For $1\leq p\leq\infty$, the $r$-th order $L^p$-modulus of smoothness is
\begin{equation}
\omega_r(u,t,\Omega)_p=\sup_{|h|\leq t}\|\Delta_h^ru\|_{L^p(\Omega_{rh})}
\end{equation}
where $\Omega_{rh}=\{x\in\Omega:[x+rh]\subset\Omega\}$,
and $\Delta_h^r$ is the $r$-th order forward difference operator defined recursively by $[\Delta_h^1u](x)=u(x+h)-u(x)$ and $\Delta_h^ku=\Delta_h^1(\Delta_h^{k-1})u$,
i.e.,
\begin{equation}
\Delta_h^ru (x) = \sum_{k=0}^r (-1)^{r+k} \binom{r}{k} u(x+kh).
\end{equation}
For $1\leq p,q\leq\infty$, $\alpha\geq0$, and $r>0$ an integer,
let us define $B^\alpha_{p,q;r}(\Omega)$ as the space of functions $u\in L^p(\Omega)$ for which
\begin{equation}
|u|_{B^\alpha_{p,q;r}(\Omega)}
= \big( \int_0^\infty|t^{-\alpha}\omega_r(u,t,\Omega)_p|^q \frac{\mathrm{d}t}t \big)^{1/q} < \infty,
\end{equation}
with the usual modifications for $q=\infty$.
If $\alpha>r$ then the space $B^\alpha_{p,q;r}$ is trivial in the sense that $B^\alpha_{p,q;r}=\mathbb{P}_{r-1}$, the polynomials of order $r$.
On the other hand, so long as $r>\alpha$, different choices of $r$ will result in norms that are equivalent to each other, and in this case we have the classical *Besov spaces* $B^\alpha_{p,q}(\Omega)=B^\alpha_{p,q;r}(\Omega)$.
The reason for this is the so-called *Marchaud inequalities*.

My question concerns the borderline case $r=\alpha$. In this case, the situation seems to depend on the index $q$. Can you please confirm or invalidate the followings? Reference suggestions are also very welcome.

- If $1\leq q<\infty$ and $\alpha=r$, then $B^\alpha_{p,q;r}=\mathbb{P}_{r-1}$.
- The case $q=\infty$ gives nontrivial spaces. Namely, we have $B^\alpha_{p,\infty;\alpha}(\Omega)=W^{\alpha,p}(\Omega)$ for $p>1$, and $B^\alpha_{1,\infty;\alpha}(\Omega)$ consists of those functions whose derivatives of order up to $\alpha-1$ are in $BV(\Omega)$.

For the one dimensional case, the second item is proved on page 53 of *Constructive Approximation* by DeVore and Lorentz.
I have not checked but suspect that the same proof works in multi-dimensions, with $BV(\Omega)=B^1_{1,\infty;1}(\Omega)$ now taken as a definition.

Now, what puzzles me is that on page 202 of *Rational approximation of real functions* by Petrushev and Popov, it is claimed that $B^\alpha_{p,p;\alpha}(\Omega)$ is nontrivial, which seems to contradict the first item above.

I am also interested in the full range $p,q>0$ involving quasi-Banach spaces. Here the borderline case would be given by $r=\alpha-\max\{0,\frac1p-1\}$.