×

Sobolev inequalities for differential forms and \(L_{q,p}\)-cohomology. (English) Zbl 1105.58008

In this paper, the relation between Sobolev inequalities for differential forms on a Riemannian manifold \((M, g)\) and the \(L_{q,p}\)-cohomology of \((M, g)\) is investigated.
The \(L_{q,p}\)-cohomology of \((M, g)\) is defined as \[ H^k_{q,p} (M) := Z^k_p (M)/d\Omega^{k-1}_{q,p} (M), \] where \(Z^k_p (M)\) is the Banach space of closed \(k\)-forms in \(L^p (M)\) and \(\Omega^{k-1}_{q,p} (M)\) is the space of all \((k - 1)\)-forms \(\phi\) in \(L^q (M)\) such that \(d\phi \in L^p\). Let \(L^1_l (M,\Lambda^k)\) be the space of differential \(k\)-forms whose coefficients are locally integrable. Let \(L^p (M,\Lambda^ k)\) be the space of differential forms \(\theta\) in \(L^1_l (M,\Lambda^ k)\) such that \[ \| \theta\|_p : =\biggl(\int_{M}|\theta|^{p} \,dx\biggr)^{\frac{1}{p}} < \infty. \] Let \(B^k_{q,p} (M) := d (L^q (M,\Lambda^{k-1}))\cap L^p (M,\Lambda^k)\) and \(\overline {B}^k_{q,p}\) the closure of \(B^k_{q,p} (M)\).
The authors prove the following results:
Let \((M, g)\) be a smooth \(n\)-dimensional compact Riemannian manifold, \(1\leq k\leq n\) and \(p, q \in (1,\infty)\).
\(\bullet\) There exists a constant \(C\) such that, for any differential form \(\theta\) of degree \(k - 1\) on \(M\) with coefficients in \(L^q\), we have \[ \inf_{\zeta \in Z^{k-1}}\| \theta - \zeta\|_{L^{q}(M)} \leq C \| d\theta\|_{L^{p}(M)} \tag{1} \] if and only if \[ \frac{1}{p} - \frac {1}{q} \leq \frac {1}{n}, \tag{2} \] where \(Z^{k-1}\) denotes the set of smooth closed \((k - 1)\)-forms on \(M\).
\(\bullet\) There exists a constant \(C\) such that, for all closed differential forms \(\omega\) of degree \(k\) on \(M\) with coefficients in \(L^p (M)\), there exists a differential form \(\theta\) of degree \(k - 1\) such that \(d\theta = \omega\) and \[ \| \theta\|_{L^{q}} \leq C \| \omega\|_{L^{p}} \tag{3} \] if and only if \(p, q\) satisfy (2) and \(H^k_{\text{de Rham}} (M) = 0\).
From now on, let \((M, g)\) be a smooth \(n\)-dimensional Riemannian manifold, \(1\leq k\leq n\) and \(p, q \in (1,\infty)\).
\(\bullet\) \(H^k_{q,p} (M) = 0\) if and only if there exists a constant \(C\) such that, for any closed \(p\)-integrable differential form \(\omega\) of degree \(k\) on \(M\), there exists a differential form \(\theta\) of degree \(k - 1\) such that \(d\theta =\omega\) and \[ \| \theta \|_{L^q} \leq C \| \omega \|_{L^p}. \]
\(\bullet\) If \(\overline {B}^k_{q,p} /B^k_{q,p} = 0\), then there exists a constant \(C\) such that, for any differential form \(\theta \in \Omega^{k-1}_{q,p}(M)\) of degree \(k - 1\), there exists a closed form \(\zeta \in Z^{k -1}_{q} (M)\) such that \[ \| \theta - \zeta \|_{L^q} \leq C \| d\theta \|_{L^p}.\tag{4} \]
\(\bullet\) If \(1 < q < \infty\), and if there exists a constant \(C\) such that, for any form \(\theta \in \Omega^{k-1}_{q,p} (M)\) of degree \(k - 1\), there exists a \(\zeta \in Z^{k - 1}_{q} (M)\) such that (4) holds, then \(\overline{B}^{k}_{q,p}/ B^k_{q,p} = 0\).
The construction is presented in a precise way.

MSC:

58J10 Differential complexes
58A12 de Rham theory in global analysis
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
35J15 Second-order elliptic equations
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Agmon, S., Douglis, A., and Nirenberg, L.Comm. Pure Appl. Math. 17, 35–92, (1964). · Zbl 0123.28706 · doi:10.1002/cpa.3160170104
[2] Brezis, H.Analyse Fonctionnelle, Théorie et Applications, Dunod, Paris, (1999).
[3] de Rham, G.Grundlehren der Mathematischen Wissenschaften,266, Springer-Verlag, Berlin, (1984).
[4] Edmunds, D.E. and Evans, W.D.Spectral theory and differential operators, Oxford Science Publications, (1987). · Zbl 0628.47017
[5] Folland, G.Real Analysis. Modern Techniques and their Applications, 2nd ed., John Wiley &amp; Sons, Inc., New York, (1999). · Zbl 0924.28001
[6] Gaffney, M. A special Stake’s Theorem for complete Riemannian manifolds,Ann. of Math. 60, 140–145, (1954). · Zbl 0055.40301 · doi:10.2307/1969703
[7] Gilbarg, D. and Trudinger, N.Elliptic Partial Differential Equations of Second Order, 2nd ed., Grundlehren der Mathematischen Wissenschaften,224, Springer-Verlag, Berlin, (1983). · Zbl 0562.35001
[8] Gol’dshtein, V.M., Kuz’minov, V.I., and Shvedov, I.A. Differential forms on Lipschitz manifolds,Siberian Math. J. 23(2), (1984).
[9] Gol’dshtein, V.M., Kuz’minov, V.I., and Shvedov, I.A. A property of De Rham regularization operators,Siberian Math. J. 25(2), (1984).
[10] Gol’dshtein, V.M., Kuz’minov, V.I., and Shvedov, I.A. Dual spaces of spaces of differential forms,Siberian Math. J. 54(1), (1986).
[11] Gromov, M. Asymptotic invariants of infinite groups, inGeometric Group Theory, Vol. 2, London Math. Soc. Lecture Notes,182, Cambridge University Press, (1992).
[12] Iwaniec, T. and Lutoborski, A. Integral estimates for null Lagrangians,Arch. Rat. Mech. Anal. 125, 25–79, (1993). · Zbl 0793.58002 · doi:10.1007/BF00411477
[13] D’Onofrio, L. and Iwaniec, T. Notes on p-harmonic analysis,Cont. Math. to appear.
[14] Pansu, P. CohomologieLp des variétés à courbure négative, cas du degré 1,Rend. Sem. Mat. Univ. Politec. Torino, Fascicolo Speciale P.D.E and Geometry, 95–119, (1989).
[15] Pansu, P. CohomologieLp, espaces homogènes et pincement, preprint, Orsay, (1999).
[16] Stein, E.M.Princeton Mathematical Series,30, Princeton University Press, Princeton, NJ, (1970).
[17] Troyanov, M. Parabolicity of manifolds,Siberian Adv. Math. 9, 125–150, (1999). · Zbl 0991.31008
[18] Warner, F. Foundations of differentiable manifolds and Lie groups,Graduate Texts in Mathematics,94, Springer-Verlag, New York-Berlin, (1983). · Zbl 0516.58001
[19] Yosida, K.Functional Analysis, reprint of the 6th ed., (1980), Classics in Mathematics, Springer-Verlag, Berlin, (1995). · Zbl 0435.46002
[20] Zucker, S. Lp-cohomology: Banach spaces and homological methods on Riemannian manifolds,Proc. Symp. Pure Math. 54(2), 637–655, (1993). · Zbl 0805.55006
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.