Analysis Seminar

[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