Harmonic Besov spaces on the ball

We initiate a detailed study of two-parameter Besov spaces on the unit ball of R-n 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

Bactericidal Efficacy of Cold Plasma at Different Depths of Infected Root Canals In Vitro

OBJECTIVES: Cold plasma (CP) has been shown to be effective even against multiresistant microorganisms. As previous investigations on the effect of CP in root canals showed promising results, the aim of the present study was to analyze the bactericidal efficacy of CP in different depths of infected dentin. METHODS: 32 standardized root canals of human mandibular premolars were infected with Enterococcus faecalis and incubated for one week. Specimens were randomly selected for one of four disinfection methods: control (5mL NaCl), 5mL chlorhexidine (CHX), CP alone (CP), and a combination of 5mL CHX and cold plasma (CHX+CP). CHX was ultrasonically activated for 30s, while cold plasma was used for 60s in the root canals. Dentin samples at depths of 300, 500 and 800 µm were obtained and diluted serially. Colony forming units (CFUs) were counted on agar plates after 24h of incubation. RESULTS: The highest overall logarithmic reduction factors (RF) were obtained from CHX+CP (log RF 3.56 p<0.01; Mann-Whitney U test), followed by CP (log RF 3.27 p<0.01) and CHX alone (log RF 2.65 p<0.01) related to the control. All disinfection methods showed significantly lower CFU counts compared to the control group in 300 µm and 800 µm (both p<0.01, Kruskal-Wallis test). DISCUSSION: The adjuvant use of CP might be beneficial in highly infected root canals to improved disinfection. However, the disinfection effect against Enterococcus faecalis of CP is comparable to ultrasonically activated CHX