In this paper, we prove contact Poincaré and Sobolev inequalities in Heisenberg groups $\mathbb{H}^{n}$, where the word ‘contact’ is meant to stress that de Rham’s exterior differential is replaced by the exterior differential of the so-called Rumin complex $(E_{0}^{\bullet },d_{c})$, which recovers the scale invariance under the group dilations associated with the stratification of the Lie algebra of $\mathbb{H}^{n}$. In addition, we construct smoothing operators for differential forms on sub-Riemannian contact manifolds with bounded geometry, which act trivially on cohomology. For instance, this allows us to replace a closed form, up to adding a controlled exact form, with a much more regular differential form.

In this paper we study Mumford–Shah-type functionals associated with doubling metric measures or strong A∞ weights in the setting of the perimeter theory in the sense of Ambrosio and Miranda in metric spaces. We prove an existence theorem in a suitably defined class of special BV functions.

We derive Sobolev--Poincaré inequalities that estimate the $L^q(d\mu)$ norm of a function on a metric ball when $\mu$ is an arbitrary Borel measure. The estimate is in terms of the $L^1(d\nu)$ norm on the ball of a vector field gradient of the function, where $d\nu/dx$ is a power of a fractional maximal function of $\mu$. We show that the estimates are sharp in several senses, and we derive isoperimetric inequalities as corollaries. 1991 Mathematics Subject Classification: 46E35, 42B25.