Abstract
In this paper, we show that, for doubling manifolds satisfiying the scaled Poincaré inequalities and \(p\in (2,\infty )\), the boundedness of the Riesz transform dΔ−1/2 on L p, is essentially equivalent to the fact that \(H_{1,d}^{p}\) is equal the L p closure of the set of L p exact harmonic 1-forms. Here, \(H_{1,d}^{p}\) is a Hardy space of exact 1 −forms, naturally associated with the Riesz transform, as defined by Auscher, McIntosh and Russ.
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References
Auscher, P., McIntosh, A., Russ, E.: Hardy spaces of differential forms on Riemannian manifolds. J. Geom. Anal. 18(1), 192–248 (2008)
Auscher, P., Coulhon, T., Duong, X.T., Hofmann, S.: Riesz transform on manifolds and heat kernel regularity. Ann. Sci. École Norm. Sup. (4) 37(6), 911–957 (2004)
Auscher, P., Coulhon, T.: Riesz transform on manifolds and Poincaré inequalities. Ann. Sc. Norm. Super. Pisa Cl. Sci.(5) 4(3), 531–555 (2005)
Coulhon, T., Devyver, B., Sikora, A.: Gaussian heat kernel estimates: from functions to forms. arxiv:1606.02423
Coulhon, T., Duong, X.T.: Riesz transforms for 1p2. Trans. Amer. Math. Soc. 351(3), 1151–1169 (1999)
Coulhon, T., Duong, X.T.: Riesz transform and related inequalities on noncompact Riemannian manifolds. Comm. Pure Appl. Math. 56(12), 1728–1751 (2003)
Coulhon, T., Zhang, Q.S.: Large time behavior of heat kernels on forms. J. Differ. Geom. 77(3), 353–384 (2007)
Coifman, R.R., Meyer, Y., Stein, E.M.: Some new function spaces and their applications to harmonic analysis. J. Funct. Anal. 62(2), 304–335 (1985)
Devyver, B.: A Gaussian estimate for the heat kernel on differential forms and application to the Riesz transform. Math. Ann. 358(1–2), 25–68 (2014)
Feller, W.: An Introduction to Probability Theory and Its Applications, vol. I. Wiley, New York (1968)
Hofmann, S., Lu, G., Mitrea, D., Mitrea, M., Yan, L.: Hardy spaces associated to non-negative self-adjoint operators satisfying Davies-Gaffney estimates, vol. 214 (2011)
Saloff-Coste, L.: Aspects of Sobolev-type Inequalities London Mathematical Society Lecture Note Series, vol. 289. Cambridge University Press, Cambridge (2002)
Shen, Z.: Bounds of Riesz transforms on L p spaces for second order elliptic operators. Ann. Inst. Fourier (Grenoble) 55(1), 173–197 (2005)
Stein, E.M.: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals Princeton Mathematical Series, vol. 43. Princeton University Press, Princeton, NJ (1993)
Stein, E.M.: Singular Integrals and Differentiability of Functions. Princeton University Press, Princeton (1970)
Yau, S.T.: Some function-theoretic properties of complete Riemannian manifold and their applications to geometry. Indiana Univ. Math. J 25(7), 659–670 (1976)
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Devyver, B. Hardy Spaces and Heat Kernel Regularity. Potential Anal 48, 1–33 (2018). https://doi.org/10.1007/s11118-017-9623-0
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DOI: https://doi.org/10.1007/s11118-017-9623-0