L1-Poincaré and Sobolev inequalities for differential forms in Euclidean spaces
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In this paper, we prove Poincaré and Sobolev inequalities for differential forms in L1(ℝn). The singular integral estimates that it is possible to use for Lp, p > 1, are replaced here with inequalities which go back to Bourgain and Brezis (2007).
Keywordsdifferential forms Sobolev-Poincaré inequalities homotopy formula
MSC(2010)58A10 26D15 46E35
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The first author and the second author were supported by Funds for Selected Research Topics from the University of Bologna, MAnET Marie Curie Initial Training Network, GNAMPA (Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni) of INdAM (Istituto Nazionale di Alta Matematica “F. Severi”), Italy, and PRIN (Progetti di ricerca di Rilevante Interesse Nazionale) of the MIUR (Ministero dell’Istruzione dell’Università e della Ricerca), Italy. The third author was supported by MAnET Marie Curie Initial Training Network, Agence Nationale de la Recherche (Grant Nos. ANR-10-BLAN 116-01 GGAA and ANR-15-CE40-0018 SRGI), and thanks the hospitality of Isaac Newton Institute, of EPSRC (Engineering and Physical Sciences Research Council) (Grant No. EP/K032208/1) and Simons Foundation.
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