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Integral inequalities of Hardy and Poincaré type


Authors: Harold P. Boas and Emil J. Straube
Journal: Proc. Amer. Math. Soc. 103 (1988), 172-176
MSC: Primary 46E35; Secondary 26D10, 35H05
DOI: https://doi.org/10.1090/S0002-9939-1988-0938664-0
MathSciNet review: 938664
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Abstract: The Poincaré inequality $ \vert\vert u\vert{\vert _p} \leq C\vert\vert\nabla u\vert{\vert _p}$ in a bounded domain holds, for instance, for compactly supported functions, for functions with mean value zero and for harmonic functions vanishing at a point. We show that it can be improved to $ \vert\vert u\vert{\vert _p} \leq C\vert\vert{\delta ^\beta }\nabla u\vert{\vert _p}$, where $ \delta $ is the distance to the boundary, and the positive exponent $ \beta $ depends on the smoothness of the boundary.


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  • [Ad] R. A. Adams, Sobolev spaces, Academic Press, New York, 1975. MR 0450957 (56:9247)
  • [An] A. Ancona, On strong barriers and an inequality of Hardy for domains in $ {{\mathbf{R}}^n}$, J. London Math. Soc. 34 (1986), 274-290. MR 856511 (87k:31004)
  • [AS] S. Axler and A. L. Shields, Univalent multipliers of the Dirichlet space, Michigan Math. J. 32 (1985), 65-80. MR 777302 (86c:30043)
  • [CH] R. Courant and D. Hilbert, Methoden der mathematischen Physik, vol. 2, Springer-Verlag, Berlin, 1937.
  • [D] J. Detraz, Classes de Bergman de fonctions harmoniques, Bull. Soc. Math. France 109 (1981), 259-268. MR 623793 (83m:31006)
  • [GT] D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Springer-Verlag, Berlin and New York, 1983. MR 737190 (86c:35035)
  • [G] P. Grisvard, Elliptic problems in nonsmooth domains, Pitman, Boston, Mass., 1985. MR 775683 (86m:35044)
  • [HLP] G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, Cambridge Univ. Press, 1934.
  • [H] J. A. Hummel, Counterexamples to the Poincaré inequality, Proc. Amer. Math. Soc. 8 (1957), 207-210. MR 0084546 (18:878c)
  • [Kd] L. D. Kudryavtsev, Direct and inverse imbedding theorems, Transl. Math. Monographs, Amer. Math. Soc. Providence, R.I., 1974.
  • [Kf] A. Kufner, Weighted Sobolev spaces, Wiley, New York, 1985. MR 802206 (86m:46033)
  • [Le] J. Lewis, Uniformly fat sets, preprint. MR 946438 (89e:31012)
  • [M] C. B. Morrey, Multiple integrals in the calculus of variations, Springer, New York, 1966. MR 0202511 (34:2380)
  • [St] E. M. Stein, Singular integrals and differentiability properties of func tions, Princeton Univ. Press, Princeton, N.J., 1970. MR 0290095 (44:7280)
  • [T] M. E. Taylor, Pseudodifferential operators, Princeton Univ. Press., Princeton, N.J., 1981. MR 618463 (82i:35172)
  • [Z] W. P. Ziemer, A Poincaré-type inequality for solutions of elliptic differential equations, Proc. Amer. Math. Soc. 97 (1986), 286-290. MR 835882 (87k:35042)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1988-0938664-0
Keywords: Poincaré's inequality, Hardy's inequality, hypoelliptic partial differential equations
Article copyright: © Copyright 1988 American Mathematical Society

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