In this article we revisit the inequalities of Kato and Ponce concerning the L r norm of the Bessel potential J s = (1 − ∆) s/2 (or Riesz potential D s = (−∆) s/2 ) of the product of two functions in terms of the product of the L p norm of one function and the L q norm of the the Bessel potential J s (resp. Riesz potential D s ) of the other function. Here the indices p, q, and r are related as in Hölder's inequality 1/p + 1/q = 1/r and they satisfy 1 ≤ p, q ≤ ∞ and 1/2 ≤ r < ∞. Also the estimate is weak-type in the case when either p or q is equal to 1. In the case when r < 1 we indicate via an example that when s ≤ n/r − n the inequality fails. Furthermore, we extend these results to the multi-parameter case.
In this article, we examine L 2 well-posedness and stabilization property of the dispersion-generalized Benjamin-Ono equation with periodic boundary conditions. The main ingredient of our proof is a development of dissipation-normalized Bourgain space, which gains smoothing properties simultaneously from dissipation and dispersion within the equation. We will establish a bilinear estimate for the derivative nonlinearity using this space and prove the linear observability inequality leading to small-data stabilization.
We prove that the "good" Boussinesq model is locally well-posed in the spaceIn the proof, we employ the method of normal forms, which allows us to explicitly extract the rougher part of the solution, while we show that the remainder is in the smoother space C([0, T ], H β (T)), β < min(1 − 3α, 1 2 − α). Thus, we establish a smoothing effect of order min(1 − 2α, 1 2 ) for the nonlinear evolution. This is new even in the previously considered cases α ∈ (0, 1 4 ).
For the Schrödinger equation u t + iu xx = ∇ β [u 2 ], β ∈ (0, 1/2), we establish local well-posedness in H β−1+ (note that if β = 0, this matches, up to an endpoint, the sharp result of Bejenaru-Tao, [4]). Our approach differs significantly from the previous ones in that we use normal form transformation to analyze the worst interacting terms in the nonlinearity and then show that the remaining terms are (much) smoother. In particular, this allows us to conclude that u − e −it∂ 2x u(0) ∈ H − 1 2 (R 1 ), even though u(0) ∈ H β−1+ . In addition and as a byproduct of our normal form analysis, we obtain a Lipschitz continuity property in H − 1 2 of the solution operator (which originally acts on H β−1+ ), which is new even in the case β = 0. As an easy corollary, we obtain local well-posedness resultsFinally, we sketch an approach to obtain similar statements for the equations u t +iu xx = ∇ β [uū] and u t + iu xx = ∇ β [ū 2 ].
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