Let $X$ be a metric space with a doubling measure, $Y$ be a boundedly compact metric space and $u:X\to Y$ be a Lebesgue precise mapping whose upper gradient $g$ belongs to the Lorentz space $L_{m,1}$, $m\ge 1$. Let $E\subset X$ be a set of measure zero. Then ${\widehat{\mathscr{H}}}_m(E\cap u^{-1}\left(y\right))=0$ for ${\mathscr{H}}_m$-a.e. $y\in Y$, where ${\mathscr{H}}_m$ is the $m$-dimensional Hausdorff measure and ${\widehat{\mathscr{H}}}_m$ is the $m$-codimensional Hausdorff measure. This property is closely related to the coarea formula and implies a version of the Eilenberg inequality. The result relies on estimates of Hausdorff content of level sets...

To each complex number $\lambda$ is associated a representation ${\pi }_{\lambda }$ of the conformal group $S{O}_0(1,n)$ on ${𝒞}^{\infty }\left(S^{n-1}\right)$ (spherical principal series). For three values ${\lambda }_1,{\lambda }_2,{\lambda }_3$, we construct a trilinear form on ${𝒞}^{\infty }\left(S^{n-1}\right)\times {𝒞}^{\infty }\left(S^{n-1}\right)\times {𝒞}^{\infty }\left(S^{n-1}\right)$, which is invariant by ${\pi }_{{\lambda }_1}\otimes {\pi }_{{\lambda }_2}\otimes {\pi }_{{\lambda }_3}$. The trilinear form, first defined for $({\lambda }_1,{\lambda }_2,{\lambda }_3)$ in an open set of ${ℂ}^3$ is extended meromorphically, with simple poles located in an explicit family of hyperplanes. For generic values of the parameters, we prove uniqueness of trilinear invariant forms.

We study continuous measures on a compact semisimple Lie group $G$ using representation theory. In Section 2 we prove a Wiener type characterization of a continuous measure. Next we construct central measures on $G$ which are related to the well known Riesz products on locally compact abelian groups. Using these measures we show in Section 3 that if $C$ is a compact set of continuous measures on $G$ there exists a singular measure $\nu$ such that $\nu *\mu$ is absolutely continuous with respect to the Haar measure on...

In this paper we generalize Wiener’s characterization of continuous measures to compact homogenous manifolds. In particular, we give necessary and sufficient conditions on probability measures on compact semisimple Lie groups and nilmanifolds to be continuous. The methods use only simple properties of heat kernels.

We give sufficient conditions on the kernel K for the convolution operator Tf = K ∗ f to be bounded on Hardy spaces $H^p\left(G\right)$, where G is a homogeneous group.