Let $\phi\in\mathcal{S}(\mathbb{R}^{n})$ be a Schwartz function, such that ${\phi}\equiv 1$ on the unit ball $|\xi|\leq 1$ and $\text{supp}({\phi})\subset B_{2}(0)$. Set $\phi_{0}=\phi$ and $\phi_{j}=\phi(2^{-j}\cdot)-\phi(2^{-j+1}\cdot)$ for $j\geq 1$. Write $P_{j}f:=(\widehat{f}{\phi}_{j})^{\vee}$. For $1<p<\infty$ and $s\in\mathbb{R}$, define the space $H^{s,p}(\mathbb{R}^{n})$ by $$H^{s,p}(\mathbb{R}^{n}):=\left\{f\in\mathcal{S}'(\mathbb{R}^{n}) : \left\|\left(\sum_{j=0}^{\infty}2^{2js}|P_{j}f|^{2}\right)^{1/2}\right\|_{L^{p}}<\infty\right\}$$ One can show that $H^{s,p}$ is equivalent to the Bessel potential space $\left\{f\in\mathcal{S}'(\mathbb{R}^{n}) : ((1+|\xi|^{2})^{s/2}\widehat{f})^{\vee}\in L^{p}(\mathbb{R}^{n})\right\}$ with an equivalence of norms. $H^{s,p}$ is a special case of the Triebel-Lizorkin space $F_{p,q}^{s}$, where the $2$ above is replace by $0<q\leq\infty$.
Similarly, for $0<p<\infty$, and $0<q<\infty$, we define the Besov space $B_{p,q}^{s}(\mathbb{R}^{n})$ by $$B_{p,q}^{s}(\mathbb{R}^{n}):=\left\{f\in\mathcal{S}'(\mathbb{R}^{n}) : \left(\sum_{j=0}^{\infty}2^{jqs}\|P_{j}f\|_{L^{p}}^{q}\right)^{1/q}<\infty\right\}, \quad B_{p,\infty}^{s}(\mathbb{R}^{n}):=\left\{f\in\mathcal{S}'(\mathbb{R}^{n}) : \sup_{j}2^{js}\|P_{j}f\|_{L^{p}}<\infty\right\}$$
Using the following lemma,
Lemma. Let $n\geq 2$. Suppose $f\in L^{p}(\mathbb{R}^{n})$ has Fourier support in the ball $|\xi|\leq R$, then $\|f\|_{L^{p}(\mathbb{R}^{n-1})}\leq R^{1/p}\|f\|_{L^{p}(\mathbb{R}^{n})}$ for $1\leq p\leq\infty$.
I am trying to show that for $f\in\mathcal{S}(\mathbb{R}^{n})$, where $n>1$, the trace $f(\cdot,0)$ on $\mathbb{R}^{n-1}$ satisfies $$\|f(\cdot,0)\|_{H^{s-\sigma,p}(\mathbb{R}^{n-1})}\lesssim_{n,p,s,}\|f\|_{H^{s,p}(\mathbb{R}^{n})}\tag{*}$$ where $1/p<\sigma\leq s$.
I can establish the lemma, and by applying the lemma to the Littlewood-Paley projections $P_{k}f$, I believe that I can show that for $f\in\mathcal{S}(\mathbb{R}^{n})$, we have $$\|f(\cdot,0)\|_{B_{p,q}^{s-1/p}(\mathbb{R}^{n-1})}\leq\|f\|_{B_{p,q}^{s}(\mathbb{R}^{n})}$$ where $1<p<\infty$, $1\leq q\leq\infty$, and $s>1/p$. Let us abuse notation, and write $f$ instead of $f(\cdot,0)$ for the trace of $f$ on the hyperplane $\mathbb{R}^{n-1}\subset\mathbb{R}^{n}$. Now we have the following embeddings, which can readily be verified from Minkowski's integral inequality and the nesting property of sequence spaces, $$B_{p,1}^{s}({\mathbb{R}^{m}})\hookrightarrow H^{s,p}(\mathbb{R}^{m})\hookrightarrow B_{p,\max\{2,p\}}^{s}(\mathbb{R}^{m}),\quad B_{p,q}^{s+\epsilon}(\mathbb{R}^{m})\hookrightarrow B_{p,r}^{s}(\mathbb{R}^{m})$$ for $1<p<\infty$, $0<q\leq \infty$, and $0<r\leq\infty$, $\epsilon>0$, and $m\geq 1$. Combining these two facts, we obtain that for any $s>\sigma>1/p$, $$H^{s,p}(\mathbb{R}^{n})\hookrightarrow B_{p,\max\{2,p\}}^{s}(\mathbb{R}^{n})\hookrightarrow B_{p,\max\{2,p\}}^{s-1/p}(\mathbb{R}^{n-1})\hookrightarrow B_{p,1}^{s-\sigma}(\mathbb{R}^{n-1})\hookrightarrow H^{s-\sigma,p}(\mathbb{R}^{n-1})$$ since $s-1/p>s-\sigma$. Note, implicitly I'm using the density of Schwartz functions in these spaces to obtain a bounded operator on the whole space, but this isn't an issue.
I feel like there is a more direct and elegant route for establishing the trace inequality that doesn't make use of embeddings between the Sobolev space $H^{s,p}$ and the Besov spaces, which I am failing to see. Unfortunately, no one is around right now due to the holiday due to discuss the matter, so I thought I turned to Math SE for thoughts.