Bourgain and Brezis established, for maps with zero average, the existence of a solution of (1) . Maz'ya proved that if, in addition, , then (1) can be solved in . Their arguments are quite different. We present an elementary property of fundamental solutions of the biharmonic operator in two dimensions. This property unifies, in two dimensions, the two approaches, and implies another (apparently unrelated) estimate of Maz'ya and Shaposhnikova. We discuss higher dimensional analogs of the above results.
Bourgain and Brezis ont montré que, si est de moyenne nulle, alors (1) a une solution . Maz'ya a prouvé que si, de plus, on a , alors il existe une solution de (1) dans . Les deux preuves sont distinctes. Dans cette note, nous présentons une propriété élémentaire des solutions fondamentales de l'opérateur biharmonique en dimension deux. Cette propriété unifie, en dimension deux, les approches de Bourgain–Brezis et Maz'ya, et implique une autre estimation de Maz'ya et Shaposhnikova (apparemment non liée aux précédentes). Nous discutons des variantes de ces résultats en dimension supérieure.
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Petru Mironescu 1
@article{CRMATH_2010__348_9-10_513_0, author = {Petru Mironescu}, title = {On some inequalities of {Bourgain,} {Brezis,} {Maz'ya,} and {Shaposhnikova} related to $ {L}^{1}$ vector fields}, journal = {Comptes Rendus. Math\'ematique}, pages = {513--515}, publisher = {Elsevier}, volume = {348}, number = {9-10}, year = {2010}, doi = {10.1016/j.crma.2010.03.019}, language = {en}, }
TY - JOUR AU - Petru Mironescu TI - On some inequalities of Bourgain, Brezis, Maz'ya, and Shaposhnikova related to $ {L}^{1}$ vector fields JO - Comptes Rendus. Mathématique PY - 2010 SP - 513 EP - 515 VL - 348 IS - 9-10 PB - Elsevier DO - 10.1016/j.crma.2010.03.019 LA - en ID - CRMATH_2010__348_9-10_513_0 ER -
Petru Mironescu. On some inequalities of Bourgain, Brezis, Maz'ya, and Shaposhnikova related to $ {L}^{1}$ vector fields. Comptes Rendus. Mathématique, Volume 348 (2010) no. 9-10, pp. 513-515. doi : 10.1016/j.crma.2010.03.019. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2010.03.019/
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