Let $S$ be a smooth hypersurface in $\mathbb{R}^{n}$ with surface measure $d\sigma$. Let $1\leq p,q\leq\infty$, $R>0$, and $\mathcal{N}_{R^{-1}}(S)$ denote the $R^{-1}$ neighborhood of $S$. Suppose we have the local restriction estimate

$$\|\widehat{f}\|_{L^{q}(\mathcal{N}_{R^{-1}}(S))}\lesssim_{p,q,\alpha,S}R^{\alpha-\frac{1}{q}}\|f\|_{L^{p}(B(x_{0},R)} \tag{1}$$

for test functions $f$ supported in the ball $B(x_{0},R)$, for some $\alpha\geq 0$. In the notes "Recent Progress on the Restriction Conjecture" by Terence Tao, Problem 2.2(b) is an outline of how (1) under the condition $q\leq p,p'$ implies the global restriction estimate

$$\|\widehat{f}\|_{L^{q}(\mathcal{N}_{R^{-1}}(S))}\lesssim_{p,q,\alpha,S}R^{\alpha-\frac{1}{q}}\|f\|_{L^{p}(\mathbb{R}^{n})} \tag{2}$$

Following the suggested outline for (1) implies (2), we use a $C^{\infty}$ partition of unity to write $f=\sum_{B}f_{B}$, where the sum ranges over a collection of a finitely overlapping balls $\mathbb{R}^{n}$ and $\mathrm{supp}(f_{B})\subset B$. Applying (1) to each $f_{B}$, we have the estimate

$$\|\widehat{f_{B}}\|_{L^{q}(\mathcal{N}_{R^{-1}}(S))} \lesssim R^{\alpha-\frac{1}{q}}\|f_{B}\|_{L^{p}(B)}$$

Now we need to sum over the pieces to somehow get (2). The suggestion is to prove an estimate of the form

$$\|\sum_{B'}F_{B'}\ast\widehat{\psi_{B'}}\|_{L^{r}(\mathbb{R}^{n})}\lesssim\left(\sum_{B'}\|F_{B'}\|_{L^{r}(\mathbb{R}^{n})}^{\min\{r,r'\}}\right)^{\frac{1}{\min\{r,r'\}}} \tag{3}$$

where $1\leq r\leq \infty$, $\psi_{B'}$ is a bump function adapted to the ball $B'$, and the sum ranges over a collection of finitely overlapping balls $B'$ of radius $R$. The $r=1,2, \infty$ cases of (3) follow from Minkowski's inequality and Young's inequality, and the rest of the $r$ follow from complex interpolation.

My problem is how to use (3) to get (2). I know I have the reproducing formula $\widehat{f_{B}}=\widehat{f_{B}}\ast\widehat{\psi_{B}}$, where $\psi_{B}$ is $1$ on $B$ and concentrated on $2B$, but I'm not sure how to apply this in conjunction with (3). Any suggestions?

Let $S$ be a smooth hypersurface in $\mathbb{R}^{n}$ with surface measure $d\sigma$. Let $1\leq p,q\leq\infty$, $R>0$, and $\mathcal{N}_{R^{-1}}(S)$ denote the $R^{-1}$ neighborhood of $S$. Suppose we have the local restriction estimate

$$\|\widehat{f}\|_{L^{q}(\mathcal{N}_{R^{-1}}(S))}\lesssim_{p,q,\alpha,S}R^{\alpha-\frac{1}{q}}\|f\|_{L^{p}(B(x_{0},R)} \tag{1}$$

for test functions $f$ supported in the ball $B(x_{0},R)$, for some $\alpha\geq 0$. In the notes "Recent Progress on the Restriction Conjecture" by Terence Tao, Problem 2.2(b) is an outline of how (1) under the condition $q\leq p,p'$ implies the global restriction estimate

$$\|\widehat{f}\|_{L^{q}(\mathcal{N}_{R^{-1}}(S))}\lesssim_{p,q,\alpha,S}R^{\alpha-\frac{1}{q}}\|f\|_{L^{p}(\mathbb{R}^{n})} \tag{2}$$

Following the suggested outline for (1) implies (2), we use a $C^{\infty}$ partition of unity to write $f=\sum_{B}f_{B}$, where the sum ranges over a collection of a finitely overlapping balls $\mathbb{R}^{n}$ and $\mathrm{supp}(f_{B})\subset B$. Applying (1) to each $f_{B}$, we have the estimate

$$\|\widehat{f_{B}}\|_{L^{q}(\mathcal{N}_{R^{-1}}(S))} \lesssim R^{\alpha-\frac{1}{q}}\|f_{B}\|_{L^{p}(B)}$$

Now we need to sum over the pieces to somehow get (2). The suggestion is to prove an estimate of the form

$$\|\sum_{B'}F_{B'}\ast\widehat{\psi_{B'}}\|_{L^{r}(\mathbb{R}^{n})}\lesssim\left(\sum_{B'}\|F_{B'}\|_{L^{r}(\mathbb{R}^{n})}^{\min\{r,r'\}}\right)^{\frac{1}{\min\{r,r'\}}} \tag{3}$$

where $1\leq r\leq \infty$, $\psi_{B'}$ is a bump function adapted to the ball $B'$, and the sum ranges over a collection of finitely overlapping balls $B'$ of radius $R$. The $r=1, \infty$ cases of (3) follow from Minkowski's inequality and Young's inequality, the $r=2$ case follows from Plancherel, and the rest of the $r$ follow from complex interpolation.

My problem is how to use (3) to get (2). I know I have the reproducing formula $\widehat{f_{B}}=\widehat{f_{B}}\ast\widehat{\psi_{B}}$, where $\psi_{B}$ is $1$ on $B$ and concentrated on $2B$, but I'm not sure how to apply this in conjunction with (3). Any suggestions?