We prove that the maximal operator associated with variable homogeneous planar curves (t, ut α ) t∈R , α = 1 positive, is bounded on L p (R 2 ) for each p > 1, under the assumption that u : R 2 → R is a Lipschitz function. Furthermore, we prove that the Hilbert transform associated with (t, ut α ) t∈R , α = 1 positive, is bounded on L p (R 2 ) for each p > 1, under the assumption that u : R 2 → R is a measurable function and is constant in the second variable. Our proofs rely on stationary phase methods, T T * arguments, local smoothing estimates and a pointwise estimate for taking averages along curves.
We consider the Cauchy problem for the one dimensional cubic nonlinear Schrödinger equation iut + uxx − |u| 2 u = 0. As the first step local well-posedness in the modulation space M2,p (2 ≤ p < ∞) is derived (see Theorem 1.4), which covers all the subcritical cases. Afterwards in order to approach the endpoint case, we will prove the almost global wellposedness in some Orlicz type space (see Theorem 1.8), which is a natural generalization of M2,p, and is almost critical from the viewpoint of scaling. The new ingredient is an endpoint version of the two dimensional restriction estimate (see Lemma 3.7).
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