Riesz-Thorin complex interpolation

Riesz-Thorin complex interpolation

Suppose $T:L^{p_1}({\mathbb{R}}^n)\to L^{q_1}({\mathbb{R}}^n)$ and $T:L^{p_2}({\mathbb{R}}^n)\to L^{q_2}({\mathbb{R}}^n)$ is an operator that is bounded and linear, then $T:L^p({\mathbb{R}}^n)\to L^q({\mathbb{R}}^n)$ whenever $\exists \theta \in (0,1)$ such that

\left(\frac{1}{p}, \frac{1}{q}\right) = (1 - \theta)\left(\frac{1}{p_1}, \frac{1}{q_1}\right) + \theta\left(\frac{1}{p_2}, \frac{1}{q_2}\right)

Corollary

Let $p^{\prime }$ be the lesbegue exponent of $p$, that is $\frac{1}{p}$ Then $\Vert \hat{f}{\Vert }_{L^{p^{\prime }}({\mathbb{R}}^n)}\le \Vert f{\Vert }_{L^p({\mathbb{R}}^n)}\forall f\in L^p({\mathbb{R}}^n)$