I am listening to the class of Tao at https://terrytao.wordpress.com/2020/04/13/247b-notes-2-decoupling-theory/. And I have trouble in the interpolation property of decoupling constant in exercise 10.
(iv) (Interpolation) Suppose that $\frac{1}{p}=\frac{1-\theta}{p_{0}}+\frac{\theta}{p_{1}}$ for some $1 \leq p_{0} \leq p \leq p_{1} \leq \infty$ and $0 \leq \theta \leq 1,$ and suppose also that $\mathcal{U}=\left\{U_{1}, \ldots, U_{n}\right\}$ is a non-empty collection of open non-empty subsets of $\mathbf{R}^{d}$ for which one has the projection bounds $$ \left\|P_{U_{j}} f\right\|_{L^{p_{i}}\left(\mathbf{R}^{d}\right)} \lesssim_{p_{i}, d}\|f\|_{L^{p_{i}}\left(\mathbf{R}^{d}\right)} $$ for all $f \in \mathcal{S}\left(\mathbf{R}^{d}\right), i=0,1,$ and $j=1, \ldots, n,$ where the Fourier multiplier $P_{U_{i}}$ is defined bv$$\widehat{P_{U_{j}} f}(\xi)=1_{U_{j}}(\xi) \hat{f}(\xi)$$ Show that $$ \operatorname{Dec}_{p}(\mathcal{U}) \lesssim_{p_{0}, p_{1}, d, \theta} \operatorname{Dec}_{p_{0}}(\mathcal{U})^{1-\theta} \operatorname{Dec}_{p_{1}}(\mathcal{U})^{\theta} $$
Any ideas will be helpful.
Define the operator $T:\ell^2(L^p(\mathbb{R}^d))\to L^p(\mathbb{R}^d)$ given by $$ T(\{f_j\}) := \sum_j P_{U_j}f_j. $$ Then $$ \begin{align} \lVert T(\{f_j\})\rVert_{p_i} &\le Dec_{p_i}(\mathcal{U})\Big(\sum_j\lVert P_{U_j}f_j\rVert_{p_i}^2\Big)^\frac{1}{2} \\\\ &\lesssim_{p_i,d}Dec_{p_i}(\mathcal{U})\Big(\sum_j\lVert f_j\rVert_{p_i}^2\Big)^\frac{1}{2}, \end{align} $$ where we used the hypothesis on $P_{U_j}$. Since $T$ is $L^{p_i}$-bounded, then by Riesz-Thorin theorem we get $$ \lVert T(\{f_j\})\rVert_{p_i} \lesssim_{p,d}Dec_{p_0}(\mathcal{U})^{1-\theta}Dec_{p_1}(\mathcal{U})^\theta\Big(\sum_j\lVert f_j\rVert_{p_i}^2\Big)^\frac{1}{2}, $$ and the exercise follows after taking $f_j$ with Fourier support in $U_j$.