A note on sharp 1-dimensional Poincaré inequalities

Chua, Seng-Kee; Wheeden, Richard

doi:10.1090/S0002-9939-06-08545-5

A note on sharp 1-dimensional Poincaré inequalities
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Proc. Amer. Math. Soc. 134 (2006), 2309-2316 Request permission

Abstract:

Let $1<p<\infty$ and $-\infty < a < b <\infty$. We show by using elementary methods that the best constant $C$ (necessarily independent of $a$ and $b$) for which the 1-dimensional Poincaré inequality \[ \|f-f_{av}\|_{{}_{\scriptstyle {L^1[a,b]}}} \le C (b-a)^{2-\frac {1}{p}} \|f’\|_{{}_{\scriptstyle {L^p[a,b]}}}\] holds for all Lipschitz continuous functions $f$, with $f_{av}=\int ^b_a f/(b-a),$ is \[ C=\frac {1}{2} (1+p’)^{-1/p’}.\]

References

Gabriel Acosta and Ricardo G. Durán, An optimal Poincaré inequality in $L^1$ for convex domains, Proc. Amer. Math. Soc. 132 (2004), no. 1, 195–202. MR 2021262, DOI 10.1090/S0002-9939-03-07004-7
Robert A. Adams, Sobolev spaces, Pure and Applied Mathematics, Vol. 65, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. MR 0450957
Frederick J. Almgren Jr. and Elliott H. Lieb, Symmetric decreasing rearrangement is sometimes continuous, J. Amer. Math. Soc. 2 (1989), no. 4, 683–773. MR 1002633, DOI 10.1090/S0894-0347-1989-1002633-4
Thierry Aubin, Problèmes isopérimétriques et espaces de Sobolev, J. Differential Geometry 11 (1976), no. 4, 573–598 (French). MR 448404
M. Bebendorf, A note on the Poincaré inequality for convex domains, Z. Anal. Anwendungen 22 (2003), no. 4, 751–756. MR 2036927, DOI 10.4171/ZAA/1170
William Beckner and Michael Pearson, On sharp Sobolev embedding and the logarithmic Sobolev inequality, Bull. London Math. Soc. 30 (1998), no. 1, 80–84. MR 1479040, DOI 10.1112/S0024609397003901
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, Extension theorems on weighted Sobolev spaces, Indiana Univ. Math. J. 41 (1992), no. 4, 1027–1076. MR 1206339, DOI 10.1512/iumj.1992.41.41053
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
Seng-Kee Chua and Richard L. Wheeden, Sharp conditions for weighted 1-dimensional Poincaré inequalities, Indiana Univ. Math. J. 49 (2000), no. 1, 143–175. MR 1777034, DOI 10.1512/iumj.2000.49.1754
S.-K. Chua and R. L. Wheeden, Estimates of best constants for weighted Poincaré inequalities on convex domains, Proc. London Math Soc., to appear.
Andrea Cianchi, A sharp form of Poincaré type inequalities on balls and spheres, Z. Angew. Math. Phys. 40 (1989), no. 4, 558–569 (English, with Italian summary). MR 1008923, DOI 10.1007/BF00944807
Andrea Cianchi, David E. Edmunds, and Petr Gurka, On weighted Poincaré inequalities, Math. Nachr. 180 (1996), 15–41. MR 1397667, DOI 10.1002/mana.3211800103
J.-M. Coron, The continuity of the rearrangement in $W^{1,p}(\textbf {R})$, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 11 (1984), no. 1, 57–85. MR 752580
David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, Classics in Mathematics, Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition. MR 1814364
Petr Gurka and Alois Kufner, A note on a two-weighted Sobolev inequality, Approximation and function spaces (Warsaw, 1986) Banach Center Publ., vol. 22, PWN, Warsaw, 1989, pp. 169–172. MR 1097190
W. K. Hayman, Some bounds for principal frequency, Applicable Anal. 7 (1977/78), no. 3, 247–254. MR 492339, DOI 10.1080/00036817808839195
Keijo Hildén, Symmetrization of functions in Sobolev spaces and the isoperimetric inequality, Manuscripta Math. 18 (1976), no. 3, 215–235. MR 409773, DOI 10.1007/BF01245917
B. Opic and A. Kufner, Hardy-type inequalities, Pitman Research Notes in Mathematics Series, vol. 219, Longman Scientific & Technical, Harlow, 1990. MR 1069756
Elliott H. Lieb, Gaussian kernels have only Gaussian maximizers, Invent. Math. 102 (1990), no. 1, 179–208. MR 1069246, DOI 10.1007/BF01233426
Elliott H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math. (2) 118 (1983), no. 2, 349–374. MR 717827, DOI 10.2307/2007032
P.-L. Lions, The concentration-compactness principle in the calculus of variations. The limit case. II, Rev. Mat. Iberoamericana 1 (1985), no. 2, 45–121. MR 850686, DOI 10.4171/RMI/12
Guozhen Lu and Richard L. Wheeden, Poincaré inequalities, isoperimetric estimates, and representation formulas on product spaces, Indiana Univ. Math. J. 47 (1998), no. 1, 123–151. MR 1631545, DOI 10.1512/iumj.1998.47.1494
Vladimir G. Maz’ja, Sobolev spaces, Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1985. Translated from the Russian by T. O. Shaposhnikova. MR 817985, DOI 10.1007/978-3-662-09922-3
L. E. Payne and H. F. Weinberger, An optimal Poincaré inequality for convex domains, Arch. Rational Mech. Anal. 5 (1960), 286–292 (1960). MR 117419, DOI 10.1007/BF00252910
Gerald Rosen, Minimum value for $c$ in the Sobolev inequality $\phi ^{3}\|\leq c\nabla \phi \|^{3}$, SIAM J. Appl. Math. 21 (1971), 30–32. MR 289739, DOI 10.1137/0121004
G. J. Sinnamon, Weighted Hardy and Opial-type inequalities, J. Math. Anal. Appl. 160 (1991), no. 2, 434–445. MR 1126128, DOI 10.1016/0022-247X(91)90316-R
Giorgio Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl. (4) 110 (1976), 353–372. MR 463908, DOI 10.1007/BF02418013
Giorgio Talenti, Some inequalities of Sobolev type on two-dimensional spheres, General inequalities, 5 (Oberwolfach, 1986) Internat. Schriftenreihe Numer. Math., vol. 80, Birkhäuser, Basel, 1987, pp. 401–408. MR 1018163