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We consider the cubic Nonlinear Schrödinger Equation (NLS) and the Korteweg-de Vries equation in one space dimension. We prove that the solutions of NLS satisfy a-priori local in time $H^s$ bounds in terms of the $H^s$ size of the initial data for $s\ge -\frac{1}{4}$ (joint work with D. Tataru, [, ]) , and the solutions to KdV satisfy global a priori estimate in $H^{-1}$ (joint work with T. Buckmaster []).

Let n ≥ 1, d = 2n, and let (x,y) ∈ ℝⁿ × ℝⁿ be a generic point in ℝ²ⁿ. The twisted Laplacian $L=-1/2{\sum }_{j=1}^n[({\partial }_{x_j}+i{y}_j)²+({\partial }_{y_j}-i{x}_j)²]$ has the spectrum n + 2k = λ²: k a nonnegative integer. Let $P_{\lambda }$ be the spectral projection onto the (infinite-dimensional) eigenspace. We find the optimal exponent ϱ(p) in the estimate $\left|\right|P_{\lambda }{u\left|\right|}_{L^p\left({ℝ}^d\right)}\lesssim {\lambda }^{\varrho \left(p\right)}{\left|\right|u\left|\right|}_{L²\left({ℝ}^d\right)}$ for all p ∈ [2,∞], improving previous partial results by Ratnakumar, Rawat and Thangavelu, and by Stempak and Zienkiewicz. The expression for ϱ(p) is ϱ(p) = 1/p -1/2 if 2 ≤ p ≤ 2(d+1)/(d-1), ϱ(p) = (d-2)/2 - d/p if 2(d+1)/(d-1)...

In [2] Kenig, Ruiz and Sogge proved $${\parallel u\parallel }_{L^{\frac{2n}{n-2}}\left({ℝ}^n\right)}\lesssim {\parallel Lu\parallel }_{L^{\frac{2n}{n+2}}\left({ℝ}^n\right)}$$ provided $n\ge 3$, $u\in C_0^{\infty }\left({ℝ}^n\right)$ and $L$ is a second order operator with constant coefficients such that the second order coefficients are real and nonsingular. As a consequence of [3] we state local versions of this inequality for operators with $C^2$ coefficients. In this paper we show how to apply these local versions to the absence of embedded eigenvalues...

We survey recent work on local well-posedness results for parabolic equations and systems with rough initial data. The design of the function spaces is guided by tools and constructions from harmonic analysis, like maximal functions, square functions and Carleson measures. We construct solutions under virtually optimal scale invariant conditions on the initial data. Applications include BMO initial data for the harmonic map heat flow and the Ricci-DeTurck flow for initial metrics with small local...