Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-23T02:24:20.675Z Has data issue: false hasContentIssue false
  • English
  • Français

Linearised Moser-Trudinger inequality

Published online by Cambridge University Press:  17 April 2009

Meelae Kim
Meelae Kim
Affiliation:
Center for Teaching and Research, Korea Polytechnic University, Jungwang-dong 3–101, Sihung-si, Kyunggi-do, Korea e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

As a limiting case of the Sobolev imbedding theorem, the Moser-Trudinger inequality was obtained for functions in with resulting exponential class integrability. Here we prove this inequality again and at the same time get sharper information for the bound. We also generalise the Linearised Moser inequality to higher dimensions, which was first introduced by Beckner for functions on the unit disc. Both of our results are obtained by using the method of Carleson and Chang. The last section introduces an analogue of each inequality for the Laplacian instead of the gradient under some restricted conditions.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 62 , Issue 3 , December 2000 , pp. 445 - 457
Copyright
Copyright © Australian Mathematical Society 2000

References

[1]Adams, D.R., ‘A sharp inequality of J. Moser for higher order derivatives’, Ann. of Math. 128 (1988), 385–398.CrossRefGoogle Scholar
[2]Beckner, W., ‘Moser-Trudinger inequality in higher dimensions’, Internat. Math. Res. Notices 1 (1991), 83–91.CrossRefGoogle Scholar
[3]Beckner, W., ‘Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality’, Ann. of Math. 138 (1993), 213–242.CrossRefGoogle Scholar
[4]Berger, M.S., ‘Riemannian structures on prescribed Gaussian curvature for compact 2-manifolds’, J. Differential Geom. 5 (1971), 325–332.CrossRefGoogle Scholar
[5]Branson, T.P., Chang, S.-Y.A. and Yang, P.C., ‘Estimates and extremals for zeta function determinants on four-manifolds’, Comm. Math Phys. 149 (1992), 241–262.CrossRefGoogle Scholar
[6]Carleson, L. and Chang, S.-Y.A., ‘On the existence of an extremal function for an inequality of J. Moser’, Bull. Sci. Math 110 (1986), 113–127.Google Scholar
[7]Jodeit, M., ‘An inequality for the indefinite integral of a function in Lq’, Studia Math. 44 (1972), 545–554.CrossRefGoogle Scholar
[8]Moser, J., ‘A sharp form of an inequality by N. Trudinger’, Indiana Univ. Math. J. 20 (1971), 1077–1092.CrossRefGoogle Scholar
[9]Moser, J., ‘On a nonlinear problem in differential geometry’, in Dynamical systems, (Peixoto, M.M., Editor) (Academic Press, New York, 1973), pp. 273–280.CrossRefGoogle Scholar
[10]Onofri, E., ‘On the positivity of the effective action in a theory of random surfaces’, Comm. Math. Phys. 86 (1982), 321–326.CrossRefGoogle Scholar
[11]Osgood, B., Phillips, R. and Sarnak, P., ‘Extremals of determinants of Laplacians’, J. Funct. Anal. 80 (1988), 148–211.CrossRefGoogle Scholar
[12]Trudinger, N.S., ‘On imbeddings into Orlicz spaces and some applications’, J. Math. Mech. 17 (1967), 473–483.Google Scholar