Bilinear Sobolev–Poincaré Inequalities and Leibniz-Type Rules

By Frédéric Bernicot, Diego Maldonado, Kabe Moen, V. Naibo. Journal of Geometric Analysis, 2011

47 citations recorded by the source corpus.

The dual purpose of this article is to establish bilinear Poincaré-type estimates associated with an approximation of the identity and to explore the connections between bilinear pseudo-differential operators and bilinear potential-type operators. The common underlying theme in both topics is their applications to Leibniz-type rules in Sobolev and Campanato–Morrey spaces under Sobolev scaling.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/4afdc8c56b3af29c0886bbff3535917d0bf40a4f

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Citations in the corpus

Compactness properties of commutators of bilinear fractional integrals (2013, background): “As observed by Bernicot, Maldonado, Moen and Naibo [2], see also [19], if 1 < s < r satisfy (1)s − 1 r = α n , then BIα is bounded on appropriate product weighted Lebesgue spaces; we have (8) BIα : Lp(w p 1)×L (w2)→ L (w1w r 2) where w1,w2 ∈ As,r, that is, for i = 1,2,”
Smoothing properties of bilinear operators and Leibniz-type rules in Lebesgue and mixed Lebesgue spaces (2017, background): “The authors in [12] showed, using a calculus similar to (10), that (13) ‖Iν(f, g)‖Lq ‖Df‖Lp1 ‖g‖Lp2 + ‖f‖Lp1 ‖Dg‖Lp2 if 1 < p1, p2 < ∞, 0 < 1 p1 + 1 p2 − s n = 1 q , and 0 < s < 2n and ν ≤ s.”