arXiv subjects
Publications and source records attributed to Zoltan Balogh.
We give alternative proofs of (unsharp) results of Gromov's on his H\''older equivalence problem: for which $α$ does there exist a $C^α$-homeomorphism of an open set of Euclidean space to an open set of a given Carnot group? We indicate a possible route to sharp bounds.
Let $\H^n$ be the Heisenberg group of topological dimension $2n+1$. We prove that if $n$ is odd, the pair of metric spaces $(\H^n, \H^n)$ does not have the Lipschitz extension property.
We study the connection between the $p$--Talagrand inequality and the $q$--logarithmic Sololev inequality for conjugate exponents $p\geq 2$, $q\leq 2$ in proper geodesic metric spaces. By means of a general Hamilton--Jacobi semigroup we prove that these are equivalent, and moreover equivalent to the hypercontractivity of the Hamilton--Jacobi semigroup. Our results generalize those of Lott and Villani. They can be applied to deduce the $p$-Talagrand inequality in the sub-Riemannian setting of the Heisenberg group.