Mathematisches Seminar

"Best constants for the Hausdorff-Young inequality on compact Lie groups", Herr Prof. Michael Cowling (University of New South Wales, Sydney)

17.07.2014 von 12:30 bis 14:00

LMS4 Raum 325 - Seminarhörsaal

Einladender: Prof. Dr. Detlef Müller

 

Abstract:

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{\large\bf Best constants for the Hausdorff-Young inequality\\ on compact Lie groups}
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Abstract:

This is joint work with Javier Parcet (ICMAT, Madrid).
Let $G$ be a connected compact Lie group. The equivalence classes of
irreducible unitary representations $\pi_\lambda$ of $G$ are
parametrised by their ``highest weights'' $\lambda$.  When $1 \leq p
\leq 2$ and $p' = p/(p-1)$, we may write the Hausdorff--Young
inequality
\[
\left( \sum_{\lambda} \dim(\pi_\lambda)  \left\| \pi_\lambda(f)
\right\|_{p'} ^{p'} \right)^{1/p'}
\leq C_p \left\| f\right\|_p
\]
for all $f \in L^p(G)$.

The best value for $C_p$ is $1$.  If we restrict the functions $f$ to
have support in a fixed open set $U$, we obtain a similar inequality
with $C_p$ replaced by $C_p(U)$, and it is known that $C_p(U) < 1$
when $U$ is small.  It is conjectured that $\lim_{U \to {e}} C_p(U)$
is the Babenko--Beckner constant $( p^{1/p} / p'^{1/p'}
)^{\dim(G)/2}$.  We report on what is known about this.




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