Toeplitz operators with distributional symbols on Bergman spaces

We study the boundedness and compactness of Toeplitz operators Ta on Bergman spaces , 1 < p < ∞. The novelty is that we allow distributional symbols. It turns out that the belonging of the symbol to a weighted Sobolev space of negative order is sufficient for the boundedness of Ta. We show the natural relation of the hyperbolic geometry of the disc and the order of the distribution. A corresponding sufficient condition for the compactness is also derived.

Toeplitz operators on Bergman spaces with locally integrable symbols

We study the boundedness of Toeplitz operators $T_a$ with locally integrable symbols on Bergman spaces $A^p(\mathbb{D})$, $1 < p < \infty$. Our main result gives a sufficient condition for the boundedness of $T_a$ in terms of some ``averages'' (related to hyperbolic rectangles) of its symbol. If the averages satisfy an ${o}$-type condition on the boundary of $\mathbb{D}$, we show that the corresponding Toeplitz operator is compact on $A^p$. Both conditions coincide with the known necessary conditions in the case of nonnegative symbols and $p=2$. We also show that Toeplitz operators with symbols of vanishing mean oscillation are Fredholm on $A^p$ provided that the averages are bounded away from zero, and derive an index formula for these operators.

Enabling sharing and reuse of scientific data

The purpose of this study was to develop an understanding of the current state of scientific data sharing that stakeholders could use to develop and implement effective data sharing strategies and policies. The study developed a conceptual model to describe the process of data sharing, and the drivers, barriers, and enablers that determine stakeholder engagement. The conceptual model was used as a framework to structure discussions and interviews with key members of all stakeholder groups. Analysis of data obtained from interviewees identified a number of themes that highlight key requirements for the development of a mature data sharing culture.

Association of vitamin D status with arterial blood pressure and hypertension risk: a mendelian randomisation study

BACKGROUND: Low plasma 25-hydroxyvitamin D (25[OH]D) concentration is associated with high arterial blood pressure and hypertension risk, but whether this association is causal is unknown. We used a mendelian randomisation approach to test whether 25(OH)D concentration is causally associated with blood pressure and hypertension risk. METHODS: In this mendelian randomisation study, we generated an allele score (25[OH]D synthesis score) based on variants of genes that affect 25(OH)D synthesis or substrate availability (CYP2R1 and DHCR7), which we used as a proxy for 25(OH)D concentration. We meta-analysed data for up to 108 173 individuals from 35 studies in the D-CarDia collaboration to investigate associations between the allele score and blood pressure measurements. We complemented these analyses with previously published summary statistics from the International Consortium on Blood Pressure (ICBP), the Cohorts for Heart and Aging Research in Genomic Epidemiology (CHARGE) consortium, and the Global Blood Pressure Genetics (Global BPGen) consortium. FINDINGS: In phenotypic analyses (up to n=49 363), increased 25(OH)D concentration was associated with decreased systolic blood pressure (β per 10% increase, -0·12 mm Hg, 95% CI -0·20 to -0·04; p=0·003) and reduced odds of hypertension (odds ratio [OR] 0·98, 95% CI 0·97-0·99; p=0·0003), but not with decreased diastolic blood pressure (β per 10% increase, -0·02 mm Hg, -0·08 to 0·03; p=0·37). In meta-analyses in which we combined data from D-CarDia and the ICBP (n=146 581, after exclusion of overlapping studies), each 25(OH)D-increasing allele of the synthesis score was associated with a change of -0·10 mm Hg in systolic blood pressure (-0·21 to -0·0001; p=0·0498) and a change of -0·08 mm Hg in diastolic blood pressure (-0·15 to -0·02; p=0·01). When D-CarDia and consortia data for hypertension were meta-analysed together (n=142 255), the synthesis score was associated with a reduced odds of hypertension (OR per allele, 0·98, 0·96-0·99; p=0·001). In instrumental variable analysis, each 10% increase in genetically instrumented 25(OH)D concentration was associated with a change of -0·29 mm Hg in diastolic blood pressure (-0·52 to -0·07; p=0·01), a change of -0·37 mm Hg in systolic blood pressure (-0·73 to 0·003; p=0·052), and an 8·1% decreased odds of hypertension (OR 0·92, 0·87-0·97; p=0·002). INTERPRETATION: Increased plasma concentrations of 25(OH)D might reduce the risk of hypertension. This finding warrants further investigation in an independent, similarly powered study.

Toeplitz operators on Dirichlet-Besov spaces

We study Toeplitz operators on the Besov spaces in the case of the open unit disk. We prove that a symbol satisfying a weak Lipschitz type condition induces a bounded Toeplitz operator. Such symbols do not need to be bounded functions or have continuous extensions to the boundary of the open unit disk. We discuss the problem of the existence of nontrivial compact Toeplitz operators, and also consider Fredholm properties and prove an index formula.