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The Riesz–Thorin Theorem

Matthias Engelmann
Foredrag for studerende
Fredag, 7 december, 2012, at 14:30-15:30, in Aud. D3 (1531-215)
Abstrakt:
I am going to present the Riesz-Thorin Theorem, a result which is often referred to as interpolation theory between $\mathrm{L}^p(\mathbb{R}^n)$ spaces. Let $T$ be a bounded linear map from $\mathrm{L}^{p_0}(\mathbb{R}^n)$ to $\mathrm{L}^{q_0}\(\mathbb{R}^n)$ and from $\mathrm{L}^{p_1}(\mathbb{R}^n)$ to $\mathrm{L}^{q_1}\(\mathbb{R}^n)$. Loosely speaking the theorem asserts that the set of all pairs of indices $(1/p,1/q)$ for which $T$ is bounded is a convex set. More precisely T is a bounded map from $\mathrm{L}^{p_t}(\mathbb{R}^n)$ to $\mathrm{L}^{q_t}\(\mathbb{R}^n)$, where $1/p_t = t/p_1 + (1-t)/p_0$ and $1/q_t = t/q_1 + (1-t)/q_0$ and $t\in[0,1]$. The main ingredients of the proof are basic results from Banach space theory, integration theory and the Hadamard three line theorem. If time allows, I will provide some applications in functional analysis.
Kontaktperson: Søren Fuglede Jørgensen