If one replaces the Poisson kernel of the unit disc by its square root, then normalised Poisson integrals of L p boundary functions converge along approach regions wider than the ordinary nontangential cones, as proved by Rönning (1 ≤ p < ∞) and Sjögren (p = 1 and p = ∞). In this paper we present new and simplified proofs of these results. We also generalise the L ∞ result to higher dimensions.

Background There is an increased interest in quantifying and characterizing epicardial fat which has been linked to various cardiovascular diseases such as coronary artery disease and atrial fibrillation. Recently, three-dimensional single-phase Dixon techniques has been used to depict the heart and to quantify the surrounding fat. The purpose of this study was to investigate the merits of a new high-resolution cine 3D Dixon technique for quantification of epicardial adipose tissue and compare it to single-phase 3D Dixon in patients with cardiovascular disease.Methods Fifteen patients referred for clinical CMR examination of known or suspected heart disease were scanned on a 1.5T scanner using single-phase Dixon and cine Dixon. Epicardial fat was segmented by three readers and intra- and inter-observer variability was calculated per slice. Cine Dixon segmentation was performed in the same cardiac phase as single-phase Dixon. Subjective image quality assessment of water and fat images were performed by three readers using a 4-point Likert scale (1 = severe; 2 = significant; 3 = mild; 4 = no blurring of cardiac structures).Results Intra-observer variability was excellent for cine Dixon images (ICC = 0.96), and higher than single-phase Dixon (ICC = 0.92). Inter-observer variability was good for cine Dixon (ICC = 0.76) and moderate for single-phase Dixon (ICC = 0.63). The intra-observer measurement error (mean ± standard deviation) per slice for cine was − 0.02 ± 0.51 ml (-0.08 ± 0.4%), and for single-phase 0.39 ± 0.72 ml (0.18 ± 0.41%). Inter-observer measurement error for cine was 0.46 ± 0.98 ml (0.11 ± 0.46%) and for single-phase 0.42 ± 1.53 ml (0.17 ± 0.47%). Visual scoring of the water image yielded median of 2 (interquartile range = [Q3-Q1] 2–2) for cine and median of 3 (interquartile range = 3 − 2) for single-phase (P < 0.05) while no significant difference was found for the fat images, both techniques yielding a median of 3 and interquartile range of 3 − 2.Conclusion Cine Dixon can be used to quantify epicardial fat with lower intra- and inter-observer variability compared to standard single-phase Dixon. The time-resolved information provided by the cine acquisition appears to support the delineation of the epicardial adipose tissue depot.

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