The question is about exercise 8.3.2 part (c) of Grafakos's book Moder Fourier Analysis.
The hint seems to be wrong and as there has been a couple of situations like this with the book I want to make sure that I do not miss anything. I have tried to find a list of mistakes in Grafakos but there does not seem to be one.
The Problem (Exercise 8.3.2 part c): Let $\Phi$ be smooth, radial, positive function on $\mathbb{R}^n$ with support in $B(0,1)$ and integral $1$. Denote $\Phi_t (x) = t^{-n} \Phi (x/t)$ and $P_t (f) = \Phi _t * f$. Now let $T$ be a continuous linear operator from the space of Schwarz functions $\mathcal{S} (\mathbb{R}^n)$ to its dual $\mathcal{S} ' (\mathbb{R}^n)$. We want to show that $P_t T P_t (f) \to 0$ in $\mathcal{S} ' (\mathbb{R}^n)$ as $t\to \infty$, where $f$ is Schwarz with $\hat f$ vanishing near the origin.
Hint(from the book): Pair with a Schwarz function $g$ and use part (a)[$P_t(f) \to 0$ in $\mathcal{S}(R^{n})$ as $t\to \infty$] and the fact that all Schwarz seminorms of $P_t(g)$ are bounded uniformly in $t>0$.
My approach: First, let us see that the hint is wrong. Let $n=1$. It is not difficult to see that the second part of the hint is true. So show $\sup_{\xi } | ( \hat{ P_t (g) } )' (\xi) |$ is not bounded in $t$ which contradicts that the Schwarz seminorms of $P_t (g)$ are bounded uniformly in $t>0$.
Standard computations give $$\mathcal{F} (P_t (g)) ' (\xi) = (\hat \Phi (t\xi) \hat g(\xi) )' = t\hat \Phi '(t\xi) \hat g (\xi) + \hat \Phi (t\xi) \hat g ' (\xi).$$ Putting $\xi = \frac{1}{4} t^{-1}$ we get that the last expression is equal to $t\hat \Phi '(1/4) \hat g (t^{-1}/4) + \hat \Phi (1/4) \hat g ' (t^{-1}/4).$ Obviously, for certain admissible choices of $g$ and $\Phi$ this expression is not bounded.
On the other hand, we can show that not only $P_t (f) \to 0$ in $\mathcal {S} (\mathbb{R} ^n)$ as $t\to 0$ but the convergence happens $o(t^M)$ for any $M>0$. Moreover, for arbitrary Schwarz $g$ we have $\rho_{\alpha, \beta} (P_t g) \leq C(g, \Phi) t^{|\alpha|}$, where $\rho$ denotes the Schwarz seminorm. In this case, we get something of the following form \begin{align*} | \langle P_t T P_t f, g \rangle | &= | \langle T P_t f, P_t g \rangle | \\ & \leq \left[ \sum_{|\alpha|, |\beta| \leq N} c_{\alpha, \beta} \rho_{\alpha, \beta} (P_t f)\right] \cdot \left[ \sum_{|\alpha|, |\beta| \leq N} c_{\alpha, \beta} \rho_{\alpha, \beta} (P_t g)\right] \\ &\leq C(f, \Phi, N) t^{-(N + 1)} \cdot C(g, \Phi, N) t^{N}. \end{align*} Thus, we get the desired result.
I'd appreciate if anyone would check this and maybe could provide any information on the existence of the list of mistakes in this book.