Complex Interpolation

This is the set of notes I wrote up for the talk I gave at the student analysis seminar on December 1 and December 8.

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Let $T:X \to Y$ be an operator between vector spaces satisfying some linearity condition. We fix Banach subspaces $A_0$ and $A_1$ fo $X$ and $B_0$ and $B_1$ of $Y$ . If the restrictions $T|_{A_0}$ and $T|_{A_1}$ are operators into $B_0$ and $B_1$ , respectively, satisfying some norm estimates, then an interpolation theorem tells us that we can often find “intermediate” Banach spaces $(A_t,B_t)$ between $(A_0,B_0)$ and $(A_1,B_1)$ such that $T|_{A_t}$ , for each $t \in [0,1]$ , is a bounded operator into $B_t$ with its norm depending canonically on the norm estimates at the endpoints.

Contents

Lecture 1 – Strong Endpoint Estimates: The theorems of Riesz-Thorin and Stein. If a linear map is bounded as an operator from to and from to , respectively, the the Riesz-Thorin interpolation theorem establishes the boundedness of as an operator from to , where and depend on the exponents and the parameter . As an immediate application, we shall prove the Hausdorff-Young inequality for the Fourier transform and the generalized Young’s inequality. We also mention the variable-parameter version of Riesz-Throin, proven by E. Stein in his 1955 thesis.
Lecture 2 – Weak Endpoint Estimates: The theorems of Marcinkiewicz and Fefferman-Stein. What if an operator just fails to be bounded at the endpoints? In the second part of the talk, we shall consider the weak-type operators, which are bounded as an operator from a Lebesgue space to a weak Lebesgue space. The prototypical examples of such operators are the Hardy-Littlewood maximal function in the theory of differentiation and the Hilbert transform in the theory of singular integrals. We also mention an extension of Stein’s interpolation theorem by C. Fefferman, which allows the endpoint-operator to be bounded from a Lebesgue space to a Hardy space.