# Inequalities for moduli of smoothness versus embeddings of function spaces

Seminar
Speaker
Prof. Walter Trebels, Technical University, Darmstadt, Germany
Date
19/03/2012 - 14:00Add to Calendar 2012-03-19 14:00:00 2012-03-19 14:00:00 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.   אוניברסיטת בר-אילן - המחלקה למתמטיקה mathoffice@math.biu.ac.il Asia/Jerusalem public
Abstract

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.

תאריך עדכון אחרון : 16/04/2012