Holomorphic linearization of commuting germs of holomorphic maps

Let $f_1, ..., f_h$ be $h\ge 2$ germs of biholomorphisms of \C^n fixing the origin. We investigate the shape a (formal) simultaneous linearization of the given germs can have, and we prove that if $f_1, ..., f_h$ commute and their linear parts are almost simultaneously Jordanizable then they are simultaneously formally linearizable. We next introduce a simultaneous Brjuno-type condition and prove that, in case the linear terms of the germs are diagonalizable, if the germs commutes and our Brjuno-type condition holds, then they are holomorphically simultaneously linerizable. This answers to a multi-dimensional version of a problem raised by Moser.Comment: 24 pages; final version with erratum (My original paper failed to cite the work of L. Stolovitch [ArXiv:math/0506052v2]); J. Geom. Anal. 201

Formal Poincare'-Dulac renormalization for holomorphic germs

We shall describe an alternative approach to a general renormalization procedure for formal self-maps, originally suggested by Chen-Della Dora and Wang-Zheng-Peng, giving formal normal forms simpler than the classical Poincare-Dulac normal form. As example of application we shall compute a complete list of normal forms for bi-dimensional superattracting germs with non-vanishing quadratic term; in most cases, our normal forms will be the simplest possible ones (in the sense of Wang-Zheng-Peng). We shall also discuss a few examples of renormalization of germs tangent to the identity, revealing interesting second-order resonance phenomena

Automorphisms of C-k with an invariant non-recurrent attracting Fatou component biholomorphic to C x (C*)(k-1)

We prove the existence of automorphisms of C-k , k >= 2, having an invariant, non-recurrent Fatou component biholomorphic to C x (C*)(k-1) which is attracting, in the sense that all the orbits converge to a fixed point on the boundary of the component. As a corollary, we obtain a Runge copy of C x (C*)(k-1) in C-k. The constructed Fatou component also avoids k analytic discs intersecting transversally at the fixed point

Fatou flowers and parabolic curves

In this survey we collect the main results known up to now (July 2015) regarding possible generalizations to several complex variables of the classical Leau-Fatou flower theorem about holomorphic parabolic dynamics

Characterizing Long COVID in Children and Adolescents

ImportanceMost research to understand postacute sequelae of SARS-CoV-2 infection (PASC), or long COVID, has focused on adults, with less known about this complex condition in children. Research is needed to characterize pediatric PASC to enable studies of underlying mechanisms that will guide future treatment.ObjectiveTo identify the most common prolonged symptoms experienced by children (aged 6 to 17 years) after SARS-CoV-2 infection, how these symptoms differ by age (school-age [6-11 years] vs adolescents [12-17 years]), how they cluster into distinct phenotypes, and what symptoms in combination could be used as an empirically derived index to assist researchers to study the likely presence of PASC.Design, setting, and participantsMulticenter longitudinal observational cohort study with participants recruited from more than 60 US health care and community settings between March 2022 and December 2023, including school-age children and adolescents with and without SARS-CoV-2 infection history.ExposureSARS-CoV-2 infection.Main outcomes and measuresPASC and 89 prolonged symptoms across 9 symptom domains.ResultsA total of 898 school-age children (751 with previous SARS-CoV-2 infection [referred to as infected] and 147 without [referred to as uninfected]; mean age, 8.6 years; 49% female; 11% were Black or African American, 34% were Hispanic, Latino, or Spanish, and 60% were White) and 4469 adolescents (3109 infected and 1360 uninfected; mean age, 14.8 years; 48% female; 13% were Black or African American, 21% were Hispanic, Latino, or Spanish, and 73% were White) were included. Median time between first infection and symptom survey was 506 days for school-age children and 556 days for adolescents. In models adjusted for sex and race and ethnicity, 14 symptoms in both school-age children and adolescents were more common in those with SARS-CoV-2 infection history compared with those without infection history, with 4 additional symptoms in school-age children only and 3 in adolescents only. These symptoms affected almost every organ system. Combinations of symptoms most associated with infection history were identified to form a PASC research index for each age group; these indices correlated with poorer overall health and quality of life. The index emphasizes neurocognitive, pain, and gastrointestinal symptoms in school-age children but change or loss in smell or taste, pain, and fatigue/malaise-related symptoms in adolescents. Clustering analyses identified 4 PASC symptom phenotypes in school-age children and 3 in adolescents.Conclusions and relevanceThis study developed research indices for characterizing PASC in children and adolescents. Symptom patterns were similar but distinguishable between the 2 groups, highlighting the importance of characterizing PASC separately for these age ranges

Researching COVID to Enhance Recovery (RECOVER) Adult Study Protocol: Rationale, Objectives, and Design

IMPORTANCE: SARS-CoV-2 infection can result in ongoing, relapsing, or new symptoms or other health effects after the acute phase of infection; termed post-acute sequelae of SARS-CoV-2 infection (PASC), or long COVID. The characteristics, prevalence, trajectory and mechanisms of PASC are ill-defined. The objectives of the Researching COVID to Enhance Recovery (RECOVER) Multi-site Observational Study of PASC in Adults (RECOVER-Adult) are to: (1) characterize PASC prevalence; (2) characterize the symptoms, organ dysfunction, natural history, and distinct phenotypes of PASC; (3) identify demographic, social and clinical risk factors for PASC onset and recovery; and (4) define the biological mechanisms underlying PASC pathogenesis. METHODS: RECOVER-Adult is a combined prospective/retrospective cohort currently planned to enroll 14,880 adults aged ≥18 years. Eligible participants either must meet WHO criteria for suspected, probable, or confirmed infection; or must have evidence of no prior infection. Recruitment occurs at 86 sites in 33 U.S. states, Washington, DC and Puerto Rico, via facility- and community-based outreach. Participants complete quarterly questionnaires about symptoms, social determinants, vaccination status, and interim SARS-CoV-2 infections. In addition, participants contribute biospecimens and undergo physical and laboratory examinations at approximately 0, 90 and 180 days from infection or negative test date, and yearly thereafter. Some participants undergo additional testing based on specific criteria or random sampling. Patient representatives provide input on all study processes. The primary study outcome is onset of PASC, measured by signs and symptoms. A paradigm for identifying PASC cases will be defined and updated using supervised and unsupervised learning approaches with cross-validation. Logistic regression and proportional hazards regression will be conducted to investigate associations between risk factors, onset, and resolution of PASC symptoms. DISCUSSION: RECOVER-Adult is the first national, prospective, longitudinal cohort of PASC among US adults. Results of this study are intended to inform public health, spur clinical trials, and expand treatment options

Severe Asthma Standard-of-Care Background Medication Reduction With Benralizumab: ANDHI in Practice Substudy

peer reviewedBackground: The phase IIIb, randomized, parallel-group, placebo-controlled ANDHI double-blind (DB) study extended understanding of the efficacy of benralizumab for patients with severe eosinophilic asthma. Patients from ANDHI DB could join the 56-week ANDHI in Practice (IP) single-arm, open-label extension substudy. Objective: Assess potential for standard-of-care background medication reductions while maintaining asthma control with benralizumab. Methods: Following ANDHI DB completion, eligible adults were enrolled in ANDHI IP. After an 8-week run-in with benralizumab, there were 5 visits to potentially reduce background asthma medications for patients achieving and maintaining protocol-defined asthma control with benralizumab. Main outcome measures for non–oral corticosteroid (OCS)-dependent patients were the proportions with at least 1 background medication reduction (ie, lower inhaled corticosteroid dose, background medication discontinuation) and the number of adapted Global Initiative for Asthma (GINA) step reductions at end of treatment (EOT). Main outcomes for OCS-dependent patients were reductions in daily OCS dosage and proportion achieving OCS dosage of 5 mg or lower at EOT. Results: For non–OCS-dependent patients, 53.3% (n = 208 of 390) achieved at least 1 background medication reduction, increasing to 72.6% (n = 130 of 179) for patients who maintained protocol-defined asthma control at EOT. A total of 41.9% (n = 163 of 389) achieved at least 1 adapted GINA step reduction, increasing to 61.8% (n = 110 of 178) for patients with protocol-defined EOT asthma control. At ANDHI IP baseline, OCS dosages were 5 mg or lower for 40.4% (n = 40 of 99) of OCS-dependent patients. Of OCS-dependent patients, 50.5% (n = 50 of 99) eliminated OCS and 74.7% (n = 74 of 99) achieved dosages of 5 mg or lower at EOT. Conclusions: These findings demonstrate benralizumab's ability to improve asthma control, thereby allowing background medication reduction. © 202

Toeplitz operators and skew carleson measures for weighted bergman spaces on strongly pseudoconvex domains

In this paper we study properties of Toeplitz operators on weighted Bergman spaces of bounded strongly pseudoconvex domains. We prove that a Toeplitz operator built using a weighted Bergman kernel of weight β and integrating against a measure μ maps continuously a weighted Bergman space Ap1α1 (D) into Ap2α2 (D) if and only if μ is a (λ, γ)-skew Carleson measure, where λ = 1 + 1/p1 - 1/p2 and γ = 1 l (β + α1/p1 - α2/p2). This generalizes results obtained by Pau and Zhao on the unit ball, and by Abate, Raissy and Saracco on a smaller class of Toeplitz operators on strongly pseudoconvex domains

Corrigendum to “Backward iteration in strongly convex domains” [Adv. Math. 228 (5) (2011) 2837–2854] (*** (2011) 228(5) (2837–2854), (S000187081100274X), (10.1016/j.aim.2011.06.044))

We correct a gap in two lemmas in [2], providing a new proof of the main results of that paper for hyperbolic and strongly elliptic self-maps of a bounded strongly convex domain with C2 boundary

Backward iteration in strongly convex domains

We prove that a backward orbit with bounded Kobayashi step for a hyperbolic, parabolic or strongly elliptic holomorphic self-map of a bounded strongly convex C^2 domain in C^d necessarily converges to a repelling or parabolic boundary fixed point, generalizing previous results obtained by Poggi-Corradini in the unit disk and by Ostapyuk in the unit ball of C^d