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I am looking for references about embeddings from the homogeneous besov space $\dot{B}^s_{p,r}(\mathbb{R}^3)$ (or $\dot{B}^s_{p,r}(\mathbb{R}^d)$ for any $d>0$) into sobolev space $\dot{W}^{\alpha,\beta}(\mathbb{R}^3)$ (or $\dot{W}^{\alpha,\beta}(\mathbb{R}^d)$ for any $d>0$).

I already looked in the known book "Fourier analysis and nonlinear partial differential equations" from Bahouri, Chemin and Danchin. There are in the book some embedding theorems but none of the desired form.

Among all the possible embeddings, I would like to know if there exists the exponents (s,p,r) such that

$$\dot{B}^{s}_{p,r}(\mathbb{R}^3)\hookrightarrow \dot{W}^{1,\infty}(\mathbb{R}^3).$$

Any help or references is welcome.

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Yes, there exists $(s,p,r)$ such that $\dot B^s_{p,r}(\mathbb{R}^d)\subset \dot W^{1,\infty}(\mathbb{R}^d)$. It is easy to check that $\dot B^s_{p,r}(\mathbb{R}^d)\subset \dot W^{1,\infty}(\mathbb{R}^d)$ when $s=1+\frac d p$ and $r=1$, as $\dot B^{1+\frac d p}_{p,1}\subset \dot B^1_{\infty,1}$ and $\dot B^1_{\infty,1}\subset \dot W^{1,\infty}$.

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