## Contact Info

Pure MathematicsUniversity of Waterloo

200 University Avenue West

Waterloo, Ontario, Canada

N2L 3G1

Departmental office: MC 5304

Phone: 519 888 4567 x43484

Fax: 519 725 0160

Email: puremath@uwaterloo.ca

Thursday, February 7, 2019 — 4:00 PM EST

**[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

University of Waterloo

200 University Avenue West

Waterloo, Ontario, Canada

N2L 3G1

Departmental office: MC 5304

Phone: 519 888 4567 x43484

Fax: 519 725 0160

Email: puremath@uwaterloo.ca

University of Waterloo

University of Waterloo

43.471468

-80.544205

200 University Avenue West

Waterloo,
ON,
Canada
N2L 3G1

The University of Waterloo acknowledges that much of our work takes place on the traditional territory of the Neutral, Anishinaabeg and Haudenosaunee peoples. Our main campus is situated on the Haldimand Tract, the land granted to the Six Nations that includes six miles on each side of the Grand River. Our active work toward reconciliation takes place across our campuses through research, learning, teaching, and community building, and is centralized within our Office of Indigenous Relations.