of Mathematics 10, Kent Ridge Crescent Singapore 0511 email: e-mail: matesk@math.nus.sg
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Abstract

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We study density and extension problems for weighted Sobolev spaces on bounded (ε, δ) domains 𝓓 when a doubling weight w satisfies the weighted Poincaré inequality on cubes near the boundary of 𝓓 and when it is in the Muckenhoupt Ap class locally in 𝓓. Moreover, when the weights wi(x) are of the form dist(x, Mi)αi, αi∈ ℝ, Mi⊂ 𝓓 that are doubling, we are able to obtain some extension theorems on (ε, ∞) domains.

Keywords

46E35Poincaré inequalitiesAp weightspower weightsdoublinglocally Ap weights(ε, δ) and (ε, ∞) domains
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 48 , Issue 3 , 01 June 1996 , pp. 527 - 541
Copyright
Copyright © Canadian Mathematical Society 1996

References

1. Adams, Richard, Sobolev Spaces, Academic Press, New York, 1975.Google Scholar
2. Brown, R.C. and Hinton, D.B., Sufficient conditions for weighted inequalities of sum form, J. Math. Anal. Appl. 112 (1985), 563–578.Google Scholar
3. Brown, R.C., Weighted interpolation inequalities of sum and product form in ℝn, Proc. London Math. Soc. (3) 56 (1988), 261–280.Google Scholar
4. Brown, R.C., Weighted interpolation inequalities and embeddings in ℝn, Canad. J. Math. (6) 62 (1990), 959–980.Google Scholar
5. Calderón, Alberto P., Lebesgue spaces of differentiable functions and distributions, Proc. Symp. Pure Math. IV( 1961), 33–49.Google Scholar
6. Chanillo, Sagun and Wheeden, Richard L., Poincaré inequalities for a class of non-Ap weights, Indiana Math. J. (3) 41 (1992), 605–623.Google Scholar
7. Chiarenza, Filippo and Frasca, Michele, A note on a weighted Sobolev inequality, Proc. Amer. Math. Soc. (4)93 (1985), 703–704.Google Scholar
8. Christ, Michael, The extension problem for certain function spaces involving fractional orders of differentiability, Ark. Mat. (1) 22 (1984), 63–81.Google Scholar
9. Chua, Seng-Kee, Extension theorems on weighted Sobolev spaces, Indiana Math. J. 41 (1992), 1027–1076.Google Scholar
10. Chua, Seng-Kee, Extension and restriction theorems on weighted Sobolev spaces, Ph.D.thesis, Rutgers University, 1991.Google Scholar
11. Chua, Seng-Kee, Some Remarks on extension theorems for weighted Sobolev spaces, Illinois J. Math., 38 (1994), 95–126.Google Scholar
12. Chua, Seng-Kee, Weighted Sobolev s inequalities on domains satisfying the chain condition, Proc. Amer. Math. Soc. 117 (1993), 449–457.Google Scholar
13. Chua, Seng-Kee, Restriction theorems on weighted Sobolev spaces of mixed norm, Real Anal. Exchange (2) 17(1991/92), 633–651.Google Scholar
14. Chua, Seng-Kee, On weighted Sobolev interpolation inequalities, Proc. Amer. Math. Soc. 121 (1994), 441–449.Google Scholar
15. Chua, Seng-Kee, Weighted Sobolev interpolation inequalities on certain domains, J.London Math. Soc. (2) 51 (1995), 532–544.Google Scholar
16. Coifman, Ronald R. and Fefferman, Charles, Weighted norm inequalities for maximal functions and singular integrals, Studia Math. 51 (1974), 241–250.Google Scholar
17. Edmunds, D.E., Alois Kufner andJiong Sun, Extension of functions in weighted Sobolev spaces, Rend. Accad. Naz. Sci. XL Mem. Mat. (5), 108° (17) 16 (1990), 327–339.Google Scholar
18. Fabes, Eugene B., Kenig, Carlos E. and Raul Serapioni, P., The local regularity of solutions of degenerate elliptic equations, Comm. Partial Differential Equations 7 (1982), 77–116.Google Scholar
19. Gol'dshtein, V.M. and Reshetnyak, Yu. G., Quasiconformal Mappings and Sobolev Spaces, Kluwer Academic Publishers, 1990.Google Scholar
20. Gutierrez, Cristian E. and Richard Wheeden, L., Sobolev interpolation inequalities with weights, Trans. Amer. Math. Soc. 323 (1991), 263–281.Google Scholar
21. Hunt, R.A.,Kurtz, D.S. and Neugebauer, C.I., A note on the equivalence of Ap and Sawyer's condition for equal weights, Conf. on Harm. Anal,in Honor of Zymund, A., Wadsworth, 1983. 156–158.Google Scholar
22. Jones, Peter, Quasiconformal mappings and extendability of functions in Sobolev spaces, Acta Math. (1-2) 147 (1981), 71–88.Google Scholar
23. Kufner, Alois, Weighted Sobolev Spaces, John Wiley & Sons Ltd, 1985.Google Scholar
24. Kufner, Alois, How to define reasonable Sobolev spaces, Comment. Math. Univ. Carolin. (3) 25 (1984), 537- 554.Google Scholar
25. Martio, Olli and Sarvas, J., Injectivity theorems in plane and space, Ann. Acad. Sci. Fenn. Series A I Math. (2) 4 (1979), 383–401.Google Scholar
26. Maz'ja, Vladimir G., Sobolev Spaces, Springer-Verlag, New York, 1985.Google Scholar
27. Muckenhoupt, Benjamin, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc. 165 (1972), 207–226.Google Scholar
28. Muckenhoupt, Benjamin and Wheeden, Richard, On the dual of weighted Hl of the half-space, Studia Math. (1)63 (1978), 57–79.Google Scholar
29. Sawyer, Eric, A characterization of a two-weighted norm inequality for maximal operators, Studia Math. 75 (1982), 1–11.Google Scholar
30. Sawyer, Eric and Wheeden, Richard L., Weighted inequalities for fractional integrals on Euclidean and homogeneous spaces, Amer. J. Math. (4) 114 (1992), 813–874.Google Scholar
31. Stein, Elias M.,Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, 1970.Google Scholar
32. Stromberg, Jan-Olov and Alberto Torchinsky,Weighted Hardy Spaces, Lecture Notes in Math. 1381, Springer, New York, 1989.Google Scholar
33. Torchinsky, Alberto,Real-Variable Methods in Harmonic Analysis, Academic Press Inc., New York, 1986.Google Scholar
34. Wheeden, Richard L. and Antoni Zygmund,Measure and Integral, Marcel Dekker Inc., New York, 1977.Google Scholar
35. Wolff, Thomas H., Restriction of Ap weights, preprint.Google Scholar