A weighted Poincaré inequality with a doubling weight
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- by Ritva Hurri-Syrjānen PDF
- Proc. Amer. Math. Soc. 126 (1998), 545-552 Request permission
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 $C=C(D,p,q,w_{1},w_{2})$.References
- B. Bojarski, Remarks on Sobolev imbedding inequalities, Complex analysis, Joensuu 1987, Lecture Notes in Math., vol. 1351, Springer, Berlin, 1988, pp. 52–68. MR 982072, DOI 10.1007/BFb0081242
- Seng-Kee Chua, Weighted Sobolev inequalities on domains satisfying the chain condition, Proc. Amer. Math. Soc. 117 (1993), no. 2, 449–457. MR 1140667, DOI 10.1090/S0002-9939-1993-1140667-2
- Chuan Yu Chen, Dynamical theorems for systems of mass points when the effective mass is a tensor, Acta Math. Sci. (Chinese) 8 (1988), no. 2, 177–184 (Chinese). MR 966459
- W. D. Evans and D. J. Harris, Sobolev embeddings for generalized ridged domains, Proc. London Math. Soc. (3) 54 (1987), no. 1, 141–175. MR 872254, DOI 10.1112/plms/s3-54.1.141
- Ritva Hurri, Poincaré domains in $\textbf {R}^n$, Ann. Acad. Sci. Fenn. Ser. A I Math. Dissertationes 71 (1988), 42. MR 978019
- Ritva Hurri-Syrjänen, Unbounded Poincaré domains, Ann. Acad. Sci. Fenn. Ser. A I Math. 17 (1992), no. 2, 409–423. MR 1190332, DOI 10.5186/aasfm.1992.1725
- Ritva Hurri-Syrjänen, An improved Poincaré inequality, Proc. Amer. Math. Soc. 120 (1994), no. 1, 213–222. MR 1169032, DOI 10.1090/S0002-9939-1994-1169032-X
- E. Sawyer and R. L. Wheeden, Weighted inequalities for fractional integrals on Euclidean and homogeneous spaces, Amer. J. Math. 114 (1992), no. 4, 813–874. MR 1175693, DOI 10.2307/2374799
- Jan-Olov Strömberg and Richard L. Wheeden, Fractional integrals on weighted $H^p$ and $L^p$ spaces, Trans. Amer. Math. Soc. 287 (1985), no. 1, 293–321. MR 766221, DOI 10.1090/S0002-9947-1985-0766221-X
- Jussi Väisälä, Exhaustions of John domains, Ann. Acad. Sci. Fenn. Ser. A I Math. 19 (1994), no. 1, 47–57. MR 1246886
Additional Information
- Ritva Hurri-Syrjānen
- 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
- Email: syrjanen@math.utexas.edu, hurrisyr@helsinki.fi
- Received by editor(s): January 5, 1996
- Received by editor(s) in revised form: August 22, 1996
- Communicated by: Palle E. T. Jorgensen
- © Copyright 1998 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 126 (1998), 545-552
- MSC (1991): Primary 46Exx, 26Dxx
- DOI: https://doi.org/10.1090/S0002-9939-98-04059-3
- MathSciNet review: 1415588