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ISSN 1088-6826(online) ISSN 0002-9939(print)



A weighted Poincaré inequality
with a doubling weight

Author: Ritva Hurri-Syrjanen
Journal: Proc. Amer. Math. Soc. 126 (1998), 545-552
MSC (1991): Primary 46Exx, 26Dxx
MathSciNet review: 1415588
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Abstract: We show that unbounded John domains (and even a larger class of domains than John domains) satisfy the weighted Poincaré inequality

\begin{equation*}\inf _{a\in \mathbb{R}} \|u(x)-a\|_{L^{q}(D,w_{1})} \le C\|\nabla u(x)\|_{L^{p}(D,w_{2})}\end{equation*}

whenever $u$ is a Lipschitz function on $D$, $w_{1}$ is a doubling weight, and weights satisfy certain cube conditions, and $C=C(D,p,q,w_{1},w_{2})$.

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  • [B] Bojarski, B., Remarks on Sobolev imbedding inequalities, Complex Analysis (Joensuu 1987), Lecture Notes in Math., Springer-Verlag, Berlin and Heidelberg 1351 (1988), 52-68. MR 90b:46068
  • [C] Chua, S.-K., Weighted Sobolev inequalities on domains satisfying the chain condition, Proc. Amer. Math. Soc. 117 (1993), 449-457. MR 93d:46050
  • [CWZ] Chen, Z.Q., R.J. Williams and Z. Zhao, A Sobolev inequality and Neumann heat kernel estimate for unbounded domains, Math. Research Letters 1 (1994), 177-184. MR 90d:70034
  • [EH] Evans, W.D. and D.J. Harris, Sobolev embeddings for generalized ridged domains, Proc. London Math. Soc. 54 (3) (1987), 141-175. MR 88b:46056
  • [H] Hurri, R., Poincaré domains in $R^{n}$, Ann. Acad. Sci. Fenn. Ser. IA, Dissertations 71 (1988), 1-41. MR 90a:30074
  • [H-S] Hurri-Syrjanen, R., Unbounded Poincaré domains, Ann. Acad. Sci. Fenn. Ser. IA 17 (1992), 409-423. MR 93k:46022
  • [H-S2] Hurri-Syrjanen, R., An improved Poincaré inequality, Proc. Amer. Math. Soc. 120 (1994), 213-222. MR 94b:46047
  • [SW] Sawyer, E. and R.L. Wheeden, Weighted inequalities for fractional integrals on Euclidean and homogeneous spaces, Amer. J. Math. 114 (1992), 813-874. MR 94i:42024
  • [StW] Strömberg, J.-O. and R.L. Wheeden, Fractional integrals on weighted $H^{p}$ and $L^{p}$ spaces, Trans. Amer. Math. Soc. 287 (1985), 293-321. MR 86f:42016
  • [V] Vaisala, J., Exhaustions of John domains, Ann. Acad. Sci. Fenn. Ser. IA 19 (1994), 47-54. MR 94i:30024

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Additional Information

Ritva Hurri-Syrjanen
Affiliation: Department of Mathematics, University of Texas, Austin, Texas 78712
Address at time of publication: Department of Mathematics, P.O. Box 4, FIN-00014 University of Helsinki, Finland

Received by editor(s): January 5, 1996
Received by editor(s) in revised form: August 22, 1996
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1998 American Mathematical Society

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