arXiv subjects
Publications and source records attributed to W. Hebisch.
We study coercive inequalities on finite dimensional metric spaces with probability measures which do not have volume doubling property. This class of inequalities includes Poincaré and Log-Sobolev inequality. Our main result is proof of Log-Sobolev inequality on Heisenberg group equipped with either heat kernel measure or "gaussian" density build from optimal control distance. As intermediate results we prove so called U-bounds.
Let $L$ denote a right-invariant sub-Laplacian on an exponential, hence solvable Lie group $G$, endowed with a left-invariant Haar measure. Depending on the structure of $G$, and possibly also that of $L$, $L$ may admit differentiable $L^p$-functional calculi, or may be of holomorphic $L^p$-type for a given $p\ne 2$. By ``holomorphic $L^p$-type'' we mean that every $L^p$-spectral multiplier for $L$ is necessarily holomorphic in a complex neighborhood of some non-isolated point of the $L^2$-spectrum of $L$. This can in fact only arise if the group algebra $L^1(G)$ is non-symmetric. Assume that $p\ne 2$. For a point $l$ in the dual $\frak g ^*$ of the Lie algebra $\frak g$ of $G$, we denote by $Ω(l)=Ad^*(G)l$ the corresponding coadjoint orbit. We prove that every sub-Laplacian on $G$ is of holomorphic $L^p$-type, provided there exists a point $l\in \frak g ^*$ satisfying ``Boidol's condition'' (which is equivalent to the non-symmetry of $L^1(G)$), such that the restriction of $Ω(l)$ to the nilradical of $\frak g$ is closed.