A family of Hardy-type spaces on nondoubling manifolds
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A family of Hardy‑type spaces on nondoubling manifolds Alessio Martini1 · Stefano Meda2 · Maria Vallarino3 Received: 27 August 2019 / Accepted: 1 February 2020 © Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We introduce a decreasing one-parameter family 𝔛𝛾 (M) , 𝛾 > 0 , of Banach subspaces of the Hardy–Goldberg space 𝔥1 (M) on certain nondoubling Riemannian manifolds with bounded geometry, and we investigate their properties. In particular, we prove that 𝔛1∕2 (M) agrees with the space of all functions in 𝔥1 (M) whose Riesz transform is in L1 (M) , and we obtain the surprising result that this space does not admit an atomic decomposition. Keywords Hardy space · Atom · Noncompact manifold · Exponential growth · Riesz transform Mathematics Subject Classification 42B20 · 42B30 · 42B35 · 58C99
1 Introduction In their seminal paper, Fefferman and Stein [22] defined the classical Hardy space H 1 (ℝn ) as follows:
Work partially supported by PRIN 2015 “Real and complex manifolds: geometry, topology and harmonic analysis” and by the EPSRC Grant EP/P002447/1. The authors are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). * Maria Vallarino [email protected] Alessio Martini [email protected] Stefano Meda [email protected] 1
School of Mathematics, University of Birmingham, Edgbaston, Birmingham B15 2TT, UK
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Dipartimento di Matematica e Applicazioni, Università di Milano-Bicocca, Via R. Cozzi 53, 20125 Milan, Italy
3
Dipartimento di Scienze Matematiche “Giuseppe Luigi Lagrange”, Dipartimento di Eccellenza 2018‑2022, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Turin, Italy
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H 1 (ℝn ) ∶= {f ∈ L1 (ℝn ) ∶ ||∇(−Δ)−1∕2 f || ∈ L1 (ℝn )};
(1.1)
here ∇ and Δ denote the Euclidean gradient and Laplacian, respectively. Following up earlier work of Burkholder et al. [6], Fefferman and Stein obtained several characterisations of H 1 (ℝn ) in terms of various maximal operators and square functions, thereby starting the real variable theory of Hardy spaces. Their analysis was complemented by Coifman [13], who showed that H 1 (ℝ) admits an atomic decomposition. This result was later extended to higher dimensions by Latter [32]. It is natural to speculate whether an analogue of the results of Fefferman–Stein, Coifman and Latter holds in different settings. In other words, one may ask what is the most appropriate way to define Hardy spaces in settings other than ℝn and whether different definitions lead to the same spaces. In this paper, we will consider this problem on a class of nondoubling Riemannian manifolds. There is a huge literature concerning this question on manifolds or on even more abstract sorts of spaces, and it is virtually impossible to give an account of the main results in the field. Thus, without any pretence of exhaustiveness, we mention just a few c
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