Riesz transforms associated with the Hodge Laplacian in Lipschitz subdomains of Riemannian manifolds

Steve Hofmann[1]; Marius Mitrea[1]; Sylvie Monniaux[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)

Annales de l’institut Fourier (2011)

  • Volume: 61, Issue: 4, page 1323-1349
  • ISSN: 0373-0956

Abstract

top
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.

How to cite

top

Hofmann, Steve, Mitrea, Marius, and Monniaux, Sylvie. "Riesz transforms associated with the Hodge Laplacian in Lipschitz subdomains of Riemannian manifolds." Annales de l’institut Fourier 61.4 (2011): 1323-1349. <http://eudml.org/doc/219773>.

@article{Hofmann2011,
abstract = {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.},
affiliation = {University of Missouri Department of Mathematics Columbia - 202 Mathematical Sciences Building Columbia, MO 65211 (USA); University of Missouri Department of Mathematics Columbia - 202 Mathematical Sciences Building Columbia, MO 65211 (USA); Université Paul Cézanne LATP - UMR 6632 Faculté des Sciences et Techniques Avenue Escadrille Normandie Niémen 13397 Marseille Cédex 20 (France)},
author = {Hofmann, Steve, Mitrea, Marius, Monniaux, Sylvie},
journal = {Annales de l’institut Fourier},
keywords = {Hodge-Laplacian; Riesz transforms; differential forms; Lipschitz domain; Riemannian manifolds; Riesz transform; Hodge Laplacian},
language = {eng},
number = {4},
pages = {1323-1349},
publisher = {Association des Annales de l’institut Fourier},
title = {Riesz transforms associated with the Hodge Laplacian in Lipschitz subdomains of Riemannian manifolds},
url = {http://eudml.org/doc/219773},
volume = {61},
year = {2011},
}

TY - JOUR
AU - Hofmann, Steve
AU - Mitrea, Marius
AU - Monniaux, Sylvie
TI - Riesz transforms associated with the Hodge Laplacian in Lipschitz subdomains of Riemannian manifolds
JO - Annales de l’institut Fourier
PY - 2011
PB - Association des Annales de l’institut Fourier
VL - 61
IS - 4
SP - 1323
EP - 1349
AB - 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.
LA - eng
KW - Hodge-Laplacian; Riesz transforms; differential forms; Lipschitz domain; Riemannian manifolds; Riesz transform; Hodge Laplacian
UR - http://eudml.org/doc/219773
ER -

References

top
  1. S. Blunck, P. C. Kunstmann, Weak type ( p , p ) estimates for Riesz transforms, Math. Z. 247 (2004), 137-148 Zbl1138.35315MR2054523
  2. Sönke Blunck, Peer Christian Kunstmann, Calderón-Zygmund theory for non-integral operators and the H ∞ functional calculus, Rev. Mat. Iberoamericana 19 (2003), 919-942 Zbl1057.42010MR2053568
  3. Michael Christ, A T ( b ) theorem with remarks on analytic capacity and the Cauchy integral, Colloq. Math. 60/61 (1990), 601-628 Zbl0758.42009MR1096400
  4. Thierry Coulhon, Xuan Thinh Duong, Riesz transforms for 1 ≤ p ≤ 2 , Trans. Amer. Math. Soc. 351 (1999), 1151-1169 Zbl0973.58018MR1458299
  5. Robert Dautray, Jacques-Louis Lions, Analyse mathématique et calcul numérique pour les sciences et les techniques. Vol. 8, (1988), Masson, Paris Zbl0749.35005MR1016605
  6. Javier Duoandikoetxea, Fourier analysis, 29 (2001), American Mathematical Society, Providence, RI Zbl0969.42001MR1800316
  7. Xuan T. Duong, Derek W. Robinson, Semigroup kernels, Poisson bounds, and holomorphic functional calculus, J. Funct. Anal. 142 (1996), 89-128 Zbl0932.47013MR1419418
  8. Xuan Thinh Duong, Alan MacIntosh, Singular integral operators with non-smooth kernels on irregular domains, Rev. Mat. Iberoamericana 15 (1999), 233-265 Zbl0980.42007MR1715407
  9. Eugene Fabes, Osvaldo Mendez, Marius Mitrea, Boundary layers on Sobolev-Besov spaces and Poisson’s equation for the Laplacian in Lipschitz domains, J. Funct. Anal. 159 (1998), 323-368 Zbl0930.35045MR1658089
  10. Waldemar Hebisch, A multiplier theorem for Schrödinger operators, Colloq. Math. 60/61 (1990), 659-664 Zbl0779.35025MR1096404
  11. Waldemar Hebisch, Functional calculus for slowly decaying kernels, (1995) 
  12. Steve Hofmann, José María Martell, L p bounds for Riesz transforms and square roots associated to second order elliptic operators, Publ. Mat. 47 (2003), 497-515 Zbl1074.35031MR2006497
  13. David Jerison, Carlos E. Kenig, The inhomogeneous Dirichlet problem in Lipschitz domains, J. Funct. Anal. 130 (1995), 161-219 Zbl0832.35034MR1331981
  14. Alf Jonsson, Hans Wallin, Function spaces on subsets of R n , Math. Rep. 2 (1984) Zbl0875.46003MR820626
  15. O. Mendez, Marius Mitrea, Finite energy solutions Complex powers of the Neumann Laplacian in Lipschitz domains, Mathematische Nachrichten 223 (2001), 77-88 Zbl0981.35014MR1817850
  16. Dorina Mitrea, Marius Mitrea, Finite energy solutions of Maxwell’s equations and constructive Hodge decompositions on nonsmooth Riemannian manifolds, J. Funct. Anal. 190 (2002), 339-417 Zbl0996.35078MR1899489
  17. Dorina Mitrea, Marius Mitrea, Sylvie Monniaux, The Poisson problem for the exterior derivative operator with Dirichlet boundary condition in nonsmooth domains, Commun. Pure Appl. Anal. 7 (2008), 1295-1333 Zbl1157.35412MR2425010
  18. Marius Mitrea, Sharp Hodge decompositions, Maxwell’s equations, and vector Poisson problems on nonsmooth, three-dimensional Riemannian manifolds, Duke Math. J. 125 (2004), 467-547 Zbl1073.31006MR2166752
  19. Marius Mitrea, Sylvie Monniaux, 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. 361 (2009), 3125-3157 Zbl1170.42010MR2485421
  20. Marius Mitrea, Michael Taylor, Potential theory on Lipschitz domains in Riemannian manifolds: Sobolev-Besov space results and the Poisson problem, J. Funct. Anal. 176 (2000), 1-79 Zbl0968.58023MR1781631
  21. A. Pazy, Semigroups of linear operators and applications to partial differential equations, 44 (1983), Springer-Verlag, New York Zbl0516.47023MR710486
  22. Zhong Wei Shen, Boundary value problems for parabolic Lamé systems and a nonstationary linearized system of Navier-Stokes equations in Lipschitz cylinders, Amer. J. Math. 113 (1991), 293-373 Zbl0734.35080MR1099449
  23. Michael E. Taylor, Partial differential equations. II, 116 (1996), Springer-Verlag, New York Zbl0869.35002MR1395149

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.