Recall from the previous lecture

Theorem (Riesz–Thorin Interpolation)Suppose is a -linear transformation andwith and . Then

where

*Proof:* It suffices to bound

Note and commutes with ; in particular, . Setting

then

- We have the inequality
and hence if , then

and similarly

- is a bounded holomorphic function in a neighborhood of .

Recalling the Hadamard three lines theorem (Phragmen–Lindelöf principle), we obtain the result (* is sub-harmonic, sub-harmonic functions obey the maximum principle, the fact that is bounded means that the part of the boundary at infinity does not contribute*).

Theorem (Stein Interpolation)Let be a family of operators depending analytically on in the strip and continuous up to the boundary. Suppose also thatfor some and . Then

where , , and were as in the statement of the Riesz–Thorin theorem.

*Proof:* Write as above, note is sub-harmonic. The result then follows from a careful revisiting of the 3-lines lemma; details to follow.

RemarkA “meta-application” of Stein is that one can show that if , , , , then . For example, (or could be for fixed , say .

TheoremIf maps continuously from to , then and (i.e., Hausdorff–Young is all the -type estimates).

*Proof:* (a) Scaling. Let be non-zero and fixed. Set . Now . Also, so . So if , then we must have . We can break this by sending or , unless .

*An interpretation:* If is measured in and , then , , , .

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