The (Weak-L 2) Boundedness of the Quadratic Carleson Operator

Abstract

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...

Maximal operators and Hilbert transforms along variable non‐flat homogeneous curves

“…Remark that the case α = 2 is a deep result due to the third author [20] which Corollary 1.7 does not encompass due to the quadratic modulation symmetries present if α = 2. The third author also proved bounds for the full polynomial Carleson operator [23], where the supremum goes over all polynomials P with real coefficients of degree less than a fixed number.…”

Polynomial Carleson Operators Along Monomial Curves in the Plane

“…We will see how to carry out these deductions from a more general argument we now give in the context of the quadratic Carleson operator C par along the parabola (defined in equation (1.14)). We prove that an inequality of the form (1.15) would imply the analogue over R of Lie's result [Lie09] on the one-variable quadratic Carleson operator C Q : Proposition 6.1. Assume the veracity of the estimate…”

The pointwise convergence of Fourier Series (II). Strong L1 case for the lacunary Carleson operator

“…We will now make our decomposition in (61) precise. For simplicity, assuming without loss of generality 28 that 27 In fact relation (60) turns out to be essentially equivalent with the lacunarity requirement; more precisely, we have that any lacunary sequence {nj }j ⊆ N obeys (60) and, conversely, any (increasing) sequence of natural numbers obeying (60) can be decomposed as union of no more thanC > 0 (depending only onC) lacunary (sub)sequences. 28 The sequence {n k } k lacunary implies lim inf k→∞ n k+1 n k = α > 1.…”