Ahlfors Bers Colloquium

May 19-22, 2005
University of Michigan
Ann Arbor, Michigan

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University of Michigan

Mathematics Department

Organizers
Dick Canary
Juha Heinonen

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Workshop Abstract Detail

Jeremy Tyson University of Illinois

Best constants for weighted Moser-Trudinger inequalities on sub-Riemannian spaces Trudinger's inequality asserts the inclusion of $W^{1,n}(\Omega)$, $\Omega\subset{\mathbb R}^n}$, in the Orlicz space $\mbox{Exp}(\alpha L^{n/(n-1)})$ for some positive $\alpha$ depending only on $n$. Moser (1971) determined the largest coefficient $\alpha$ for which Trudinger's inequality holds. We will discuss the problem of best constants in the Moser Trudinger (MT) inequality on Carnot groups and general Carnot-Carath\'eodory spaces. Using explicit representation formulas arising from nonlinear potential theory and a convolution rearrangement argument inspired by Adams (1988), Cohn and Lu (2001) and Balogh, Manfredi and Tyson (2003) obtained sharp MT inequalities in the Heisenberg group and general Carnot groups, respectively. As a further extension of these results, we establish sharp MT inequalities on groups of Heisenberg type relative to some subradial power weights, and use the rotational symmetry of the Heisenberg group to deduce sharp weighted MT inequalities in the Grushin plane.

Topic: Conformal Analysis and Geometric Function Theory