Hardy spaces of differential forms on Riemannian manifolds. (English) Zbl 1217.42043
The paper deals with the consideration of Hardy spaces of differential forms on \(M\), \(\Lambda^kT^{*}M\), being \(M\) a complete connected Riemannian manifold whose corresponding measure is doubling. The authors show that several characterizations can be obtained as equivalent: via tent spaces, maximal functions or atomic decomposition.
As a consequence of all characterizations, a complete comparison between \(H^p(\Lambda^kT^{*}M)\) and \(L^p(\Lambda^kT^{*}M)\) is obtained. More precisely, the action of the Riesz transform \(D\Delta^{-1/2}\) as a bounded operator on \(H^p(\Lambda^kT^{*}M)\), \(1\leq p \leq +\infty\), is proved.
Duality and interpolation results are also obtained.
As a consequence of all characterizations, a complete comparison between \(H^p(\Lambda^kT^{*}M)\) and \(L^p(\Lambda^kT^{*}M)\) is obtained. More precisely, the action of the Riesz transform \(D\Delta^{-1/2}\) as a bounded operator on \(H^p(\Lambda^kT^{*}M)\), \(1\leq p \leq +\infty\), is proved.
Duality and interpolation results are also obtained.
Reviewer: Santiago Boza (Barcelona)
MSC:
| 42B30 | \(H^p\)-spaces |
| 58J05 | Elliptic equations on manifolds, general theory |
| 47A60 | Functional calculus for linear operators |
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