**
[Talk
rescheduled
from
February
6,
2019]**

**
Sang-Gyun
Youn,
Queen's
University**

"Sobolev embedding properties on compact matrix quantum groups"

One
of
the
equivalent
statements
of
the
*
fractional
Sobolev
embedding
theorem* on
$\mathbb{T}^d
$
with
respect
to
the
Laplacian
operator
$\Delta$
is
that

\[

\left\|(1-\Delta)^{-\frac{d}{2}(\frac{1}{p}-\frac{1}{q})}\right\|_{L^q(\mathbb{T}^d)}\lesssim
\|
f
\|_{L^p(\mathbb{T}^d)}

\]

for
any
$1<p<q<\infty$
and
$f\in
L^p(\mathbb{T}^d)$.
The
above
inequality
will
be
discussed
within
the
category
of
compact
quantum
groups
and
main
targets
are
compact
Lie
groups,
duals
of
discrete
groups
and
free
orthogonal
quantum
groups.
One
of
the
main
aims
of
this
talk
is
to
explain
the
following *sharp* inequality

\[

\|T(\lambda(f))\|_{L^q(\mathcal{L}(\mathbb{F}_N))}\lesssim
\|\lambda(f)\|_{L^p(\mathcal{L}(\mathbb{F}_N))}~(1<p<q<\infty)

\]

where
$\lambda(f)\sim
\displaystyle
\sum_{x\in
\mathbb{F}_N}f(x)\lambda_x\in
L^p(\mathcal{L}(\mathbb{F}_N))$
and
$T(\lambda(f))\sim
\displaystyle
\sum_{x\in
\mathbb{F}_N}\frac{f(x)}{(1+|x|)^{3(\frac{1}{p}-\frac{1}{q})}}\lambda_x$.

MC 5417