Interpolation and harmonic majorants in big Hardy-Orlicz spaces

Free interpolation in Hardy spaces is caracterized by the well-known Carleson condition. The result extends to Hardy-Orlicz spaces contained in the scale of classical Hardy spaces $H^p$, $p>0$. For the Smirnov and the Nevanlinna classes, interpolating sequences have been characterized in a recent paper in terms of the existence of harmonic majorants (quasi-bounded in the case of the Smirnov class). Since the Smirnov class can be regarded as the union over all Hardy-Orlicz spaces associated with a so-called strongly convex function, it is natural to ask how the condition changes from the Carleson condition in classical Hardy spaces to harmonic majorants in the Smirnov class. The aim of this paper is to narrow down this gap from the Smirnov class to ``big'' Hardy-Orlicz spaces. More precisely, we characterize interpolating sequences for a class of Hardy-Orlicz spaces that carry an algebraic structure and that are strictly bigger than $\bigcup_{p>0} H^p$. It turns out that the interpolating sequences are again characterized by the existence of quasi-bounded majorants, but now the weights of the majorants have to be in suitable Orlicz spaces. The existence of harmonic majorants in such Orlicz spaces will also be discussed in the general situation. We finish the paper with an example of a separated Blaschke sequence that is interpolating for certain Hardy-Orlicz spaces without being interpolating for slightly smaller ones.Comment: 19 pages, 2 figure

Normal Cones and Thompson Metric

The aim of this paper is to study the basic properties of the Thompson metric $d_T$ in the general case of a real linear space $X$ ordered by a cone $K$. We show that $d_T$ has monotonicity properties which make it compatible with the linear structure. We also prove several convexity properties of $d_T$ and some results concerning the topology of $d_T$, including a brief study of the $d_T$-convergence of monotone sequences. It is shown most of the results are true without any assumption of an Archimedean-type property for $K$. One considers various completeness properties and one studies the relations between them. Since $d_T$ is defined in the context of a generic ordered linear space, with no need of an underlying topological structure, one expects to express its completeness in terms of properties of the ordering, with respect to the linear structure. This is done in this paper and, to the best of our knowledge, this has not been done yet. The Thompson metric $d_T$ and order-unit (semi)norms $|\cdot|_u$ are strongly related and share important properties, as both are defined in terms of the ordered linear structure. Although $d_T$ and $|\cdot|_u$ are only topological (and not metrical) equivalent on $K_u$, we prove that the completeness is a common feature. One proves the completeness of the Thompson metric on a sequentially complete normal cone in a locally convex space. At the end of the paper, it is shown that, in the case of a Banach space, the normality of the cone is also necessary for the completeness of the Thompson metric.Comment: 36 page

Identification of potential non-invasive biomarkers in diastrophic dysplasia.

Diastrophic dysplasia (DTD) is a recessive chondrodysplasia caused by pathogenic variants in the SLC26A2 gene encoding for a cell membrane sulfate/chloride antiporter crucial for sulfate uptake and glycosaminoglycan (GAG) sulfation. Research on a DTD animal model has suggested possible pharmacological treatment approaches. In view of future clinical trials, the identification of non-invasive biomarkers is crucial to assess the efficacy of treatments. Urinary GAG composition has been analyzed in several metabolic disorders including mucopolysaccharidoses. Moreover, the N-terminal fragment of collagen X, known as collagen X marker (CXM), is considered a real-time marker of endochondral ossification and growth velocity and was studied in individuals with achondroplasia and osteogenesis imperfecta. In this work, urinary GAG sulfation and blood CXM levels were investigated as potential biomarkers for individuals affected by DTD. Chondroitin sulfate disaccharide analysis was performed on GAGs isolated from urine by HPLC after GAG digestion with chondroitinase ABC and ACII, while CXM was assessed in dried blood spots. Results from DTD patients were compared with an age-matched control population. Undersulfation of urinary GAGs was observed in DTD patients with some relationship to the clinical severity and underlying SLC26A2 variants. Lower than normal CXM levels were observed in most patients, even if the marker did not show a clear pattern in our small patient cohort because CXM values are highly dependent on age, gender and growth velocity. In summary, both non-invasive biomarkers are promising assays targeting various aspects of the disorder including overall metabolism of sulfated GAGs and endochondral ossification