Inequalities for moduli of smoothness versus embeddings of function spaces

Define on $\, L^p({\mathbb R}^n),\, p\ge 1,$  moduli of smoothness of order
$\, r,\, r \in {\mathbb N},$ by
\[
\omega_r(t,f)_p:=\sup _{|h| <t} \| \Delta_h^rf\|_p\, ,\quad t>0,\; \;
\Delta_hf(\cdot)= f(\cdot +h)-f(\cdot),\; \Delta_h^r=\Delta_h \Delta^{r-1}_h .
\]
Trivially one has $\, \omega_r(t,f)_p \lesssim \omega_k(t,f)_p\, ,\;
k<r.$ Its converse is known as Marchaud inequality. M.F. Timan 1958
proved a sharpening of the converse, nowadays called
{\it sharp Marchaud inequality}, which in the present context takes the
form,
\[
\omega_k(t,f)_p \lesssim t^k \left( \int_{t}^{\infty} [s^{-k}
\omega_r(u,f)_p]^q \frac{du}{u} \right)^{1/q},\qquad  t>0,\quad k<r.
\]
where $\, q:=\min (p,2),\, 1<p<\infty.$
Here we will show that the sharp Marchaud inequality as well as further
sharp inequalities for moduli of smoothness like Ulyanov  and Kolyada type
ones  are equivalent to  (known) embeddings
between Besov and potential spaces.\\
 To this end  one has to make
use of moduli of smoothness of fractional order which can be
characterized by Peetre's (modified) $\, K$-functional, living on $\,
L^p$ and associated Riesz potential spaces. Limit cases  of
the Holmstedt formula (connecting different $\, K$-functionals) show
that the embeddings imply the desired inequalities.
Conversely, the embeddings result from the inequalities for moduli of
smoothness by limit procedures.