Abstract. We consider both the defocusing and focusing cubic nonlinear Klein-Gordon equations utt − ∆u + u ± u 3 = 0 in two space dimensions for real-valued initial data u(0) ∈ H 1 x and ut(0) ∈ L 2 x . We show that in the defocusing case, solutions are global and have finite global L 4 t,x spacetime bounds. In the focusing case, we characterize the dichotomy between this behaviour and blowup for initial data with energy less than that of the ground state.These results rely on analogous statements for the two-dimensional cubic nonlinear Schrödinger equation, which are known in the defocusing case and for spherically-symmetric initial data in the focusing case. Thus, our results are mostly unconditional.It was previously shown by Nakanishi that spacetime bounds for KleinGordon equations imply the same for nonlinear Schrödinger equations.

We analyze the blowup behaviour of solutions to the focusing nonlinear Klein-Gordon equation in spatial dimensions d ≥ 2. We obtain upper bounds on the blowup rate, both globally in space and in light cones. The results are sharp in the conformal and sub-conformal cases. The argument relies on Lyapunov functionals derived from the dilation identity. We also prove that the critical Sobolev norm diverges near the blowup time.

We prove that convolution with affine arclength measure on the curve parametrized by h(t) := (t, t 2 , . . . , t n ) is a bounded operator from L p (R n ) to L q (R n ) for the full conjectured range of exponents, improving on a result due to M. Christ. We also obtain nearly sharp Lorentz space bounds.

Given a finite collection of C 1 vector fields on a C 2 manifold which span the tangent space at every point, we consider the question of when there is locally a coordinate system in which these vector fields have a higher level of smoothness. For example, when is there a coordinate system in which the vector fields are smooth, or real analytic, or have Zygmund regularity of some finite order? We address this question in a quantitative way, which strengthens and generalizes previous works on the quantitative theory of sub-Riemannian (aka Carnot-Carathéodory) geometry due to Nagel, Stein, and Wainger, Tao and Wright, the second author, and others. Furthermore, we provide a diffeomorphism invariant version of these theories. This is the first part in a three part series of papers. In this paper, we study a particular coordinate system adapted to a collection of vector fields (sometimes called canonical coordinates) and present results related to the above questions which are not quite sharp; these results form the backbone of the series. The methods of this paper are based on techniques from ODEs. In the second paper, we use additional methods from PDEs to obtain the sharp results. In the third paper, we prove results concerning real analyticity and use methods from ODEs.For the remainder of the section, fix χ ∈ (0, ξ] as in Proposition 9.31.Lemma 9.32. For m ∈ N, s ∈ [0, 1], J ∈ I(n, q),m−1,m−1,s 1, (9.49) and for s ∈ (0, ∞),{s−1,s−1} 1. (9.50) + C m K∈I0(n,q) g K j,J0 C m−1,s X J 0 (BX J 0 (x0,ξ)) X J X J0 C m−1,s X J 0 (BX J 0 (x0,χ)) X K X J0 C m−1,s X J 0 (BX J 0 (x0,χ)) m−1,m−1,s 1,

We characterize (up to endpoints) the k-tuples (p 1 , . . . , p k ) for which certain k-linear generalized Radon transforms map the productThis generalizes a result of Tao and Wright.