Hardy’s inequality for 𝑊^{1,𝑝}₀-functions on Riemannian manifolds

Miklyukov, Vladimir; Vuorinen, Matti

doi:10.1090/S0002-9939-99-04849-2

Hardy’s inequality for $W^{1,p}_0$-functions on Riemannian manifolds
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by Vladimir M. Miklyukov and Matti K. Vuorinen PDF

Proc. Amer. Math. Soc. 127 (1999), 2745-2754 Request permission

Abstract:

We prove that for every Riemannian manifold $\mathcal {X}$ with the isoperimetric profile of particular type there holds an inequality of Hardy type for functions of the class $W_0^{1,p}( \mathcal {X})$. We also study manifolds satisfying Hardy’s inequality and, in particular, we establish an estimate for the rate of growth of the weighted volume of the noncompact part of such a manifold.

References

L. Ahlfors, Zur theorie der Überlagerungsflächen, Acta Math., 65 (1935), 157-194.
Alano Ancona, On strong barriers and an inequality of Hardy for domains in $\textbf {R}^n$, J. London Math. Soc. (2) 34 (1986), no. 2, 274–290. MR 856511, DOI 10.1112/jlms/s2-34.2.274
D. H. Armitage and Ü. Kuran, The convexity of a domain and the superharmonicity of the signed distance function, Proc. Amer. Math. Soc. 93 (1985), no. 4, 598–600. MR 776186, DOI 10.1090/S0002-9939-1985-0776186-8
Itai Benjamini and Jianguo Cao, A new isoperimetric comparison theorem for surfaces of variable curvature, Duke Math. J. 85 (1996), no. 2, 359–396. MR 1417620, DOI 10.1215/S0012-7094-96-08515-4
Serguei G. Bobkov and Christian Houdré, Some connections between isoperimetric and Sobolev-type inequalities, Mem. Amer. Math. Soc. 129 (1997), no. 616, viii+111. MR 1396954, DOI 10.1090/memo/0616
Ju. D. Burago and V. A. Zalgaller, Geometricheskie neravenstva, “Nauka” Leningrad. Otdel., Leningrad, 1980 (Russian). MR 602952
Christopher B. Croke, Some isoperimetric inequalities and eigenvalue estimates, Ann. Sci. École Norm. Sup. (4) 13 (1980), no. 4, 419–435. MR 608287, DOI 10.24033/asens.1390
B. A. Dubrovin, S. P. Novikov, and A. T. Fomenko, Sovremennaya geometriya, “Nauka”, Moscow, 1979 (Russian). Metody i prilozheniya. [Methods and applications]. MR 566582
Herbert Federer and Wendell H. Fleming, Normal and integral currents, Ann. of Math. (2) 72 (1960), 458–520. MR 123260, DOI 10.2307/1970227
C. J. Everett Jr., Annihilator ideals and representation iteration for abstract rings, Duke Math. J. 5 (1939), 623–627. MR 13
E. Hebey, Sobolev Spaces on Riemannian Manifolds, Lectures Notes in Math., 1635, Springer-Verlag, Berlin-Heidelberg-New York, 1996.
David Hoffman and Joel Spruck, Sobolev and isoperimetric inequalities for Riemannian submanifolds, Comm. Pure Appl. Math. 27 (1974), 715–727. MR 365424, DOI 10.1002/cpa.3160270601
J. Kinnunen and O. Martio, Maximal inequalities for Sobolev functions, Math. Research Letters 4 (1997), 489-500.
André Lambert, Quelques théorèmes de décomposition des ultradistributions, Ann. Inst. Fourier (Grenoble) 29 (1979), no. 3, x, 57–100 (French, with English summary). MR 552960
John L. Lewis, Uniformly fat sets, Trans. Amer. Math. Soc. 308 (1988), no. 1, 177–196. MR 946438, DOI 10.1090/S0002-9947-1988-0946438-4
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
Pasi Mikkonen, On the Wolff potential and quasilinear elliptic equations involving measures, Ann. Acad. Sci. Fenn. Math. Diss. 104 (1996), 71. MR 1386213
Andreas Wannebo, Hardy inequalities, Proc. Amer. Math. Soc. 109 (1990), no. 1, 85–95. MR 1010807, DOI 10.1090/S0002-9939-1990-1010807-1
Shing Tung Yau, Isoperimetric constants and the first eigenvalue of a compact Riemannian manifold, Ann. Sci. École Norm. Sup. (4) 8 (1975), no. 4, 487–507. MR 397619, DOI 10.24033/asens.1299