The Hilbert transform along the parabola, the polynomial Carleson theorem and oscillatory singular integrals
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Mathematische Annalen
The Hilbert transform along the parabola, the polynomial Carleson theorem and oscillatory singular integrals João P. G. Ramos1 Received: 30 July 2020 / Revised: 13 August 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We make progress on an interesting problem on the boundedness of maximal modulations of the Hilbert transform along the parabola. Namely, if we consider the multiplier arising from it and restrict it to lines, we prove uniform L p bounds for maximal modulations of the associated operators. Our methods consist of identifying where to use effectively the polynomial Carleson theorem, and where we can take advantage of the presence of oscillation to obtain decay through the T T ∗ method. Mathematics Subject Classification 42B20 · 42B25 · 44A12
1 Introduction 1.1 Historical background For an arbitrary function f ∈ L 1 (T), a seminal problem is that of determining sharp additional constraints such that its Fourier series converge pointwise to f : S N f (x) =
N
f (n)e2πinx → f (x),
k=−N
1/2 where we define its Fourier coefficients as f (n) = −1/2 f (x)e−2πinx dx. Classical results state that, if f ∈ BV (T) is of bounded variation, then the Fourier series S N f (x) converges back to f (x) for all x such that f is continuous at x.
Communicated by Loukas Grafakos.
B 1
João P. G. Ramos [email protected] Instituto Nacional de Matemática Pura e Aplicada, Estrada Dona Castorina 110, Rio de Janeiro, RJ 22460-320, Brazil
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J. P. G. Ramos
On the other hand, Kolmogorov’s example [11] shows that convergence cannot be expected even at a single point for general f ∈ L 1 (T). The problem of determining what happens in between the case of regular functions f ∈ BV and of f ∈ L 1 has been started with Luzin’s conjecture, which states that, for any f ∈ L 2 (T), S N f (x) → f (x) for almost every x ∈ T. This was an open problem for half a century, until Carleson [1], considering the maximal function C f (x) = sup |S N f (x)|, N ∈N
proved its answer to be affirmative. Carleson’s proof involves decompositions of the function and the operator C simultaneously, in order to encompass all symmetries of the operator. Differently from a regular singular integral operator, C commutes not only with translations and dilations, but has an additional modulation symmetry. Indeed, instead of bounding C itself, one goes about bounding the operator +∞ 2πikx f (k)e C˜ f (x) = sup , N ∈Z k=N
Carleson’s methods [1] were the first to propose such a decomposition. Following the initial proof of Carleson, these results have been generalized to general f ∈ L p (T) by Hunt [10], and later Fefferman [4] elucidated the general procedure in a more streamlined way. One curious feature of Fefferman’s proof is that he in fact does not prove the L 2 → L 2,∞ bound of the Carleson operator directly, but rather goes about proving that C f belongs to L p , 1 ≤ p < 2 for all f ∈ L 2 (T). The weak-type inequality was instead only explicitly proven in [13], where
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