In particular, I'm hoping there is an embedding/inequality relating elements of the two.

For background, I have a particular function, $f \in W^{\infty,p}(\mathbb{R}^2)$, for all $p \in [1,\infty]$, which is a solution to a particular PDE. I am currently investigating the decay properties fo rthis function, and have reached a point where I have the two norms:

$||f||\_{\dot{B}^{-1}\_{p,1}}$ and $||\nabla f||\_{\dot{B}^{-1}\_{p,1}}$

All I wish to do is show that these two norms are **finite**. As I am very new to Besov spaces, I am not sure how I can do this.

I have been given a hint that the norm should be finite for $p>2$, but I have no idea why this might be the case.

I am aware of the following embedding:

$ \dot{B}^{0}\_{p,1} \subseteq L^p \subseteq \dot{B}^{0}\_{p,\infty} $

But I am completely unaware of what we can say for Besov spaces of the form $\dot{B}^{s}\_{p,\infty}$ where $s < 0$.

The $\dot{B}^{s}\_{p,q}$-norm is defined as follows:

$ ||f||\_{\dot{B}^{s}\_{p,q}} = || \{ 2^{sj}||\phi\_j \ast f||\_{L^p(\mathbb{R}^2)} \} ||\_{\mathcal{l}^q}$, where $\phi\_j$ are defined as the inverse FT of terms from the Littlewood-Paley decomposition. That is:

$ \hat{\phi\_j}(\xi) := \hat{\phi\_0}(2^{-j}\xi)$, $\ \ \text{Supp}(\hat{\phi\_j}) \subseteq \{ \xi\in \mathbb{R}^2 \ | \ |2^{j-1}| \leq |\xi| \leq |2^{j+1}| \}$, $\ \ \Sigma\_{j \in \mathbb{Z}} \hat{\phi\_j}(\xi) = 1$, for all $\xi \neq 0.$

Thank you.