Abstract. We prove that the generalized Carleson operator with polynomial phase function of degree two is of weak type (2,2). For this, we introduce a new approach to the time-frequency analysis of the quadratic phase. IntroductionThe historical motivation for the subject of this paper is rooted in Luzin's conjecture (1913), which asserts that the Fourier series of a function f ∈ L 2 (T) converges pointwise to f Lebesgue-almost everywhere. In 1966, L. Carleson gave a positive answer to this conjecture in the celebrated paper [C]. His result can be essentially reformulated, via [S1], as follows:(In the sequel, we will always omit principal value notation.) Theorem 0. If for f ∈ C 1 (T) we define the expressionwhere here, by convention, T = [−1/2, 1/2] and A > 0 is an absolute constant.In addition to Carleson's proof, we point out two more proofs of the above result: one due to Fefferman [F], using a very beautiful geometric combinatorial argument, and the other due to Lacey and Thiele [LT1], inspired by the subtle techniques they developed for proving the Calderón conjecture ([LT2] and [LT3]). Now, given the statement of Theorem 0, it is natural to hope that this result may be set in a broader context. Following this direction, Stein conjectured that the generalized Carleson operator defined by(here d ∈ Z, d ≥ 2, P is a polynomial of degree d, and f ∈ C 1 (T)) obeys the same bounds as C. GAFA Geometric And Functional Analysis V. LIE GAFAIn [S2] he proved this conjecture, subject to the key restriction that the supremum in (3) be taken in the class of quadratic polynomials with no linear term. Further, using the T T * method and a variant of van der Corput's lemma, Stein and Wainger [SW] extended this result for polynomials of any degree, but again without the first degree term.Our aim in this paper is to provide a positive answer to this conjecture for the case d = 2. (We mention here that, using similar techniques to those in [LT1], M. Lacey published in [L2] a proof of this result, but as was revealed later by A. Ionescu, this was incorrect -for details see [L1].)The main result of the article is given byCombining this result with the techniques developed by Stein in [S1], we easily deduce Corollary 1. T is of weak type (2, 2).The proof of Theorem 1 is a combination of analytic and geometric facts; it relies on a new perspective of the time-frequency localization of the quadratic phase to which we adapt the techniques presented in [F].One particular feature of this paper is that it presents for the first time a timefrequency proof of the boundedness of a maximal operator which is invariant under quadratic modulations.Another novelty of this paper is that we show that one can prove the (quadratic) Carleson theorem using a single dyadic grid partition, on each axis defining the timefrequency plane. (For this, we will involve in our reasoning certain dilation factors of our tiles; see section 7 -forest decomposition algorithm.)Finally, given the powerful geometric intuition developed in Fefferman's paper, and als...

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 show that "splash" singularities cannot develop in the case of locally smooth solutions of the two-fluid interfaces in two dimensions. More precisely, we show that the scenario of formation of singularities discovered by Castro-Córdoba-Fefferman-Gancedo-Gómez-Serrano [6] in the case of the water waves system, in which the interface remains locally smooth but self-intersects in finite time, is completely prevented in the case of two-fluid interfaces with positive densities.1 There is a slight imprecision in these equations, namely the function z(, .t) is determined only up to a reparametrization. This is consistent, of course, with the physical interpretation of the equations. To eliminate this imprecision, there are at least two natural parametrizations one could use: Eulerian coordinates in the case when the interface is a graph above a Euclidean space (notice that the vertical direction is determined invariantly by the choice of the vector-field e2 in the gravity term in (1.2)), or Lagrangian coordinates relative to one of the fluids. In this paper we work invariantly, i.e. we do not make any specific choice of coordinates on the interface. 2 We remark that the surface tension coefficient σ has to be strictly positive in order to prevent Kelvin-Helmholtz instabilities and have local well-posedness, in the case of two-fluid interfaces ρ 1 > 0, ρ 2 > 0.

We are proving L 2 (R) × L 2 (R) → L 1 (R) bounds for the bilinear Hilbert transform HΓ along curves Γ = (t, −γ(t)) with γ being a smooth "non-flat" curve near zero and infinity.

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