# 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:

### 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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Last changed: Jan 13, 2020