Usually the Bessel potential spaces are defined as $$ H^{s, p}(\mathbb{R}^n)=\{u\in\mathscr{S}^{\prime}(\mathbb{R}^n):\mathcal{F}^{-1}(\langle\cdot\rangle^s\hat{u})\in L^p(\mathbb{R}^n)\} $$ only for exponents $1<p<\infty$. I was wondering why the cases $p=1$ and $p=\infty$ are not considered. For $p=1$ it is easy to show that $H^{s, 1}(\mathbb{R}^n)$ embeds into $L^2$-based Sobolev spaces, i.e. $$ H^{s, 1}(\mathbb{R}^n)\hookrightarrow H^{\frac{2s-n-\epsilon}{2}}(\mathbb{R}^n) $$ for any $\epsilon>0$. This follows since the Fourier transform is a continuous map $\mathcal{F}\colon L^1(\mathbb{R}^n)\rightarrow L^{\infty}(\mathbb{R}^n)$. Therefore the definition of $H^{s, 1}(\mathbb{R}^n)$ brings no extra structure. How about if $p=\infty$? Is there any reason why the space $H^{s, \infty}(\mathbb{R}^n)$ is not usually studied? I know that the Mikhlin multiplier theorem works only when $1<p<\infty$ but is this the only reason? Does the space $H^{s, \infty}(\mathbb{R}^n)$ have any good characterization like $H^{s, 1}(\mathbb{R}^n)$?
My second question considers the embeddings between Bessel potentials. I know that $H^{s, p}(\mathbb{R}^n)\hookrightarrow H^{r, p}(\mathbb{R}^n)$ whenever $s\geq r$. Also I can embed $H^{s_1, p_1}(\mathbb{R}^n)$ into $H^{s_2, p_2}(\mathbb{R}^n)$ when I have more derivatives and less integrability in the first space, i.e. $s_1\geq s_2$ and $p_1\leq p_2$. Is there any way to do the embedding from higher integrability to lower integrability? For example if I know that $$ u\in\bigcap_{s\geq 0, p\geq p^{\prime}}H^{s, p}(\mathbb{R}^n) $$ can I deduce that $u\in H^{s^{\prime}, p}(\mathbb{R}^n)$ (does not have to embed continuously) for some $s^{\prime}\in\mathbb{R}$ and $p<p^{\prime}$? Is there any characterization for the intersection like in the $L^2$ case?
References for the theory of Bessel potentials are also appreciated.