Let me start with a couple of notational reminders. For $\xi\in \mathbb R^n$, $$ 1=\varphi_{0}(\xi)+\sum_{\nu \ge 1}\varphi_{\nu}(\xi),\quad \varphi_{0}\in C^\infty_c(\mathbb R^{n}),\quad \varphi_{\nu}=\varphi(\frac{\xi}{2^{\nu}}), $$ where $\varphi\in C^\infty_c(\mathbb R^{n}), \text{supp}\varphi=\{1/4\le \vert \xi\vert \le 4\}\supset\varphi^{-1}(\{1\})= \{1/2\le \vert \xi\vert \le 2\}$. We have \begin{align} \Vert{u}\Vert_{B^{s}_{p,q}}=\bigl\Vert{(2^{\nu s}\Vert{\varphi_{\nu}(D) u}\Vert_{L^{p}})_{\nu\ge 0}}\bigr\Vert_{\ell^{q}(\mathbb N)} \quad\text{(Besov spaces)},\\ \Vert{u}\Vert_{F^{s}_{p,q}}=\bigl\Vert{ \Vert(2^{\nu s}\vert{\varphi_{\nu}(D) u}\vert )_{\nu\ge 0}\Vert_{ \ell^{q}(\mathbb N) }}\bigr\Vert_{L^{p}(\mathbb R^n)} \quad\text{(Triebel-Lizorkin spaces).} \end{align} It is easy to see that for $1<q\le p<+\infty$, we have $ B^s_{p,q}\subset F^s_{p,q}\subset B^s_{p,p} $ and thus $F^s_{p,p}=B^s_{p,p}$.

My question: is it true that $$ B^s_{p,p}=W^{s,p}=\{u\in \mathscr S'(\mathbb R^n), (1+\vert D\vert^2)^{s/2} u\in L^p\} ? $$


1 Answer 1


Your space $W^{s,p}$ is the same as $F^s_{p2}$, not $F^s_{pp}$.

See these lecture notes. The definition of $H^{s,p}$ (which you call $W^{s,p}$) is given on page 34 and the result is on page 52.


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