A family of Hardy type spaces on nondoubling manifolds

Abstract

We introduce a decreasing one-parameter family ${\mathfrak{X}}^{\gamma }(M)$, $\gamma >0$, of Banach subspaces of the Hardy-Goldberg space ${\mathfrak{h}}^1(M)$ on certain nondoubling Riemannian manifolds with bounded geometry and we investigate their properties. In particular, we prove that ${\mathfrak{X}}^{1/2}(M)$ agrees with the space of all functions in ${\mathfrak{h}}^1(M)$ whose Riesz transform is in $L^1(M)$, and we obtain the surprising result that this space does not admit an atomic decomposition.

arXiv comment21 pages

References in corpus (5)

• Gaussian heat kernel upper bounds via Phragmén-Lindelöf theorem
• Hardy type spaces on certain noncompact manifolds and applications
• H^1 and BMO for certain locally doubling metric measure spaces of finite measure
• Spectral multipliers for sub-Laplacians on solvable extensions of stratified groups
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Cited by in corpus (3)

• Maximal characterisation of local Hardy spaces on locally doubling manifolds
• Inclusions and noninclusions of Hardy type spaces on certain nondoubling manifolds
• Hardy spaces and Riesz transforms on a Lie group of exponential growth