Hardy Spaces of Differential Forms on Riemannian Manifolds

By P. Auscher, A. Mcintosh, E. Russ. 2006

247 citations recorded by the source corpus.

Abstract Let M be a complete connected Riemannian manifold. Assuming that the Riemannian measure is doubling, we define Hardy spaces Hp of differential forms on M and give various characterizations of them, including an atomic decomposition. As a consequence, we derive the Hp-boundedness for Riesz transforms on M, generalizing previously known results. Further applications, in particular to H∞ functional calculus and Hodge decomposition, are given.

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/e3a4a20d2f9855a45cfe7270f729ab00ee9ea57b

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In Applied Mathematics

1. Dachun Yang
2. Xuan Thinh Duong
3. Renjin Jiang
4. Sibei Yang
5. Yuan Zhou
6. S. Hofmann
7. Lixin Yan
8. Svitlana Mayboroda

Citations in the corpus

• A characterization of Hajłasz–Sobolev and Triebel–Lizorkin spaces via grand Littlewood–Paley functions☆ (2009, background): “2 apply in a wide range of settings, for instance, to Ahlfors n-regular metric measure spaces (see [22]), d-spaces (see [35]), Lie groups of the polynomial volume growth (see [36, 37, 29, 2]), the complete connected noncompact manifolds with a doubling measure (see [4, 5]), compact Carnot-Carathéodo”
• Orlicz-Hardy Spaces Associated with Operators Satisfying Davies-Gaffney Estimates (2009, background): “In recent years, the study of Hardy spaces and BMO spaces associated with different operators inspired great interests; see, for example, [1, 2, 3, 11, 12, 13, 14, 18, 17, 21, 33] and their references.”
• Musielak-Orlicz-Hardy Spaces Associated with Operators Satisfying Reinforced Off-Diagonal Estimates (2013, unknown): “[8], Hofmann and Mayboroda [40] and Hofmann et al.”
• New Orlicz–Hardy spaces associated with divergence form elliptic operators (2009, background): “In this section, we recall some notions and notation on the divergence form elliptic operator, and present some basic properties on Orlicz functions and also describe some basic assumptions on them.”
• Calderón reproducing formulas and new Besov spaces associated with operators (2012, methodology): “We will develop the theme in the works [17,18,1,2] and define the Besov norm Ḃ p,q via quadratic norms of L.”
• Maximal function characterizations for Hardy spaces on spaces of homogeneous type with finite measure and applications (2018, background): “See for example [17, 2, 23, 22, 24] and the references therein.”
• Musielak–Orlicz–Hardy Spaces Associated with Operators and Their Applications (2012, methodology): “We prove this lemma by borrowing some ideas from [8, 7, 95].”
• Sharp weighted norm estimates beyond Calderón–Zygmund theory (2015, unknown): “Finally, for large enough N , by adapting [Auscher et al. 2008, Lemma 3.6] one can show that '.”
• Hardy Spaces Associated to Non-Negative Self-Adjoint Operators Satisfying Davies-Gaffney Estimates (2011, methodology): “Recently, in [ AMR ] and in [HM], the authors treated Hardy spaces (and in the latter paper, BMO spaces) adapted, respectively, to the Hodge Laplacian on a Riemannian manifold with doubling measure, or to a second order divergence form elliptic operator on Rn with complex coe‐cients, in which”
• Hardy spaces associated to operators satisfying Davies–Gaffney estimates and bounded holomorphic functional calculus☆ (2013, unknown): “If p1 < p < p2, then T is bounded from H L (X ) into L (X ) and ‖Tf ‖Lp(X ) ≤ C‖f ‖Hp L (X ), (11)”
• A MAXIMAL FUNCTION CHARACTERIZATION FOR HARDY SPACES ASSOCIATED TO NONNEGATIVE SELF-ADJOINT OPERATORS SATISFYING GAUSSIAN ESTIMATES (2015, background): “This topic has attracte d lot of attention in the last decades, and has been a very active research topic in harmoni c analysis – see for example, [2, 3, 4, 10, 12, 13, 16, 17, 18, 19, 25].”
• Second order elliptic operators with complex bounded measurable coefficients in L p , Sobolev and Hardy spaces (2010, methodology): “9Although as mentioned above, our tent space/square function definition of adapted Hp spaces with p > 1 follows that given in [10].”
• Hardy and BMO spaces associated to divergence form elliptic operators (2006, methodology): “In [3] we develop a theory ofH1 (Hardy type) and BMO spaces adapted to a second order, divergence form elliptic (aka accretive) operator L in Rn, with complex, L∞ coefficents.”