[proofplan]
The Besov norm is the $\ell^q$-norm in $j$ of the dyadic pieces $2^{js}\|\Delta_j f\|_{L^p}$. Two indices change between source and target spaces: the regularity ($s_0 \to s_1$) and the integrability ($p_0 \to p_1$). We match them piece-by-piece for each frequency level $j$ using **Bernstein's inequality for Littlewood–Paley pieces**: for a function frequency-localised at scale $2^j$,
\begin{align*}
\|\Delta_j f\|_{L^{p_1}} \le C\,2^{jn(1/p_0 - 1/p_1)}\,\|\Delta_j f\|_{L^{p_0}},
\end{align*}
which exchanges integrability for a factor of $2^{jn(1/p_0 - 1/p_1)}$. The scaling hypothesis $s_0 - n/p_0 = s_1 - n/p_1$ rewrites as $s_1 + n(1/p_0 - 1/p_1) = s_0$, exactly the exponent absorbed by Bernstein. The two norms therefore agree term by term up to a uniform constant, and the $\ell^q$-norms in $j$ match.
[/proofplan]
[step:Recall the Besov norm and Bernstein's inequality]
Fix a Littlewood–Paley resolution of unity as in the previous proofs: $\varphi_0, \varphi \in C^\infty_c(\mathbb{R}^n)$ with $\varphi_0$ supported in $\{|\xi|\le 2\}$, $\varphi$ supported in $\{1/2\le|\xi|\le 2\}$, $\varphi_0 + \sum_{j\ge 1}\varphi(2^{-j}\cdot) = 1$. Set $\varphi_j(\xi) := \varphi(2^{-j}\xi)$ and define the Littlewood–Paley projectors
\begin{align*}
\Delta_j : \mathcal{S}'(\mathbb{R}^n) &\to \mathcal{S}'(\mathbb{R}^n) \\
f &\mapsto \mathcal{F}^{-1}(\varphi_j\,\hat f\,), \qquad j \ge 0.
\end{align*}
The Besov norm for parameters $(s, p, q)$ is
\begin{align*}
\|f\|_{B^s_{p,q}(\mathbb{R}^n)} := \Bigl(\sum_{j=0}^\infty 2^{jsq}\,\|\Delta_j f\|_{L^p(\mathbb{R}^n)}^q\Bigr)^{1/q}, \qquad 1 \le q < \infty,
\end{align*}
with the obvious modification $\sup_{j\ge 0} 2^{js}\,\|\Delta_j f\|_{L^p}$ for $q = \infty$.
Recall the [Bernstein Inequality for Littlewood–Paley Pieces](/theorems/???) in its $L^p$-to-$L^{p_1}$ form: if $g \in \mathcal{S}'(\mathbb{R}^n)$ has $\hat g$ supported in the ball $\{|\xi| \le R\}$ and $g \in L^{p_0}$ for some $1 \le p_0 \le \infty$, then for any $p_1$ with $p_0 \le p_1 \le \infty$,
\begin{align*}
\|g\|_{L^{p_1}(\mathbb{R}^n)} \le C_{n,p_0,p_1}\,R^{n(1/p_0 - 1/p_1)}\,\|g\|_{L^{p_0}(\mathbb{R}^n)}.
\end{align*}
Apply this to $\Delta_j f$ for each $j \ge 0$. The function $\Delta_j f$ has Fourier support in $\{|\xi|\le 2^{j+1}\}$ (in the ball of radius $2^{j+1}$), so $R = 2^{j+1}$. Hence
\begin{align*}
\|\Delta_j f\|_{L^{p_1}} \le C_{n,p_0,p_1}\,2^{(j+1)n(1/p_0 - 1/p_1)}\,\|\Delta_j f\|_{L^{p_0}} \le C'_{n,p_0,p_1}\,2^{jn(1/p_0 - 1/p_1)}\,\|\Delta_j f\|_{L^{p_0}}
\end{align*}
where $C'_{n,p_0,p_1} := 2^{n(1/p_0-1/p_1)}\,C_{n,p_0,p_1}$ absorbs the constant factor $2^{n(1/p_0 - 1/p_1)}$ from the $j+1$ shift.
[/step]
[step:Translate the scaling identity into the matching exponent]
The hypothesis $s_0 - n/p_0 = s_1 - n/p_1$ rearranges as
\begin{align*}
s_0 = s_1 + n\Bigl(\frac{1}{p_0} - \frac{1}{p_1}\Bigr).
\end{align*}
Note $1/p_0 - 1/p_1 \ge 0$ since $p_0 \le p_1$, and equality holds iff $p_0 = p_1$ (in which case $s_0 = s_1$ and the embedding is the identity). Multiplying through by $j$ and exponentiating base 2,
\begin{align*}
2^{js_0} = 2^{js_1}\,2^{jn(1/p_0 - 1/p_1)}.
\end{align*}
[/step]
[step:Bound each Besov term of the target norm by the corresponding term of the source norm]
Combine the displayed Bernstein estimate with the scaling identity. For each $j \ge 0$,
\begin{align*}
2^{js_1}\,\|\Delta_j f\|_{L^{p_1}} \le 2^{js_1}\,C'_{n,p_0,p_1}\,2^{jn(1/p_0 - 1/p_1)}\,\|\Delta_j f\|_{L^{p_0}} = C'_{n,p_0,p_1}\,2^{js_1 + jn(1/p_0 - 1/p_1)}\,\|\Delta_j f\|_{L^{p_0}} = C'_{n,p_0,p_1}\,2^{js_0}\,\|\Delta_j f\|_{L^{p_0}}.
\end{align*}
This is a *pointwise* inequality between the $j$-th term of $\|f\|_{B^{s_1}_{p_1,q}}$ and the $j$-th term of $\|f\|_{B^{s_0}_{p_0,q}}$, with constant independent of $j$.
[/step]
[step:Take the $\ell^q$-norm in $j$ to conclude]
Raise the per-$j$ inequality to the $q$-th power (for $1 \le q < \infty$) and sum:
\begin{align*}
\sum_{j=0}^\infty 2^{js_1 q}\,\|\Delta_j f\|_{L^{p_1}}^q \le (C'_{n,p_0,p_1})^q\,\sum_{j=0}^\infty 2^{js_0 q}\,\|\Delta_j f\|_{L^{p_0}}^q.
\end{align*}
Taking $q$-th roots,
\begin{align*}
\|f\|_{B^{s_1}_{p_1,q}(\mathbb{R}^n)} \le C'_{n,p_0,p_1}\,\|f\|_{B^{s_0}_{p_0,q}(\mathbb{R}^n)}.
\end{align*}
For $q = \infty$, take the supremum over $j$ of the per-$j$ bound to obtain the same conclusion with the same constant. In either case, the linear map $f \mapsto f$ is bounded from $B^{s_0}_{p_0,q}(\mathbb{R}^n)$ to $B^{s_1}_{p_1,q}(\mathbb{R}^n)$ with operator norm at most $C'_{n,p_0,p_1}$, depending only on $n$, $p_0$, $p_1$, and the resolution of unity (with the constant from Bernstein's theorem). This is the asserted continuous embedding.
[/step]
