Abstract
We develop the theory of the “local” Hardy space \(\mathfrak{h}^{1}(M)\) and John-Nirenberg space \(\mathop{\mathrm{bmo}}(M)\) when M is a Riemannian manifold with bounded geometry, building on the classic work of Fefferman-Stein and subsequent material, particularly of Goldberg and Ionescu. Results include \(\mathfrak{h}^{1}\) – \(\mathop{\mathrm{bmo}}\) duality, L p estimates on an appropriate variant of the sharp maximal function, \(\mathfrak{h}^{1}\) and bmo-Sobolev spaces, and action of a natural class of pseudodifferential operators, including a natural class of functions of the Laplace operator, in a setting that unifies these results with results on L p-Sobolev spaces. We apply results on these topics to some interpolation theorems, motivated in part by the search for dispersive estimates for wave equations.
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References
Carbonaro, A., Mauceri, G., Meda, S.: H 1 and BMO for certain nondoubling measured metric spaces. Preprint (2008)
Chang, D., Krantz, S., Stein, E.: H p theory on a smooth domain in ℝN and applications to partial differential equations. J. Funct. Anal. 114, 286–347 (1993)
Cheeger, J., Gromov, M., Taylor, M.: Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds. J. Differ. Geom. 17, 15–53 (1982)
Coifman, R., Weiss, G.: Analyse Harmonique Non-Commutative sur Certains Espaces Homogenes. LNM, vol. 242. Springer, New York (1971)
Coifman, R., Weiss, G.: Extensions of Hardy spaces and their use in analysis. Bull. AMS 83, 569–645 (1977)
Fefferman, C., Stein, E.: H p spaces of several variables. Acta Math. 129, 137–193 (1972)
Goldberg, D.: A local version of real Hardy spaces. Duke Math. J. 46, 27–42 (1979)
Helgason, S.: Lie Groups and Geometric Analysis. Academic Press, New York (1984)
Ionescu, A.: Fourier integral operators on noncompact symmetric spaces of real rank one. J. Funct. Anal. 174, 274–300 (2000)
John, F., Nirenberg, L.: On functions of bounded mean oscillation. Commun. Pure Appl. Math. 24, 415–426 (1961)
Mitrea, M., Taylor, M.: Potential theory in Lipschitz domains in Riemannian manifolds: L p, Hardy, and Hölder space results. Commun. Anal. Geom. 9, 369–421 (2001)
Sarason, D.: Functions of vanishing mean oscillation. Trans. Am. Math. Soc. 207, 391–405 (1975)
Semmes, S.: A primer on Hardy spaces, and some remarks on a theorem of Evans and Müller. Commun. Partial Differ. Equ. 19, 277–319 (1994)
Stein, E.: Topics in Harmonic Analysis Related to the Littlewood-Paley Theory. Princeton Univ. Press, Princeton (1970)
Stein, E.: Harmonic Analysis. Princeton Univ. Press, Princeton (1993)
Strichartz, R.: The Hardy space H 1 on manifolds and submanifolds. Can. J. Math. 24, 915–925 (1972)
Taylor, M.: Fourier integral operators and harmonic analysis on compact manifolds. Proc. Symp. Pure Math. 35(2), 115–136 (1979)
Taylor, M.: Pseudodifferential Operators. Princeton Univ. Press, Princeton (1981)
Taylor, M.: L p estimates on functions of the Laplace operator. Duke Math. J. 58, 773–793 (1989)
Triebel, H.: Theory of Function Spaces II. Birkhauser, Boston (1992)
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Communicated by Steven Kranz.
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Taylor, M. Hardy Spaces and bmo on Manifolds with Bounded Geometry. J Geom Anal 19, 137–190 (2009). https://doi.org/10.1007/s12220-008-9054-7
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DOI: https://doi.org/10.1007/s12220-008-9054-7