Let f : G → H be a Lipschitz map between two Carnot groups. We show that if B is a ball of G, then there exists a subset Z ⊂ B, whose image in H under f has small Hausdorff content, such that BZcan be decomposed into a controlled number of pieces, the restriction of f on each of which is quantitatively biLipschitz. This extends a result of [14], which proved the same result, but with the restriction that G has an appropriate discretization. We provide an example of a Carnot group not admitting such...

We obtain a blow-up theorem for regular submanifolds in the Heisenberg group, where intrinsic dilations are used. Main consequence of this result is an explicit formula for the density of (p+1)-dimensional spherical Hausdorff measure restricted to a p-dimensional submanifold with respect to the Riemannian surface measure. We explicitly compute this formula in some simple examples and we present a lower semicontinuity result for the spherical Hausdorff measure with respect to the weak convergence...

Let $X$ be an arbitrary hyperbolic geodesic metric space and let $\Gamma$ be a countable subgroup of the isometry group $\mathrm{Iso}\left(X\right)$ of $X$. We show that if $\Gamma$ is non-elementary and weakly acylindrical (this is a weak properness condition) then the second bounded cohomology groups $H_b^2(\Gamma ,ℝ)$, $H_b^2(\Gamma ,{\ell }^p\left(\Gamma \right))$$(1<...$

We prove that the natural map $H_{\text{b}}^2\left(\Gamma \right)\to H^2\left(\Gamma \right)$ from bounded to usual cohomology is injective if $\Gamma$ is an irreducible cocompact lattice in a higher rank Lie group. This result holds also for nontrivial unitary coefficients, and implies finiteness results for $\Gamma$: the stable commutator length vanishes and any $C^1$–action on the circle is almost trivial. We introduce the continuous bounded cohomology of a locally compact group and prove our statements by relating $H^{*}\text{b}\left(\Gamma \right)$ to the continuous bounded cohomology of the ambient group...

Let $G$ be the Lie group ${ℝ}^2⋉{ℝ}^{+}$ endowed with the Riemannian symmetric space structure. Let $X_0,X_1,X_2$ be a distinguished basis of left-invariant vector fields of the Lie algebra of $G$ and define the Laplacian $\Delta =-(X_0^2+X_1^2+X_2^2)$. In this paper we consider the first order Riesz transforms $R_i=X_i{\Delta }^{-1/2}$ and $S_i={\Delta }^{-1/2}{X}_i$, for $i=0,1,2$. We prove that the operators $R_i$, but not the $S_i$, are bounded from the Hardy space $H^1$ to $L^1$. We also show that the second-order Riesz transforms $T_{ij}=X_i{\Delta }^{-1}{X}_j$ are bounded from $H^1$ to $L^1$, while the transforms $S_{ij}={\Delta }^{-1}{X}_i{X}_j$ and $R_{ij}=X_i{X}_j{\Delta }^{-1}$, for $i,j=0,1,2$, are not.

We develop a new method to bound the hyperbolic and spherical Fourier coefficients of Maass forms defined with respect to arbitrary uniform lattices.