On an endpoint Kato-Ponce inequality

By Jean Bourgain, Dong Li. Differential and Integral Equations, 2014

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On Kato–Ponce and fractional Leibniz (2016, background): “The end-point case r = ∞ was conjectured in Grafakos, Maldonado and Naibo [12] and solved in recent [4].”
Smoothing properties of bilinear operators and Leibniz-type rules in Lebesgue and mixed Lebesgue spaces (2017, unknown): “The case p1 = p2 = p = ∞ was then further considered by Grafakos–Maldonado–Naibo [35] and completely resolved by Bourgain–Li [13].”
Strong illposedness of the incompressible Euler equation in integer Cm spaces (2014, background): “kgk∞, 1 <p<∞. (1.2) NORM INFLATION IN Cm 3 (For p = ∞ a version of the L∞ end-point Kato-Ponce inequality (conjectured in [18]) and several new Kato-Ponce type inequalities are proved in recent [7].) The aforementioned Sobolev spaces Ws,p(Rd), s>d/p+ 1 are sometimes called subcritical/non-b”
Strong ill-posedness of the incompressible Euler equation in borderline Sobolev spaces (2013, background): “lim sup t→T ∗ ‖u(t, ·)‖Hs(Rd ) = +∞, 1 The L∞ end-point Kato-Ponce inequality (conjectured in [14]) and several new Kato-Ponce type inequalities are proved in recent [2].”
Sharp commutator estimates via harmonic extensions (2016, methodology): “This commutator has been estimated by Kenig-Ponce-Vega [27] in L-spaces, see also [21, 3].”