## Publicacions Matemàtiques

### Potential Maps, Hardy Spaces, and Tent Spaces on Domains

#### Abstract

Suppose that $\Omega$ is the open region in $\mathbb{R}^n$ above a Lipschitz graph and let $d$ denote the exterior derivative on $\mathbb{R}^n$. We construct a convolution operator $T$ which preserves support in $\overline\Omega$, is smoothing of order $1$ on the homogeneous function spaces, and is a potential map in the sense that $dT$ is the identity on spaces of exact forms with support in $\overline\Omega$. Thus if $f$ is exact and supported in $\overline\Omega$, then there is a potential~$u$, given by $u=Tf$, of optimal regularity and supported in $\overline\Omega$, such that $du=f$. This has implications for the regularity in homogeneous function spaces of the de Rham complex on $\Omega$ with or without boundary conditions. The operator $T$ is used to obtain an atomic characterisation of Hardy spaces $H^p$ of exact forms with support in $\overline\Omega$ when $n/(n+1)<\leq 1$. This is done via an atomic decomposition of functions in the tent spaces $\mathcal T^p(\mathbb{R}^n\times\mathbb{R}^+)$ with support in a tent $T(\Omega)$ as a sum of atoms with support away from the boundary of $\Omega$. This new decomposition of tent spaces is useful, even for scalar valued functions.

#### Article information

Source
Publ. Mat., Volume 57, Number 2 (2013), 295-331.

Dates
First available in Project Euclid: 12 December 2013

https://projecteuclid.org/euclid.pm/1386857698

Mathematical Reviews number (MathSciNet)
MR3114771

Zentralblatt MATH identifier
1282.35113

#### Citation

Costabel, Martin; McIntosh, Alan; Taggart, Robert J. Potential Maps, Hardy Spaces, and Tent Spaces on Domains. Publ. Mat. 57 (2013), no. 2, 295--331. https://projecteuclid.org/euclid.pm/1386857698

#### References

• M. E. Bogovskiĭ, Solution of the first boundary value problem for an equation of continuity of an incompressible medium, (Russian), Dokl. Akad. Nauk SSSR 248(5) (1979), 1037\Ndash1040.
• M. E. Bogovskiĭ, Solutions of some problems of vector analysis, associated with the operators $\operatorname{div}$ and $\operatorname{grad}$, (Russian), in: “Theory of cubature formulas and the application of functional analysis to problems of mathematical physics”, Trudy Sem. S. L. Soboleva 1, Akad. Nauk SSSR Sibirsk. Otdel. Inst. Mat., Novosibirsk, 1980, pp. 5\Ndash40, 149.
• D.-C. Chang, S. G. Krantz, E. M. Stein, $H^{p}$ theory on a smooth domain in $\mathbb{R}^{n}$ and elliptic boundary value problems, J.Funct. Anal. 114(2) (1993), 286\Ndash347. \small\tt DOI: 10.1006/jfan.1993. \small\tt 1069.
• R. R. Coifman, A real variable characterization of $H^{p}$, Studia Math. 51 (1974), 269\Ndash274.
• R. R. Coifman, Y. Meyer, and E. M. Stein, Some new function spaces and their applications to harmonic analysis, J. Funct. Anal. 62(2) (1985), 304\Ndash335. \small\tt DOI: 10.1016/0022-1236(85)90007-2.
• M. Costabel and A. McIntosh, On Bogovskiĭ and regularized Poincaré integral operators for de Rham complexes on Lipschitzdomains, Math. Z. 265(2) (2010), 297\Ndash320. \small\tt DOI:\!\! 10.1007/s00209- \small\tt 009-0517-8.
• G. P. Galdi, “An introduction to the mathematical theory of the Navier-Stokes equations. Vol. I. Linearized steady problems”, Springer Tracts in Natural Philosophy 38, Springer-Verlag, New York, 1994. \small\tt DOI: 10.1007/978-1-4612-5364-8.
• R. H. Latter, A characterization of $H^p(\mathbb{R}^n)$ in terms of atoms, Studia Math. 62(1) (1978), 93\Ndash101.
• Z. Lou and A. McIntosh, Hardy spaces of exact forms on Lipschitz domains in $\mathbb{R}^{N}$, Indiana Univ. Math. J. 5(2)3 (2004), 583\Ndash611. \small\tt DOI: 10.1512/iumj.2004.53.2395.
• Z. Lou and A. McIntosh, Hardy space of exact forms on $\mathbb{R}^{N}$,Trans. Amer. Math. Soc. 357(4) (2005), 1469\Ndash1496. \small\tt DOI: 10.1090/ \small\tt S0002-9947-04-03535-4.
• D. Mitrea, M. Mitrea, and S. Monniaux, The Poisson problem for the exterior derivative operator with Dirichlet boundary condition in nonsmooth domains, Commun. Pure Appl. Anal. 7(6) (2008), 1295\Ndash1333. \small\tt DOI: 10.3934/cpaa.2008.7.1295.
• M. Mitrea, Sharp Hodge decompositions, Maxwell's equations, and vector Poisson problems on nonsmooth, three-dimensional Riemannian manifolds, Duke Math. J. 125(3) (2004), 467\Ndash547. \small\tt DOI: 10.1215/S0012-7094-04-12322-1.
• J. Nečas, “Les méthodes directes en théorie des equations elliptiques”, Masson et Cie, Éditeurs, Paris; Academia, Éditeurs, Prague, 1967.
• E. Russ, The atomic decomposition for tent spaces on spaces of homogeneous type, in: CMA/AMSI Research Symposium “Asymptotic Geometric Analysis, Harmonic Analysis, and Related Topics”, Proc. Centre Math. Appl. Austral. Nat. Univ. 42, Austral. Nat. Univ., Canberra, 2007, pp. 125\Ndash135.
• E. M. Stein, “Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals”, With the assistance of Timothy S. Murphy, Princeton Mathematical Series 43, Monographs in Harmonic Analysis III, Princeton University Press, Princeton, NJ, 1993.
• H. Triebel, “Theory of function spaces”, Monographs in Mathematics 78, Birkhäuser Verlag, Basel, 1983.