Abstract The main result of the present article is a Rademacher-type theorem for intrinsic Lipschitz graphs codimension $k\leq n$ in sub-Riemannian Heisenberg groups ${\mathbb H}^{n}$ . For purpose proving such result, we settle several related questions pertaining both to theory and one currents. First, prove an extension as well uniform approximation by means smooth graphs: these results stem...

We give a systematic study on the Hardy spaces of functions with values in the non-commutative L-spaces associated with a semifinite von Neumann algebra M. This is motivated by the works on matrix valued Harmonic Analysis (operator weighted norm inequalities, operator Hilbert transform), and on the other hand, by the recent development on the non-commutative martingale inequalities. Our non-com...

This paper presents a systematic study of the prehistory of the traditional subsystems of second-order arithmetic that feature prominently in the reverse mathematics program of Friedman and Simpson. We look in particular at: (i) the long arc from Poincaré to Feferman as concerns arithmetic definability and provability, (ii) the interplay between finitism and the formalization of analysis in the...

While there is a well-established notion of what a computable ordinal is, the question which functions on the countable ordinals ought to be computable has received less attention so far. We propose a notion of computability on the space of countable ordinals via a representation in the sense of computable analysis. The computability structure is characterized by the computability of four speci...

Abstract The seminal work of DiPerna and Lions (Invent Math 98(3):511–547, 1989) guarantees the existence uniqueness regular Lagrangian flows for Sobolev vector fields. latter is a suitable selection trajectories related ODE satisfying additional compressibility/semigroup properties. A long-standing open question whether flow corollary trajectory a.e. initial datum. Using Ambrosio’s superpositi...