Riesz transforms associated with the Hodge Laplacian in Lipschitz subdomains of Riemannian manifolds
Annales de l'Institut Fourier, Volume 61 (2011) no. 4, pp. 1323-1349.

We prove L p -bounds for the Riesz transforms associated to the Hodge-Laplacian equipped with absolute and relative boundary conditions in a Lipschitz subdomain of a (smooth) Riemannian manifold for p in a certain interval depending on the Lipschitz character of the domain.

Nous prouvons des estimations L p pour les transformées de Riesz associées au Laplacien de Hodge muni de conditions au bord absolues et relatives dans un domaine lipschitzien d’une variété riemannienne (lisse) pour p dans un intervalle dépendant des constantes lipschitziennes du domaine.

DOI: 10.5802/aif.2642
Classification: 42B20, 58J32, 42B25, 58J05
Keywords: Hodge-Laplacian, Riesz transforms, differential forms, Lipschitz domain, Riemannian manifolds
Mot clés : Laplacien de Hodge, transformées de Riesz, formes différentielles, domaines lipschitziens

Hofmann, Steve 1; Mitrea, Marius 1; Monniaux, Sylvie 2

1 University of Missouri Department of Mathematics Columbia - 202 Mathematical Sciences Building Columbia, MO 65211 (USA)
2 Université Paul Cézanne LATP - UMR 6632 Faculté des Sciences et Techniques Avenue Escadrille Normandie Niémen 13397 Marseille Cédex 20 (France)
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Hofmann, Steve; Mitrea, Marius; Monniaux, Sylvie. Riesz transforms associated with the Hodge Laplacian in Lipschitz subdomains of Riemannian manifolds. Annales de l'Institut Fourier, Volume 61 (2011) no. 4, pp. 1323-1349. doi : 10.5802/aif.2642. https://aif.centre-mersenne.org/articles/10.5802/aif.2642/

[1] Blunck, S.; Kunstmann, P. C. Weak type (p,p) estimates for Riesz transforms, Math. Z., Volume 247 (2004) no. 1, pp. 137-148 | DOI | MR | Zbl

[2] Blunck, Sönke; Kunstmann, Peer Christian Calderón-Zygmund theory for non-integral operators and the H functional calculus, Rev. Mat. Iberoamericana, Volume 19 (2003) no. 3, pp. 919-942 | DOI | MR | Zbl

[3] Christ, Michael A T(b) theorem with remarks on analytic capacity and the Cauchy integral, Colloq. Math., Volume 60/61 (1990) no. 2, pp. 601-628 | MR | Zbl

[4] Coulhon, Thierry; Duong, Xuan Thinh Riesz transforms for 1p2, Trans. Amer. Math. Soc., Volume 351 (1999) no. 3, pp. 1151-1169 | DOI | MR | Zbl

[5] Dautray, Robert; Lions, Jacques-Louis Analyse mathématique et calcul numérique pour les sciences et les techniques. Vol. 8, INSTN: Collection Enseignement. [INSTN: Teaching Collection], Masson, Paris, 1988 (Évolution: semi-groupe, variationnel. [Evolution: semigroups, variational methods], Reprint of the 1985 edition) | MR | Zbl

[6] Duoandikoetxea, Javier Fourier analysis, Graduate Studies in Mathematics, 29, American Mathematical Society, Providence, RI, 2001 (Translated and revised from the 1995 Spanish original by David Cruz-Uribe) | MR | Zbl

[7] Duong, Xuan T.; Robinson, Derek W. Semigroup kernels, Poisson bounds, and holomorphic functional calculus, J. Funct. Anal., Volume 142 (1996) no. 1, pp. 89-128 | DOI | MR | Zbl

[8] Duong, Xuan Thinh; MacIntosh, Alan Singular integral operators with non-smooth kernels on irregular domains, Rev. Mat. Iberoamericana, Volume 15 (1999) no. 2, pp. 233-265 | DOI | MR | Zbl

[9] Fabes, Eugene; Mendez, Osvaldo; Mitrea, Marius Boundary layers on Sobolev-Besov spaces and Poisson’s equation for the Laplacian in Lipschitz domains, J. Funct. Anal., Volume 159 (1998) no. 2, pp. 323-368 | DOI | MR | Zbl

[10] Hebisch, Waldemar A multiplier theorem for Schrödinger operators, Colloq. Math., Volume 60/61 (1990) no. 2, pp. 659-664 | MR | Zbl

[11] Hebisch, Waldemar Functional calculus for slowly decaying kernels (1995) (preprint)

[12] Hofmann, Steve; Martell, José María L p bounds for Riesz transforms and square roots associated to second order elliptic operators, Publ. Mat., Volume 47 (2003) no. 2, pp. 497-515 | MR | Zbl

[13] Jerison, David; Kenig, Carlos E. The inhomogeneous Dirichlet problem in Lipschitz domains, J. Funct. Anal., Volume 130 (1995) no. 1, pp. 161-219 | DOI | MR | Zbl

[14] Jonsson, Alf; Wallin, Hans Function spaces on subsets of R n , Math. Rep., Volume 2 (1984) no. 1, pp. xiv+221 | MR | Zbl

[15] Mendez, O.; Mitrea, Marius Finite energy solutions Complex powers of the Neumann Laplacian in Lipschitz domains, Mathematische Nachrichten, Volume 223 (2001), pp. 77-88 | DOI | MR | Zbl

[16] Mitrea, Dorina; Mitrea, Marius Finite energy solutions of Maxwell’s equations and constructive Hodge decompositions on nonsmooth Riemannian manifolds, J. Funct. Anal., Volume 190 (2002) no. 2, pp. 339-417 | DOI | MR | Zbl

[17] Mitrea, Dorina; Mitrea, Marius; Monniaux, Sylvie The Poisson problem for the exterior derivative operator with Dirichlet boundary condition in nonsmooth domains, Commun. Pure Appl. Anal., Volume 7 (2008) no. 6, pp. 1295-1333 | DOI | MR

[18] Mitrea, Marius Sharp Hodge decompositions, Maxwell’s equations, and vector Poisson problems on nonsmooth, three-dimensional Riemannian manifolds, Duke Math. J., Volume 125 (2004) no. 3, pp. 467-547 | DOI | MR | Zbl

[19] Mitrea, Marius; Monniaux, Sylvie On the analyticity of the semigroup generated by the Stokes operator with Neumann-type boundary conditions on Lipschitz subdomains of Riemannian manifolds, Trans. Amer. Math. Soc., Volume 361 (2009) no. 6, pp. 3125-3157 | DOI | MR | Zbl

[20] Mitrea, Marius; Taylor, Michael Potential theory on Lipschitz domains in Riemannian manifolds: Sobolev-Besov space results and the Poisson problem, J. Funct. Anal., Volume 176 (2000) no. 1, pp. 1-79 | DOI | MR | Zbl

[21] Pazy, A. Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983 | MR | Zbl

[22] Shen, Zhong Wei Boundary value problems for parabolic Lamé systems and a nonstationary linearized system of Navier-Stokes equations in Lipschitz cylinders, Amer. J. Math., Volume 113 (1991) no. 2, pp. 293-373 | DOI | MR | Zbl

[23] Taylor, Michael E. Partial differential equations. II, Applied Mathematical Sciences, 116, Springer-Verlag, New York, 1996 (Qualitative studies of linear equations) | MR | Zbl

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