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Análise / Equações Diferenciais Parciais

 Título The Hilbert transform along curves, anisotropically homogeneous multipliers and the polynomial Carleson theorem Expositor João Pedro Ramos University of Bonn Data Quinta-feira, 14 de março de 2019, 15:30 Local Sala 232 Resumo

In 1966, Fabes and Riviere investigated $L^p$ bounds for Calderón-Zygmund operators with suitable anisotropic homogeinity, in relation to regularity questions for parabolic equations. Inspired by that, E. Stein asked when the Hilbert transform along a smooth curve is $L^p$-bounded, in terms of fundamental curvature properties of the curve.

Many works address this question, but we shall focus on a celebrated result with many different proofs: the (2-dimensional) parabolic Hilbert transform

$$Tf(x,y) = p.v. \int_{\mathbb{R}} f(x-t,y-t^2) \frac{dt}{t}$$

is bounded in every $L^p$ space for $1<p<+\infty$.

In the multiplier side, its symbol satisfies a quadratic anisotropic $0$-homogeneity relationship. In connection to the classical Carleson theorem, one may ask whether Carleson-like operators associated to this transformation share similar bounds, as the usual Carleson symbol belongs to a linear $0$-homogeneous class.

This has been an open problem in time-frequency analysis for the past decade, with intense activity in the past 3 years. In this talk, we shall discuss recent developments, as well as draw connections between this problem and the celebrated polynomial Carleson theorem of Zorin-Kranich. As a by-product, we obtain a new proof of a special case of the Carleson theorem.