$L^p$ Convergence of Riesz Means for Schwartz functions

Question

For $\delta > 0$ and $\lambda>0$, define the Riesz means of $f$ by $$ S_\lambda^\delta f(x)= (2\pi)^{-n}\int_{\mathbb{R}^n} e^{i\langle x,\xi\rangle}(1 - |\xi/\lambda|)_+^\delta\hat{f}(\xi)d\xi, $$ where $t_+^\delta = t^\delta$ for $t>0$ and zero otherwise. Define the critical index for $L^p(\mathbb R^n)$ $$ \delta(p) = \max\left\{n\left|\frac{1}{2}- \frac{1}{p}\right| - \frac{1}{2},0\right\}. $$

Assume that (1) $n\geq 3$ and $p \in [1,\frac{2(n+1)}{n+3}]\cup [\frac{2(n+1)}{n-1},\infty],$ or (2) $n=2$ and $1\leq p \leq \infty.$ Suppose we know that $$\|S_\lambda^\delta f\|_{L^p(\mathbb R^n)} \leq C_{p,\delta}\|f\|_{L^p(\mathbb R^n)}$$ for all $\lambda>0$, where $f\in L^p(\mathbb R^n)$ and $\delta > \delta(p).$

Let $g \in \mathcal{S}(\mathbb R^n),$ i.e., a Schwartz function. Show that $S_\lambda^\delta g \to g$ in $L^p$ as $\lambda \to \infty$ for $\delta > \delta(p).$

I already knew that $S_\lambda^\delta g \to g$ pointwise everywhere, by the Fourier inversion formula. I tried to show this Lemma, but it turns out to be false.

Context: I am studying $\S$ 2.3 Riesz Means in $\mathbb R^n$ in Fourier Integrals In Classical Analysis, 2nd Edition by Sogge. The above assertion is needed in the proof of Corollary 2.3.2.