# 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.

- Last modified: 16/04/2012