An integral representation for Besov and Lipschitz spaces

It is well known that functions in the analytic Besov space $B_1$ on the unit disk \D admits an integral representation f(z)=\ind\frac{z-w}{1-z\bar w}\,d\mu(w), where $\mu$ is a complex Borel measure with |\mu|(\D)<\infty. We generalize this result to all Besov spaces $B_p$ with $0<p\le1$ and all Lipschitz spaces $\Lambda_t$ with $t>1$. We also obtain a version for Bergman and Fock spaces

Kernel decompositions for Schur functions on the polydisk

A certain kernel (sometimes called the Pick kernel) associated to Schur functions on the disk is always positive semi-definite. A generalization of this fact is well-known for Schur functions on the polydisk. In this article, we show that the Pick kernel on the polydisk has a great deal of structure beyond being positive semi-definite. It can always be split into two kernels possessing certain shift invariance properties.Comment: 18 page

de Branges-Rovnyak spaces: basics and theory

For $S$ a contractive analytic operator-valued function on the unit disk ${\mathbb D}$, de Branges and Rovnyak associate a Hilbert space of analytic functions ${\mathcal H}(S)$ and related extension space ${\mathcal D(S)}$ consisting of pairs of analytic functions on the unit disk ${\mathbb D}$. This survey describes three equivalent formulations (the original geometric de Branges-Rovnyak definition, the Toeplitz operator characterization, and the characterization as a reproducing kernel Hilbert space) of the de Branges-Rovnyak space ${\mathcal H}(S)$, as well as its role as the underlying Hilbert space for the modeling of completely non-isometric Hilbert-space contraction operators. Also examined is the extension of these ideas to handle the modeling of the more general class of completely nonunitary contraction operators, where the more general two-component de Branges-Rovnyak model space ${\mathcal D}(S)$ and associated overlapping spaces play key roles. Connections with other function theory problems and applications are also discussed. More recent applications to a variety of subsequent applications are given in a companion survey article

Disseminated gonococcal infection in a Japanese man with complement 7 deficiency with compound heterozygous variants A case report

Rationale: Complement deficiency are known to be predisposed to disseminated gonococcal infection (DGI). We herein present a case of DGI involving a Japanese man who latently had a complement 7 deficiency with compound heterozygous variants. Patient concerns: A previously healthy 51-year-old Japanese man complained of sudden-onset high fever. Physical examination revealed various skin lesions including red papules on his trunk and extremities, an impetigo-like pustule on left forearm, and tendinitis of his right forefinger. Diagnosis: Blood culture testing detected gram-negative cocci, which was confirmed to be Neisseria gonorrhoeae based on mass spectrometry and a pathogen-specific PCR test. Interventions: Screening tests for underlying immunocompromised factors uncovered that complement activities (CH50) was undetectable. With a suspicion of a congenital complement deficiency, genetic analysis revealed rare single nucleotide variants in complement 7 (C7), including c.281-1G>T and a novel variant c.1454C>T (p.A485V). CH50 was normally recovered by adding purified human C7 to the patient's serum, supporting that the patient has C7 deficiency with compound heterozygous variants. Outcomes: Under a diagnosis of DGI, the patient underwent an antibiotic treatment with cefotaxime for a week and was discharged without any sequela. Lessons: DGI is a rare sexually-transmitted infection that potentially induces systemic complications. Complement immunity usually defeats N. gonorrhoeae and prevents the organism from causing DGI. This case highlighted the importance of suspecting a complement deficiency when a person develops DGI

Coagulation Factor XIII-A Subunit Missense Mutation in the Pathobiology of Autosomal Dominant Multiple Dermatofibromas

Dermatofibromas are common benign skin lesions, the etiology of which is poorly understood. We identified two unrelated pedigrees in which there was autosomal dominant transmission of multiple dermatofibromas. Whole exome sequencing revealed a rare shared heterozygous missense variant in F13A1 gene encoding factor XIII subunit A, a transglutaminase involved in hemostasis, wound healing, tumor growth, and apoptosis. The variant (p.Lys679Met) has an allele frequency of 0.0002 and is predicted to be a damaging mutation. Recombinant human Lys679Met FXIII-A demonstrated reduced fibrin crosslinking activity in vitro. Of note, treatment of fibroblasts with media containing Lys679Met FXIII-A led to enhanced adhesion, proliferation and type I collagen synthesis. Immunostaining revealed co-localization between FXIII-A and α4β1 integrins, more prominently for Lys679Met FXIII-A than wild-type. In addition, both the α4β1 inhibitors and the mutation of the FXIII-A Isoleucine-Leucine-Aspartate-Threonine (ILDT) motif prevented Lys679Met FXIII-A-dependent proliferation and collagen synthesis of fibroblasts. Our data suggest that the Lys679Met mutation may leads to a conformational change in the FXIII-A protein that enhances α4-integrin binding and provide insight into an unexpected role for FXIII-A in the pathobiology of familial dermatofibroma

Boundary continuity of holomorphic functions in the ball

It is shown that any holomorphic function on the unit ball of C n {{\mathbf {C}}^n} with n n th partial derivatives in the Hardy class H 1 {H^1} has a continuous extension to the closed unit ball, and that the restriction to any real analytic curve in the boundary which is nowhere complex tangential is absolutely continuous.</p