We develop the theory of invariant structure preserving and free functions on a general structured topological space. We show that an invariant structure preserving function is pointwise approximiable by the appropriate analog of polynomials in the strong topology and therefore a free function. Moreover, if a domain of operators on a Hilbert space is polynomially convex, the set of free functions satisfies a Oka-Weil Kaplansky density type theorem-- contractive functions can be approximated by contractive polynomials.

In this paper we investigate two important properties of metric measure spaces satisfying the reduced curvature-dimension condition for negative values of the dimension parameter: the existence of a transport map between two suitable marginals and the so-called local-to-global property.

In this paper, we prove that a complete noncompact weighted manifold supporting the weighted Sobolev inequality has at least linear weighted volume growth. We also obtain vanishing results and finiteness theorems for the weighted $L^2_f$ $f$-harmonic 1-forms on a complete noncompact weighted manifold supporting the weighted Sobolev inequality.

We prove that a bicontact metric three-manifold either is flat or admits some positive sectional curvature at any point. In particular, the Blair's conjecture is true for a bicontact metric three-manifold. Moreover, we prove that a contact metric three-manifold for which the contact distribution cannot be decomposed as a sum of two one-dimensional distributions, admits some positive sectional curvature; this extends the main result of [4].

This paper investigates the weighted martingale Hardy-Lorentz spaces $\Lambda_q^s(\omega)$, where $0<q<\infty$, $\omega$ is a weight and $s$ is the conditioned square function. By taking different weights, these spaces reduce to the usual martingale Hardy-Lorentz spaces, martingale Hardy-Lorentz-Karamata spaces, martingale Hardy-Orlicz-Lorentz spaces, etc. We establish various martingale inequalities in the weighted martingale Hardy-Lorentz spaces. We also characterize the dual spaces of $\Lambda_q^s(\omega)$. To this end, we need to introduce two generalized BMO martingale spaces which are defined by stopping times or stopping time sequences. We obtain as well the John-Nirenberg inequalities for these generalized BMO martingale spaces. Our main results strengthen the work [20]. We highlight that our study demonstrates a unified approach to martingale Hardy-Lorentz type spaces. A variety of known and new results on martingale Hardy-Lorentz type spaces can be inferred from our paper.

In this work, we study the convergence of the normalized Yamabe flow with positive Yamabe constant on a class of pseudo-manifolds that includes stratified spaces with iterated cone-edge metrics. We establish convergence under a low-energy condition. We also prove a concentration--compactness dichotomy, and investigate what the alternatives to convergence are. We end by investigating a non-convergent example due to Viaclovsky in more detail.