Relationship between Sobolev-Slobodeckij spaces and Besov spaces

Question

I am trying to understand these two different ways of defining fractional Sobolev spaces. In particular, I want to determine embeddings or equality between the Besov spaces $B^{s}_{p,p}$ and the Slobodeckij spaces $W^{s,p}$. I have read somewhere that they should be equal, but I cannot find a proof in any of the classic texts. The Besov space is defined through the Littlewood-Paley operators as the completion of the righthand side

$$B^{s}_{p,p} = \left\{f \in \mathcal{S}(\mathbb{R}^d) \ : \ \left(\int_{\mathbb{R}^d} \sum_{j=0}^\infty 2^{jsp}|P_jf(x)|^p \ dx \right)^{1/p}<\infty\right\}.$$

While the Slobodeckij space is the completion of the righthand side of $$W^{s,p} = \left\{f \in \mathcal{S}(\mathbb{R}^d) \ : \ \left(\int \int \frac{|f(x+h)-f(x)|^p}{|h|^{d+sp}}\ dh \ dx\right)^{1/p} <\infty\right\}.$$

So, I have tried to prove that these two spaces are equal. I have managed to show that

$$\|f\|_{B^s_{p,p}} \lesssim \|f\|_{W^{r,q}}$$ where $s \le r$ and $p \le q$, with at least one of them being strict. To do this, let $\varphi_j = \check{\psi_j}$ where $\psi_j$ is the Fourier multiplier for $P_j$. For $j \ge1$, $\psi_j$ is localized to the annulus $2^{j-1} <|\xi|<2^{j+1}$, so $\int \varphi_j = \hat{\varphi}_j(0) = \psi_j(0)=0.$

$$P_jf(x) = \int f(x-h) \varphi_j(h) \ dh = \int [f(x-h) - f(x)]\varphi_j(h) \ dh$$ $$\Rightarrow |P_jf(x)| \le \int \frac{|f(x-h) -f(x)|}{|h|^{s+d/p}} |\varphi(h)| |h|^{s+d/p} \ dh$$ $$\le \left\|\frac{f(x+ \cdot) - f(x)}{|\cdot|^{s+d/p}}\right\|_{L^p} \left(\int |\varphi_j(h)|^{p'} |h|^{p'(s + d/p)} \ dh\right)^{1/p'}.$$

We estimate that last integral through scaling to get the result I claimed. So, if it is true that $B^s_{p,p} = W^{s,p}$, this method is not quite enough; we need a sharper estimate.

Is the equality true? Do both embeddings hold? What kind of embedding can I get of the form $B_{q,q}^r \subset W^{s,p} $? When $p \neq 2$, I have difficulty controlling the difference quotient in the $W^{s,p}$ norm by the Littlewood-Paley projections.

I would also appreciate any references. I have looked in Triebel's books, in one of which it is claimed that the equivalence of norms is true. But this sends me on a quest through references, each one referring to another book and I cannot find the actual proof.

Answer 1

My professor showed me how to make the first direction sharp, and I managed to get the other direction as well. I will post the broad strokes of the proof. Apply Holder or Jensen's inequality to $|P_jf(x)|$ to get $$\|f\|_{B_{p,p}^s}^p \lesssim \|f\|_p^p + \|\check{\psi}_1\|_1^{p-1} \int_{\mathbb{R}^d} \sum_{j=1}^\infty 2^{jsp}\int_{\mathbb{R}^d}|\check{\psi}_j(x-y)||f(x)-f(y)|^p \ dy \ dx.$$ Now it amounts to bounding the kernel $\sum_{j=1}^\infty 2^{jsp}|\check{\psi}_j(x-y)|$ directly. Owing to the fact that $\check{\psi}_1 \in \mathcal{S}(\mathbb{R}^d)$, it is legitimate to think of $\check{\psi}_1$ being compactly supported in some ball $B_R$. This part can be made rigorous. Then writing $\check{\psi}_j(x-y) = 2^{jd}\check{\psi}_1(2^j(x-y))$, we find the restriction $$2^j|x-y|\le R \Rightarrow 1\le j \le \lg\left(\frac{R}{|x-y|}\right).$$ Then we bound the sum as $$\sum_{j=1}^\infty 2^{jsp}|\check{\psi}_j(x-y)| \le \sum_{j=1}^{\lg(R/|x-y|)} 2^{j(sp+d)} \simeq |x-y|^{-d-sp}$$ to conclude $$\|f\|_{B^s_{p,p}} \lesssim \|f\|_{W^{s,p}}.$$ The original proof in my question is not optimal because it does not exploit the "almost orthogonality" of the projections $P_j$ (while it is not true that $\|f\|_p^p \lesssim \sum \|P_jf\|_p^p$, they still enjoy some nice summability in $\ell^p$).

For the other direction, we make careful use of the Bernstein type estimate $$\|P_jf -\tau_hP_jf\|_p \lesssim \min(1,|h|)\|\nabla P_jf\|_p \simeq \min(1, 2^j|h|)\|P_jf\|_p$$ i.e., we only want to use it when $|h|<< 2^{-j}$, otherwise we bound by $2\|P_jf\|_p$. Now the problem amounts to a simultaneous decomposition of physical space and frequency space and using the optimal estimate on each block. Define $\omega_f(r)= \sup_{|h| \le r} \|f-\tau_hf\|_p$, which is increasing. Then

$$\|f\|_{W^{s,p}}^p = \int_{\mathbb{R}^d} |h|^{-d-sp} \|f-\tau_hf\|_p^p \lesssim_d \int_0^\infty r^{-sp-1}\omega_f(r)^p \ dr$$ $$= \sum_{\ell=-\infty}^\infty \int_{2^{-\ell-1}}^{2^{-\ell}} r^{-sp-1} \omega_f(r)^p \ dr$$ $$\lesssim \sum_{\ell = -\infty}^\infty 2^{\ell s p} \omega_f(2^{-\ell})^p.$$

From the Bernstein estimate, $$\omega_f(2^{-\ell}) \le \sup_{|h|\le 2^{-\ell}} \left\|\sum_{j=0}^\infty P_jf - \tau_h P_j f\right\|_p$$ $$\lesssim \sup_{|h| \le 2^{-\ell}} \sum_{j=0}^\infty \min(1, 2^j|h|) \|P_jf\|_p \simeq \sum_{j=0}^\infty \min(1,2^{j-\ell})\|P_jf\|_p.$$

Therefore, $$\|f\|_{W^{s,p}} \lesssim \left(\sum_{\ell=-\infty}^\infty \left(2^{\ell s} \omega_f(2^{-\ell})\right)^p\right)^{1/p}$$ $$\lesssim \left(\sum_{\ell=-\infty}^\infty \left(\sum_{j=0}^\infty2^{\ell s} \min(1,2^{j-\ell}) \|P_jf\|_p\right)^p\right)^{1/p}$$ $$=\left(\sum_{\ell=-\infty}^\infty \left(\sum_{j=0}^\infty2^{(\ell-j) s} \min(1,2^{j-\ell}) 2^{js}\|P_jf\|_p\right)^p\right)^{1/p}$$ $$=\|\alpha \ast \beta\|_{\ell^p}$$ where $$\alpha_j \equiv \begin{cases} 2^{js}\|P_jf\|_p & j\ge 0 \\ 0 & j<0\end{cases}$$ $$\beta_\ell \equiv 2^{\ell s}\min(1,2^{-\ell}), \ \ \ \ell \in \mathbb{Z}.$$ Observe $\|\alpha\|_{\ell^p} = \|f\|_{B^s_{p,p}}$, and that $\beta \in \ell^1$ to conclude. I guess this was pretty much the entire proof.