Harmonic Besov spaces on the ball

We initiate a detailed study of two-parameter Besov spaces on the unit ball of ℝn consisting of harmonic functions whose sufficiently high-order radial derivatives lie in harmonic Bergman spaces. We compute the reproducing kernels of those Besov spaces that are Hilbert spaces. The kernels are weighted infinite sums of zonal harmonics and natural radial fractional derivatives of the Poisson kernel. Estimates of the growth of kernels lead to characterization of integral transformations on Lebesgue classes. The transformations allow us to conclude that the order of the radial derivative is not a characteristic of a Besov space as long as it is above a certain threshold. Using kernels, we define generalized Bergman projections and characterize those that are bounded from Lebesgue classes onto Besov spaces. The projections provide integral representations for the functions in these spaces and also lead to characterizations of the functions in the spaces using partial derivatives. Several other applications follow from the integral representations such as atomic decomposition, growth at the boundary and of Fourier coefficients, inclusions among them, duality and interpolation relations, and a solution to the Gleason problem. © 2016 World Scientific Publishing Company

On the Bohr inequality

The Bohr inequality, first introduced by Harald Bohr in 1914, deals with finding the largest radius $r$, $0<r<1$, such that $\sum_{n=0}^\infty |a_n|r^n \leq 1$ holds whenever $|\sum_{n=0}^\infty a_nz^n|\leq 1$ in the unit disk $\mathbb{D}$ of the complex plane. The exact value of this largest radius, known as the \emph{Bohr radius}, has been established to be $1/3.$ This paper surveys recent advances and generalizations on the Bohr inequality. It discusses the Bohr radius for certain power series in $\mathbb{D},$ as well as for analytic functions from $\mathbb{D}$ into particular domains. These domains include the punctured unit disk, the exterior of the closed unit disk, and concave wedge-domains. The analogous Bohr radius is also studied for harmonic and starlike logharmonic mappings in $\mathbb{D}.$ The Bohr phenomenon which is described in terms of the Euclidean distance is further investigated using the spherical chordal metric and the hyperbolic metric. The exposition concludes with a discussion on the $n$-dimensional Bohr radius

Hamiltonian Quantization of Effective Lagrangians with Massive Vector Fields

Effective Lagrangians containing arbitrary interactions of massive vector fields are quantized within the Hamiltonian path integral formalism. It is proven that correct Hamiltonian quantization of these models yields the same result as naive Lagrangian quantization (Matthews's theorem). This theorem holds for models without gauge freedom as well as for (linearly or nonlinearly realized) spontaneously broken gauge theories. The Stueckelberg formalism, a procedure to rewrite effective Lagrangians in a gauge invariant way, is reformulated within the Hamiltonian formalism as a transition from a second class constrained theory to an equivalent first class constrained theory. The relations between linearly and nonlinearly realized spontaneously broken gauge theories are discussed. The quartically divergent Higgs self interaction is derived from the Hamiltonian path integral.Comment: 16 pages LaTeX, BI-TP 93/1

Are C-reactive protein and homocysteine cardiovascular risk factors in obese children and adolescents?

Background: Several prospective epidemiological studies have demonstrated that high-sensitivity C-reactive protein (hsCRP) and plasma homocysteine (hcy) are predictors of future coronary events among healthy men and women. The aim of the present study was therefore to investigate a possible relationship between hsCRP, hcy levels and body mass index (BMI), relative weight (RW), serum leptin levels, and cardiovascular risk factors in obese children and adolescents. Methods: The study involved 28 obese children and adolescents (13 girls, 15 boys; BMI>95‰ for age and sex), 4.5-15 years of age (mean 10.7 ± 0.6 years), who attended hospital for a basic obesity check-up. The association between hsCRP, hcy levels and BMI, RW, serum leptin levels, and cardiovascular risk factors such as blood pressure (BP), lipid profile, serum fasting insulin levels, and insulin resistance indexes, was investigated. Results: Serum hsCRP level was positively correlated with BMI (r = 0.512, P < 0.01), RW (r = 0.438, P < 0.05), systolic and diastolic BP (r = 0.498, P < 0.01), serum leptin levels (r = 0.457, P < 0.05), but not with serum lipid, glucose, fasting insulin, plasma hcy levels or insulin resistance indexes. For hcy level, in contrast, no correlation was found with BMI, RW, systolic and diastolic BP, serum lipid levels, leptin, hsCRP, glucose, fasting insulin levels, or insulin resistance indexes. Conclusions: hsCRP is correlated with BMI, RW, BP and leptin, which are risk factors for coronary heart disease, which supports the relationship between obesity, inflammation and atherosclerosis. hsCRP in childhood obesity might be a useful index to predict possible atherosclerotic events. © 2008 Japan Pediatric Society