Eskil Rydhe: On Laplace--Carleson embeddings, and some aspects of the Fourier transform

Time: Wed 2020-02-05 15.30

Lecturer: Eskil Rydhe, Lunds universitet

Location: Kräftriket, house 5, room 31

Abstract

By the Paley--Wiener theorem, the Laplace transform \(\mathcal{L}\)  sends \(L^2(0,\infty)\) isometrically onto the Hardy space \(H^2(\mathbb{C}_+)\). With this in mind, the Carleson embedding theorem yields a simple characterization of measures \(\mu\) such that \(\mathcal{L}\colon L^2(0,\infty)\to L^2(\mathbb{C}_+,d \mu)\). By the Hausdorff--Young inequality and the Carleson--Duren embedding theorem, one obtains a similar characterization for \(\mathcal{L}\colon L^p(0,\infty)\to L^q(\mathbb{C}_+,d \mu)\), where \(1\le p\le 2\) and \(q\ge p'\). In this talk, I will focus on the case \(q\ge p>2\). A key step is to find a suitable replacement for the Hausdorff--Young inequality. This involves rediscovering/reinterpreting some ancient results by Hardy and Littlewood.

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