Besov regularity of càdlàg functions?

Question

Let $D(\mathbb{R})$ be the space of functions from $\mathbb{R}$ to $\mathbb{R}$ that are right continuous with left limits (also referred to as càdlàg functions). $D(\mathbb{R})$ is often called the Skorokhod space and plays a crucial role for the theory of Lévy processes.

I am curious about embedding relations between this space and the family of local Besov spaces $B_{p,q}^{s,\mathrm{loc}}(\mathbb{R})$ with $0<p,q\leq \infty$ and $s\in \mathbb{R}$. To be more precise, I have two questions.

• In which Besov spaces $B_{p,q}^{s,\mathrm{loc}}(\mathbb{R})$ is the space of càdlàg functions $D(\mathbb{R})$ embedded?
• Which Besov spaces $B_{p,q}^{s,\mathrm{loc}}(\mathbb{R})$ are included in $D(\mathbb{R})$?

Answer 1

(This is not a complete answer, someone more experience in Besov spaces is welcome to improve it).

I do not think there is much relation between these two concepts.

Obviously, $D(\mathbb{R})$ only contains locally bounded functions. However, the $L^p$-modulus of continuity of a càdlàg function can be as bad as you like, so $D(\mathbb{R})$ is not contained in $B^{s,\mathrm{loc}}_{p,q}$ for any $s > 0$ and $p, q > 0$.

On the other hand, if I am not mistaken, unless automatically continuous ($s p > 1$), functions in the Besov space $B^{s,\mathrm{loc}}_{p,q}$ need not have one-sided limits. Something like $\operatorname{sign}(\sin(\log |x|))$ should work as a counter-example, but I did not check the details.